Value Distribution Theory and Related Topics (Advances in Complex Analysis and Its Applications) [1 ed.] 1402079508, 9781402079504, 9781402079511

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VALUE DISTRIBUTION THEORY AND RELATED TOPICS

Advances in Complex Analysis and Its Applications Volume 3

Series Editor: C.C. Yang The Hong Kong University of Science& Technology, Hong Kong

Advisory Board: Walter Bergweiler Keil University, Germany George Csordas University of Hawaii, U.S.A. Paul Gauthier University of Montreal, Canada Phillip Griffiths Princeton, U.S.A. Irwin Kra State University of New York, U.S.A. Armen G. Sergeev Steklov Institute of Mathematics, Russia Wolfgang Tutschke University of Graz, Austria

VALUE DISTRIBUTION THEORY AND RELATED TOPICS

edited by

G. Barsegian National Academy of Sciences of Armenia Yerevan, Armenia

I. Laine University of Joensuu Joensuu, Finland

C.C. Yang Hong Kong University of Science and Technology Hong Kong, China

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

1-4020-7951-6 1-4020-7950-8

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CONTENTS Preface

vii

Geometric value distribution theory Barsegian, G.: A new program of investigations in analysis: Gamma-lines approaches Sukiasyan, G.: On level sets of quasiconformal mappings

1 75

Classical value distribution theory Alonso, A., Fernández, A. and Pérez, J.: On the unintegrated Nevanlinna fundamental inequality for meromorphic functions of slow growth

93

Barsegian, G. and Yang, C.-C.: On some new concept of exceptional values

105

Ciechanowicz, E. and Marchenko, I.: Maximum modulus points, deviations and spreads of meromorphic functions

117

Craven, T. and Csordas, G.: Composition theorems, multiplier sequences and complex zero decreasing sequences Korhonen, R.: Nevanlinna theory in an annulus

131 167

Marchenko, I. and Nikolenko, I.: On strong asymptotic tracts of functions holomorphic in a disk

181

Complex differential and functional equations Barsegian, G., Sarkisian, A. and Yang, C.-C.: A new trend in complex differential equations: quasimeromorphic solutions

189

Ha, H.K. and Yang, C.-C.: On the functional equation 201

He, Y.: Value distribution of the higher order analogues of the first Painlevé equation

209

Yang, C.-C. and Li, P.: Some further results on the functional equation

219

vi

Several variables theory Aihara, Y.: Recent topics in uniqueness problem for meromorphic mappings

233

Berenstein, C. and Li, B.Q.: On interpolation problems in

265

Hu, P.-C. and Yang, C.-C.: Jet bundles and its applications in value distribution of holomorphic mappings Tu, Z.-H.: Normal families of meromorphic mappings of several complex variables into the complex projective space

281 321

PREFACE

The Nevanlinna theory of value distribution of meromorphic functions, one of the milestones of complex analysis during the last century, was created to extend the classical results concerning the distribution of of entire functions to the more general setting of meromorphic functions. Later on, a similar reasoning has been applied to algebroid functions, subharmonic functions and meromorphic functions on Riemann surfaces as well as to analytic functions of several complex variables, holomorphic and meromorphic mappings and to the theory of minimal surfaces. Moreover, several applications of the theory have been exploited, including complex differential and functional equations, complex dynamics and Diophantine equations. The main emphasis of this collection is to direct attention to a number of recently developed novel ideas and generalizations that relate to the development of value distribution theory and its applications. In particular, we mean a recent theory that replaces the conventional consideration of counting within a disc by an analysis of their geometric locations. Another such example is presented by the generalizations of the second main theorem to higher dimensional cases by using the jet theory. Moreover, similar ideas apparently may be applied to several related areas as well, such as to partial differential equations and to differential geometry. Indeed, most of these applications go back to the problem of analyzing zeros of certain complex or real functions, meaning in fact to investigate level sets or level surfaces. The articles in this collection have been organized in four groups. The first group of articles present various aspects of the geometric value distribution theory. The focus of this group is an extensive article presenting a research program covering numerous potential applications of the recent theory of gamma-lines. The second group of articles is focusing on some more classical aspects in value distribution theory, where several interesting, unsolved problems may still be found. The third group of papers present applications of the value distribution into complex differential and functional equations, while the final group concentrates into the several variables theory. We gratefully acknowledge support from several agencies. In particular, we mention the INTAS project 99-00089 (New Trends in Complex Analysis and Potential Theory), a NATO Advanced Research Workshop grant, a grant from the International Mathematical Union as well as research grants from the Academy of Finland (grant 50981), and from the Research Council of Hong Kong. All these grants helped us to collaborate, making possible to issue this volume. We are also grateful to Riitta Sinkkonen and Minna Pylkkönen at the University of Joensuu for their important assistance. Yerevan, Joensuu and Hong Kong, September 2003

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A NEW PROGRAM OF INVESTIGATIONS IN ANALYSIS: GAMMA-LINES APPROACHES

G. BARSEGIAN Institute of Mathematics of National Academy of Sciences of Armenia 24-b Marshal Bagramian ave., Yerevan 375019, Armenia, [email protected]

Abstract. A new program of mathematical studies primarily based on the theory of Gamma-lines and ideas of the Nevanlinna value distribution theory is presented. This program establishes new connections between a variety of mathematical fields: real and complex analysis, ordinary, partial and complex differential equations, differential geometry, real and complex algebraic geometry, and Hilbert’s topological problem 16. Preliminary results, related to some of the problems posed, are given. In addition, the usefulness of this program in applications will be discussed. Mathematics Subject Classification 2000: 14, 26C, 30, 34, 35, 53. Key words and phrases: analysis, algebraic geometry, differential geometry, ordinary differential equations, partial differential equations, complex differential equations, Hilbert’s problem, Nevanlinna theory, proximity property of Gamma-lines

Introduction

The leading idea of this program may be described as follows: Many core concepts in pure mathematics deal with zeros of real functions (level sets) or zeros of complex functions or mappings widely studied, mainly quantitatively, in analysis, geometry and topology. In analysis, the study of these zeros arises in various contexts: as (in complex analysis), as tangent lines, equilibrium sets in autonomous systems of equations (in ordinary differential equations), and as boundary conditions or parabolic lines, (in partial differential equations). In differential geometry, they ap-

© 2004 Kluwer Academic Publishers

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pear in the study of families of curves. In algebraic geometry and algebraic topology they are semialgebraic sets. Indeed, several of classical Hilbert problems are dealing with or are related to investigating of such zeros. It is well known that in applied areas, level sets play a pivotal role. For example, in physics, special cases of level sets are, among others, the isotherms, isobars, equipotential lines, stream lines etc. Heretofore, zeros (or the of various classes of complex functions have been intensively investigated, but only quantitatively. Today, the fundamental problem is the determination of the geometric locations of these zeros. In fact, many core concepts in analysis are level sets, although these notions have not been studied from this point of view. Furthermore, the zeros of many classes of functions important in physical applications, have not been studied in pure mathematics. Recently, some new tools such as Gamma-lines [6], [7], [16] and catastrophe lines [17], have been developed in complex analysis. These tools enable us to study the zeros of larger classes of functions than those considered classically. In particular, these tools provide a machinery to describing not only the quantity but also the geometry of these zeros or a-points. These developments have led to a number of novel concepts such as the proximity property and the closeness property of in the theory of meromorphic functions. Unlike the classical Nevanlinna value distribution theory [58] dealing with the numbers of these new tools can be used to describe geometric locations of as well. For related ideas and some applications, we refer to [4], [7]-[11]. Other developments, such as the principle of partitioning of meromorphic functions, the comparability principle of and the comparability of derivatives may be found in [12]–[15], [16], [28], [31], [32]. Finally, new characteristic sets and deficiencies, used to analyze geometric locations of of meromorphic functions (similar to the Nevanlinna characteristic function and deficiency), will be described in Appendix of the present paper. The following program of investigations in analysis (and other fields of mathematics) now arise naturally. By utilizing the aforementioned new tools, we attempt to present here some preliminary approaches, results and open problems related to novel type of investigations of zeros in some old and new fields, also passing to study the geometry of zeros (instead of quantities) in some classical fields. In reference to some of the open problems to be considered in this work, it is my pleasure to acknowledge here my collaboration with H. Begehr, G. Csordas, I. Laine, D. T. Le, C. C. Yang and with the members of my research group, G. Sukiasian and A. Sarkisyan. I express my heartfelt gratitude to them. My special thanks to G. Csordas and I. Laine for many valuables suggestions improving the paper.

3 The following list now offers a preliminary idea of the fields of problems and topics we intend to consider in this paper: Field 1: Level sets of real functions in real analysis; level sets of solutions of partial differential equations; level sets in scientific calculations. Field 2: The crossroad of generalized value distribution theory and generalized real algebraic geometry. Field 3: The crossroad of generalized closeness property and generalized real algebraic geometry. Field 4: A subtopic in real algebraic geometry studying regularities of Nevanlinna theory type. Field 5: A subtopic in real algebraic geometry studying regularities of closeness property type. Field 6: A subtopic in complex algebraic geometry studying regularities of closeness property type. Field 7: The crossroad of generalized value distribution theory, generalized closeness property and autonomous systems of ordinary differential equations. Field 8: A subtopic in complex differential equations studying regularities of closeness property type. Field 9: A subtopic in complex functional equations studying regularities of closeness property type. Field 10: The generalized Hilbert’s problem 16 as a particular problem in generalized value distribution theory. Field 11: Hilbert’s problem 16 and the geometry of connected components of level sets of polynomials. Field 12: Gamma-lines and level sets of non-analytic classes of functions. Field 13: Gamma-lines of quasiconformal functions and catastrophe sets as tools to investigating Gamma-lines of general classes of functions. Field 14: Gamma-lines in studying regularities of closeness property type for general classes of functions. Field 15: Gamma-lines of solutions of systems of partial differential equations. Field 16: Gamma-lines of algebraic and algebroid functions. Field 17: Geometry of of algebraic and algebroid functions. Field 18: Gamma-lines in real algebraic geometry. Field 19: Gamma-lines in studying the influence of geometry in complex interpolation problems. Field 20: Gamma-lines and interpolation problems in algebraic geometry. Field 21: Transfer of concepts and results of complex analysis in differential geometry. Field 22: Gamma-lines and proximity property in differential geometry with emphasis on minimal surfaces.

4 Field 23: Differential level sets of real functions generalizing the concept of level sets, their connection with partial differential equations and with boundary value problems. Field 24: Random as a geometric point of view to partial differential equations. Field 25: Differential differential Gamma-lines, and their connections with different topics in pure and applied mathematics. Field 26: Generalizations in the theory of quasiregular functions, generalized value distribution theory and complex differential equations with quasimeromorphic solutions. Field 27: Complex differential equations with generalized quasiregular solutions. Field 28: Generalizations of analytic functions associated with a given polynomial. Field 29: Topological properties of multi-valued functions and multi-valued solutions of some classes of complex differential equations. Field 30: Global multi-valued functions (and solutions of differential equations) from the point of view of value distribution and the closeness property. Field 31: An approach to reducing investigations of solutions of differential equations to a new type of problems in value distribution theory. Field 32: Gamma-lines of polynomials and the proximity property of of polynomials. Appendix. Characteristic sets and deficiencies describing geometric locations of of meromorphic functions. Passing to more detailed discussions, I would like to mention two observations first: Sometimes problems having a very similar wording lie, in fact, in different fields and sometimes objects in different fields are almost identical. An example (that first surprised me) is the Nevanlinna value distribution theory (in complex analysis) and real algebraic geometry, both studying similar objects. Then why not thinking about the mutual penetration of methods and problems in these different, but in fact close topics? Further, as level sets of real functions have numerous important interpretations in many subfields of applied mathematics, why not strengthening pure mathematical investigations of level sets of corresponding classes of functions, as the importance of such investigations is obvious? This could result in new trends in some applied fields while making use of new tools developed recently. In what follows, we start with the last observation. Below, some old and new topics and problems, types of regularities, and their interplay will be discussed according to the list above.

5

Field 1: Level sets of real functions in real analysis; level sets of solutions of partial differential equations; level sets in scientific calculations

1.1. Level sets of functions in real analysis. Everywhere in nature, theoretical and applied investigations one can meet phenomena and problems, which are described by or connected with level sets, i.e. zeros of functions or solutions of equations

where is a real function “enough smooth”. Hence, the great importance of the following problem is self-evident: Problem 1.1. To establish methods for studying the geometry (length, curvature and so on) of level sets of large classes of real functions We mean here that the geometry of level sets should be described in terms of the given class of functions. Remark 1.1. The Hilbert problem 16(a) sounds qualitatively quite similar: To study the “topology” (number of connected components) of level sets for particular classes of functions where are real polynomials. This is one of the less investigated Hilbert problems. Despite of the fact that level sets have been used in applied sciences and scientific calculations during the last two centuries, they have been not much studied in analysis. Only recently (at the end of 1970’s) the length of level sets has been were studied for the standard class of harmonic functions in arbitrary domains in the theory of Gamma-lines. The results appeared to be analogous to the main results in the classical Nevanlinna theory of see [7], and for a more detailed version in [16]). We show that the methods of Gamma-lines can be easily adjusted to studying level sets of larger classes of functions. Therefore the research topic posed here seems to be quite easily accessible (see below). Clearly these two problems, the Hilbert problem and Problem 1.1 above offer pertinent additions to each other (see below). 1.2. Level sets of solutions of partial differential equations. As a very important subtask of Problem 1.1, we propose to study level sets of solutions of a given partial differential equation

and in particular, to study these sets when is a solution of a given boundary value problem. The level sets are visual which fact is their great advantage. Indeed, we may observe such level sets in the nature for instance, say related to

6

physical phenomena described by various types of equations (say, in various hydrodynamic experiments). Therefore, to get an idea about the distribution of level sets for a solution of a given equation is sometimes even more important than to have an idea about the solution itself. If the solutions are unknown, information about their level sets becomes more important. On the other hand, these solutions often are quite complicated and it is difficult to make a qualitative conclusion about their behavior; again information about the level sets becomes valuable. Thus we propose that investigations of level sets of solutions of equations may appear to be one of the central problems in partial differential equations (PDE) in the future: Problem 1.2. To study the geometry of level sets of solutions of different classes of partial differential equations in terms of the equations itself and of the corresponding boundary value problems. Remark 1.2 (what we already know). For the Gamma-lines of meromorphic and quasiconformal functions we already have a theory analogous to the Nevanlinna theory, see [16] or Field 12 below. A particular case of Gamma-lines is the level sets of real functions When is an analytic function its real part Re is a harmonic function. This means that in this particular case many results established for Gamma-lines of are true also for level sets of harmonic functions: the last ones are solutions of the standard Laplace equation. Moreover, thanks to this interplay we immediately get a kind of Nevanlinna theory for the level sets of harmonic functions. Thus we already have an example to be followed. So, we can try to study Problem 1.2 for different classes of partial differential equations. Remark 1.3 (point of view of oscillations). It is pertinent to indicate here a parallel between this problem and the oscillation theory of ordinary differential equations playing a crucial role in many physical problems, where these zeros frequently are of primary interest. Respectively, asymptotic properties of zeros of the solutions, distances between them etc. have been studied and are studied intensively. Note that by posing the last Problem 1.2 we, in fact, challenge to transfer oscillation problems from ordinary to partial differential equations: just instead of zeros of solutions of ordinary differential equations we should now consider zeros of solutions of partial differential equations. 1.3. Two generalities in the study of the geometry of level sets. Below we present a general inequality, see Barsegian and Sukiasyan [29], giving upper bounds for the length of level sets for large classes functions smooth enough. This inequality generalizes the tangent variation principle in the theory of Gamma-lines [16].

7

We use the following notations: is a bounded domain with a piecewise smooth boundary is the length of is the class of twice continuously differentiable functions is the total length of the level set of a function Through each point passes a level curve

whose curvature at the point we denote by We also assume that for each except probably for isolated points. In their neighborhoods, the integrals in Theorem 1.1 below are assumed to be convergent. The following result now gives upper bounds for the length in terms of the length and of double integrals of the curvature Theorem 1.1. For any function

where

and any

we have

is the standard area element.

Hence the quantity

will be considered as the the characteristic for the length and the inequality (1.4) will be referred to as the length and curvature principle. Observe that the second integral is bounded by Another approach uses a technique developed for Gamma-lines and can be applied to arbitrary functions in for which we just suppose that and are continuous in Thus the function can be also identical to zero in some subdomains of D. Respectively, solutions of may consist of some curves as well as of some subdomains By the total length we mean the total length of all plus the total length of all boundaries lying inside D. Denote

8 and The function can be identically equal to zero on some intervals Denote by the number of zeros of which are either isolated zeros of or are zeros at the end-points of the above intervals where Similarly, we define substituting by Theorem 1.2 (Part of Theorem 2.1 in [29]). Let be an arbitrary smooth function in the closure of a given plane domain D. Then

Theorem 1.2 implies the following corollary: For an arbitrary polynomial an arbitrary domain D with smooth boundary and an arbitrary

where resp. is the highest degree of P with respect to resp. and is the diameter of D. If where is a complex polynomial, then (1.5) implies that

1.4. Level sets in scientific calculations. Almost any scientific calculation deals with level sets. Respectively, any usual computer software offers some numerical and graphical support for related calculations. However, the software usually describes the level sets of functions “sufficiently simple” only. In complicated cases (which are usually the most important) one needs to analyze the behavior of these functions before passing to calculations. Here the following type of information seems to be useful: (a) First we need to know whether the length of level sets of our function in a given domain D remains bounded. Clearly, the software will not calculate those cases where Thus, these cases should be studied theoretically first. To study the last problem, it is reasonable to indicate those subdomains where the curves are of infinite length. Thus we should study the following problem: (b) Where are the subdomains of D such that Finally, we consider the case usual in complicated calculations when we know in advance that the length is finite although we have no idea about how large is this length. Clearly, it will be very useful (and almost necessary) to know how large are these level sets in order to perform the

9 related calculations in real computer time, 1 to select the step length in calculations, and to discuss the sharpness of the results obtained. Thus, working with complicated level sets we need to study the following problem: (c) How long can a level set be in D, or else in a subdomain of D? So, it would be highly useful if before computing of the lengths the software would first calculate related theoretical bounds for Then, the software might apply these bounds to decide whether the computer should start the actual calculation or whether it should return the problem to the customer for a theoretical revision. If, finally last, it is clear that the problem is solvable, then these bounds could be used by a proper adjustment of the step selection for the calculation and for testing of the sharpness of final calculations. Hence, it seems clear that our theorems above and similar investigations giving methods to estimate the length of level sets are obvious tools to study the above problems (a)–(c). So, the actual problem now is to introduce these methods into a software. Field 2: The crossroad of generalized value distribution theory and generalized real algebraic geometry Polynomials (real, complex analytic, complex non analytic) are one of the most studied concepts in pure and applied mathematics. The Gamma-lines technique permits us easily to obtain some new and general results related to zeros of polynomials. Now it would be pertinent to look at their applications, in particular in algebraic geometry and in relation to the Hilbert problem 16, as zeros (level sets) of polynomials play a crucial role in these topics. Algebraic geometry (AG) studies solutions of equations

with complex polynomials and real algebraic geometry (RAG) restricts itself to studying solutions of equations

1 Clearly, any software describes level sets approximately only by giving in fact only a “possible” curvilinear strip or curvilinear region, where the level sets sought are situated. These strips or regions (that definitely depend on the step or on the prescribed sharpness of calculations) may include several level sets of total length larger than the approximate length of the strip.

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with real polynomials is

where of

and

only. A particular case of the system (2.1)

is an ordinary complex polynomial (supposed to be of degree Solutions of this last equation are, of course, zeros of Solutions that is, of P are then solutions of equation

Let us, by analogy, call

of the system (2.2) the solutions of

If we replace the polynomial by a meromorphic function (2.3), we have to deal with solutions of

in

that is with of which are widely studied in complex analysis. In particular, in the case of meromorphic functions in we have the classical Nevanlinna value distribution theory (VDT) and the Ahlfors theory of covering surfaces. Due to the main theorem of algebra (MTA), the number of solutions of (2.3) for an arbitrary is equal to The main conclusion of VDT is similar to MTA but there is an essential difference: In the case of meromorphic functions necessarily arise deficient values (sets) may appear. These are exceptional in the sense that there are at most countably many of them. Hence, the first main conclusion of VDT is that the number of and of of are asymptotically equal (close to each other) if and are not exceptional (deficient). At present, we have a more general regularity, called the proximity, or closeness, property of (CP) describing the geometry of these points, instead of their number only, as in VDT. The CP asserts that the and should be geometrically close to each other. Moreover, this new property includes the main assertions of VDT and can be expanded to arbitrary analytic functions given in an arbitrary domain unlike VDT. Thus we have the following qualitative situation that CP contains VDT and VDT contains MTA. Since RAG studies a generalized version of MTA, it is pertinent to ask what are those topics that can enlarge RAG similarly as VDT and CP enlarge MTA?

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Therefore, of course, we should deal with quantitative investigations of solutions of

where and U and V are functions of two real variables smooth enough. However, trivial but important, these solutions are just of the following complex (but not necessarily analytic) functions

Thus we observe that quantitative investigations of solutions of (2.6) with different values (or what is the same, quantitative investigations of of solutions of (2.6)) should compose a field of study which may be considered as a generalized VDT (GVDT) for solutions of (2.6) or VDT in a generalized RAG (GRAG). The same topics may also be considered as a “value distribution theory” for general classes of complex functions (2.7) or GVDT. In both cases, we propose to study quantitatively and of functions (2.7). We also propose to deal with much wider classes of functions U and V in (2.6) or complex functions in (2.7) than is the situation in classical cases. Problem 2. To construct VDT in GRAG, that is to construct VDT for solutions of (2.6) with U, V belonging to large (but applicable) classes of functions. The problem is equivalent to constructing VDT for classes of complex functions Field 3: The crossroad of generalized closeness property and generalized real algebraic geometry We believe that a kind of the proximity property of is valid for some generalizations of meromorphic functions as well. Therefore, we pose Problem 3. Prove the closeness property of for solutions of (2.6) with U, V belonging to classes of non-analytic real functions? The problem is equivalent to proving the closeness property for large classes of complex functions not necessarily analytic. Field 4: A subtopic in real algebraic geometry studying regularities of Nevanlinna theory type Let us consider a particular case of Problem 2, dealing with (2.4) instead of (2.6). In this case, the object to be studied remains in the field of RAG and we just propose to establish a new type of theory in RAG studying

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numbers of different of the system (2.4), possibly being similar to the Nevanlinna theory. This would be equivalent to studying VDT for complex non-analytic polynomials

in other words, we would deal with VDT in RAG. Thus we in fact have posed the following Problem 4. To construct VDT in RAG. Remark 4.1. Here it should be mentioned again that the Nevanlinna value distribution theory as well as the proximity property of complex (analytic) polynomials are almost trivial in this case: polynomials have no deficient values in the sense of Nevanlinna, while making use of the MTA, we immediately get a Nevanlinna type conclusion. However, VDT in RAG for more general classes of complex (non-analytic) polynomials may appear to be more complicated. Despite of the fact that solutions of (2.3) and (2.4) should have many similar properties they should have also essential differences. Indeed, solutions of (2.3) are always pointwise while solutions of (2.4) may not. Therefore it is likely that in constructing of VDT for solutions of (2.4) (although we are still dealing with polynomials), new types of deficient values or some new types of peculiarities may appear which might have no analogue in the classical Nevanlinna VDT for meromorphic functions. We believe in fact that we are now speaking about a very interesting theory for the future. Field 5: A subtopic in real algebraic geometry studying egularities of closeness property type Similarly as to Field 4, we now consider the following particular case: Problem 5. Prove the closeness property of analytic) polynomials

for complex (non-

Below we offer a solution of Problem 5 in a particular case. However, we should first establish this property for ordinary (analytic) polynomials.2 5.1. Proximity property for ordinary (analytic) polynomials. The simplest and weakest version of the property is as follows: 2

The polynomial case has been mistakenly ignored while establishing the proximity property for meromorphic functions in the complex plane [8], [9], [16]. In fact, to transfer the proximity property to non-analytic polynomial mappings and then to compare the results obtained with the property in the standard case of analytic polynomials, we should first have the property for these analytic polynomials.

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Theorem 5.1. For an arbitrary analytic polynomial and for arbitrary real values one can enumerate of P, counting multiplicities) such that

Here depends on the coefficients of P only and (but is independent on the number q of values For it is true that

Comment 5.1. We first note that the result is sharp up to constants in (5.1). Indeed, if we take and then all lie on segments arg Consequently, prescribing to lying on segment, we have Therefore, while we have so that Therefore, due to (5.1), This shows that Theorem 1 is sharp up to a constant. Comment 5.2. The inequality (5.1) is analogous to the second fundamental theorem of Nevanlinna since in the left hand side of (5.1), q can be as large as we please but the right hand side does not depend on q. Comment 5.3. This last circumstance permits us to derive a conclusion somehow analogous to the deficiency relation in Nevanlinna theory. In fact, this conclusion reflects nothing but the proximity property of a-points for polynomials. To derive the property for polynomials, we show that if q is large, then the distances should be small for a majority of indices and To show this, note first that the total number of indices and is equal to and that there can be not more than distances in (5.1) (here is the integer part of for which holds. Consequently, for the other distances we have the inequality that reflects the smallness of the distances (proximity property). Now we should only show that the last inequality takes place for a majority of indices and This immediately follows from the fact that the total number of these indices is at least equal to Recall that we supposed q to be large. Therefore the quantity

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can be considered to be a majority with respect to the total number of indices 5.2. Proximity property of ordinary (analytic) polynomials in real algebraic geometry. Let and be real polynomials with Jacobian and let be the set of solutions of the following system of equations

Z* be the set of solutions of

and Z** be the set of solutions of

We refer to and as non- generating polynomials if there is a finite number of solutions of (5.3)–(5.5) only. The set of all solutions belonging to and lying in a disk can be decomposed into some subsets satisfying the following properties: for arbitrary values and belong to for arbitrary and values and belong to and with neighboring values K, Y, so that either K = Y + 1 or K = Y – 1. Such a splitting of the set into sets is indeed possible: Let us call the collection of sets a proper splitting. Then the following result [25] follows, reflecting a weak version of the proximity property of in real algebraic geometry: Theorem 5.2. For arbitrary real values for arbitrary real non generating polynomials and and for an arbitrary real positive value R we can enumerate the solutions of (5.3)–(5.5) such that

Field 6: A subtopic in complex algebraic geometry studying regularities of closeness property type

We next consider complex algebraic geometry, that is, solutions of the equations

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with complex variables and let us call these solutions of the equation (2.1). Unlike real algebraic geometry where a counterpart to VDT is not clear, now we are able to obtain (see the last section) a kind of closeness property in complex algebraic geometry, a clear counterpart to VDT. Indeed, this is the Bezout Theorem. However, we do not know at present how to obtain the closeness property. Respectively, we pose Problem 6. Prove the closeness property for algebraic geometry.

of (2.1) in complex

There can be numerous fields (but RAG), where such a generalized VDT or closeness property can be applied. Below we speak about one of these fields in ordinary differential equations. Field 7: The crossroad of generalized value distribution theory, generalized closeness property and autonomous systems of ordinary differential equations It is easy to see that any result in GVDT can be considered as a result in autonomous systems of differential equations

Indeed, the solutions of (2.6) with Re a = Im a = 0, that is, zeros of (2.7) mean an important equilibrium set of the equation (7.1). Respectively, by describing of in GVDT, we simultaneously describe the equilibrium points of a “distorted” equation

which we now call of the equation (7.2). In other words, the equilibrium sets of the distorted equation (7.2) are just of the function (2.7). Therefore, investigating equilibrium sets of distorted autonomous systems of differential equations becomes a subfield of the GVDT. Respectively, we pose Problem 7. To study equilibrium sets of distorted autonomous systems of differential equations using GVDT and GCP for functions (2.7). In fact, there exist already results which may be considered from the point of view of these differential equations. In fact, any existing result in ordinary and generalized VDT or CP can be rephrased in terms of these equilibrium sets. For example, any result in CP, say, considers quantity or

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location of of (2.7). However, such are equilibrium sets of a distorted equation. So, we may now rephrase Theorem 5.1: Theorem 7.1. Take where is a polynomial For arbitrary real values one can enumerate of the equation (7.2), with counting of multiplicities so that

where depends on the coefficients of P and of only, but not on the number q of values Thus, by rephrasing, we get a closeness (or Nevanlinna) type result for the equilibrium sets of an autonomous equation. Field 8: A subtopic in complex differential equations studying regularities of closeness property type

Quantities of of meromorphic solutions of complex differential equations have been intensively investigated during the last decades [52]. In [23], a new idea to describe the geometry of of meromorphic solutions of algebraic differential equations

has been described, giving some additions to classical investigations in this field. Indeed, it turns out, see [23], that for meromorphic solution of (8.1) there is a sequence and a sequence belonging to a finite totality of some rational numbers and depending on the equation (8.1) and the degrees of its coefficients such that for the distances of most of it is true that

where may be increasing as slowly as we please. Thus, mutual locations of most of the solutions of first order algebraic differential equations are completely determined by the equation, describing the closeness property of the solutions of (8.1). Now it is pertinent to put Problem 8. Prove the closeness property for of solutions of different classes of complex (algebraic as well as non-algebraic) differential equations, including higher order equations.

17

Field 9: A subtopic in complex functional equations studying regularities of closeness property type Quantities of of meromorphic solutions of complex functional equations have been investigated only recently. We are not aware about investigations related to the geometry of of such solutions. One may expect that meromorphic solutions of complex functional equations should have an interesting geometry of locations of their Hence, we may pose Problem 9. Prove the closeness property for of solutions of different classes of complex functional equations? Similarly, we may ask for the closeness property for functional-differential equations. Field 10: The generalized Hilbert’s problem 16 as a particular problem in generalized value distribution theory Particular classes of solutions of (2.6) in GVDT can be applied to the Hilbert problem 16(a), see [49], [50], [37], which asks about number, form, or positions of connected components of solutions of

that is, about connected components of level sets of real polynomials (more generally of level surfaces of This is a purely topological problem by Hilbert’s words. Hilbert presents this topic as a further development of the following known result from 1876, due to Harnack, see ([47], 1876) which says that the the number N(P) of closed connected components of solutions of (10.1) is less or equal to the genus of the polynomial P plus one. Usually, this result is cited as follows:

where denotes the degree of the polynomial P. This old result may be considered as the only comprehensive result in this field up to now. The corresponding case of three (or more) variables is esentially less investigated. 3 We mean here the following Subtask (a): To find the number of closed connected components of solutions of where P is a real polynomial. More attention attracted the following 3

The latest information in this field I have met is a short survey by Arnold, see [37], p. 50–51, indicating that the problem is still open.

18

Subtask (b): To find the number of closed connected components of these level sets that are subsets of another such closed connected component meaning that is contained in a domain with the boundary We remark that even in the case with of two variables the Subtask (b) has some particular solutions only at present. Therefore, it will be somehow banal to say: Let us consider a generalized Hilbert problem for larger classes of non-polynomial functions or let us consider the interplay between connected components of solutions of (6.1) or else connected components of solutions of distorted equations

if we want to remain in the frame of old approaches to the Hilbert problem. However, it turns out that this topological problem can be easily transformed into a problem in GVDT. To discuss this transformation we need first to cite the following main result in algebraic geometry: Bezout Theorem (see, for instance [38]). For

of degrees

polynomial equations

in variables, there exists “in general” common solutions, unless there are infinitely many

of such solutions. Here the words “in general” mean that the conclusion of the theorem is true for all polynomial systems (10.4) except for some exceptional cases for which the above number is infinite. These exceptional cases are described comprehensively in algebraic geometry as well. The following lemma reduces the subtask (b) above to the Bezout theorem: Lemma 10.1. The number N(P) of closed connected components of solutions of at most equal to the number of solutions of the equation

where R is an arbitrary real number. Proof. Suppose first that the closed connected components are smooth. Then the angle between the tangent to and the real axis accepts all

19

values in Respectively, for any real number R there is at least one point on this such that tan i.e. This means that is a solution of (10.5). On the other hand, if on a given closed connected components there is a point where the components is not smooth then we have and simultaneously so that is a solution of (10.5). Thus any involves a solution of (10.5) so that Lemma 10.1 follows. Thus we will be able to study such a topological concept as the number of closed connected components of the level sets of if we get estimates for the number of solutions of (10.5). By applying the Bezout theorem we immediately get the following addition to the Harnack theorem. Theorem 10.1. If P is a non- exceptional polynomial (in the above sense), then for an arbitrary the number N(P) of closed connected components of solutions of (10.1) is at most equal to where Moreover, Actually, it is possible that for some polynomials P and some values R,

In any such a case, Theorem 10.1 improves the Harnack estimate (10.2). Corollary 10.1. As a particular case, we have

Similarly, we may give upper bounds for the number of closed connected components of level sets of a non-exceptional polynomial or what is the same, the number of closed connected components of solutions of Indeed, arguing as above we obtain Lemma 10.2. The number N(P) of closed connected components of solutions of is at most equal to the number of solutions of the equation

20

where Therefore, by applying the Bezout theorem, we immediately get the following addition to the Harnack theorem, giving a solution of the subtask (a) above in a rather general case, see (10.5) above. Theorem 10.2. If is a non exceptional polynomial then for arbitrary real numbers the number N(P) of closed connected components of solutions of (10.7) is at most equal to Corollary 10.2. As a particular case, we have

where Thus, we see that the topological Hilbert problem 16 can be easily transformed into the following problem in AG: To study the number of solutions of with a given value R. Surely, it would be interesting to have results giving not only but also to compare and where and are given values. We observe that this new problem is just a modification of Problem 2 posed, see Field 2 above. To see clearly the difference, we put now together the corresponding equations to be studied in Problem 2 and in this new problem:

for

and

for Above we meant finding of Nevanlinna type regularities for the numbers of solutions of (10.9) with different values, say and Clearly, if we find new information about the mutual locations of solutions (a kind of proximity property), it will simultaneously give new information about the locations of closed components in the Hilbert problem, as these components involve the solutions. Summing up the above reasoning we may pose the following problem: Problem 10. To study value distribution and the proximity property for solutions of (10.9). Such studies would immediately result in conclusions related to the number and location of closed connected components in the Hilbert problem.

21

The problem stated here for the case of two variables only can obviously be rephrased for several variables considering then solutions of with

Field 11: Hilbert’s problem 16 and the geometry of connected components of level sets of polynomials In Field 1 above, we considered the interplay between topological and geometric investigations of level sets of non-analytic functions and in particular polynomials Below, we shall consider closed connected components of level sets of P. Since the subtask (b) in the Hilbert problem, see the last section, on the topology of closed connected components of level sets is of great interest we present now the following subtask of Problem 1 related to the geometry of these components. Problem 11.1. To study the geometry of closed connected components of level sets of polynomials By analogy with the subtask (b) in the Hilbert problem above, we now consider now closed connected components of the level sets contained in another closed component, see the previous Field 10. By considering these components from the point of view Gamma-lines we easily prove the following result:

Theorem 11.1. Suppose that are some closed connected components of level sets of a given polynomial and lie in Then

The above simple geometric result dealing with the topological objects of the Hilbert problem seems to offer a pertinent addition to the topic of closed connected components of level sets of polynomials. Further questions naturally arise: Why should we consider the geometry of the connected components for polynomials only? What about the case of rational functions and even of more general functions? Usefulness of such a problem is clear, say via systems of ordinary differential equations

Even more generally, what about wider classes of functions Clearly, the number of connected components can be infinite for more general classes. In this case, the question should be modified and instead of the

22

number of components, we could now ask about their asymptotic number. A standard approach in several branches of analysis including the Nevanlinna theory is to consider the asymptotic of number of zeros, or more generally of of meromorphic functions in the disks Following this idea, we may consider the geometry of all connected components of level sets lying entirely in the disk and then by taking try to get an estimate of these components in whole plane. Thus we naturally come to the following Problem 11.2. To study the geometry of connected components for general (interesting) classes of functions Further, it would be quite reasonable to study together with the total length of also the total curvature of that is where k (z, f ) is the curvature of to study a curve at the points Of course, investigating the curvature is a natural part of the study of the geometry. But it is also interesting since the knowledge of is connected with the Hilbert problem. Indeed, for any connected component

holds. Therefore, is at least equal to the number of all connected components lying entirely in In particular, in the case of polynomials P, we pose Problem 11.3. To show that

for a finite number polynomial P.

depending on the degree and the coefficients of the

As a modified version of Problem 11.3, we also put Problem 11.4. To show that

where denotes the total spherical length of curves composing lying in the disk and is a finite number depending on the degree and the coefficients of the polynomial P. Problem 11.5. To show that for a polynomial

23

where is a finite number depending on the degree and the coefficients of the polynomial P. Clearly, if the last conjecture is true, then from (11.5) it immediately follows that the number of closed connected components of P is at most equal to as one can see by comparing with the Harnack theorem). However, if we know for a given value we conclude that the number of connected components in the disk. is at most equal to So, we then get an idea about the comparative distribution of connected components in and out of Importance of such an information in the frame of the Hilbert problem is obvious. Field 12: Gamma-lines and level sets of non-analytic classes of functions

In the preceding fields, we have shown how generalized VDT and CP can be applied to study of level sets of functions, which are solutions of an equation. In the inverse direction one should be able to apply level sets to studying that are intersections of two level sets. We expect this idea to be equally important as well. Indeed, analytic and geometric investigations of level sets for general classes of functions are not less important than their topology. However despite of the wide applicability, this subject that undergone little investigation up to now. Anyway, we assume to have now enough developed tools to pass to a more detailed consideration of level sets for quite large classes of functions. A general result related to level sets is given in Field 1 above. This inequality (1.4) is just a further development of ideas coming from the theory of Gamma-lines, constructed for meromorphic (and quasimeromorphic) functions at the end of 1970’s. Gamma-lines are, in particular, level sets of real functions that is of solutions of To discuss the interplay between these notions and other topics, we need to sketch the key results in the theory of Gamma-lines, and in particular its connections with the Nevanlinna value distribution theory (VDT) as well as the Ahlfors theory of covering surfaces. These theories mainly study the number of of meromorphic functions, that is of solutions of

or, what is the same, the cardinality of the preimages In particular, the Nevanlinna theory establishes an equibalance between the number of of for “good, non-deficient complex values”. Similarly, the theory of Gamma-lines considers preimages (we call them Gammalines), where is an curve “good enough” and establishes an equibalance between the length of of for “good, non-deficient curves”

24

and The deficiency relation established in the theory of Gamma-lines is quite similar to the deficiency relation in the Nevanlinna theory. We now recall the following analogue of the Nevanlinna second fundamental theorem and the deficiency relation (see [16], Ch. 4) for of meromorphic functions. Theorem 12.1. Let be a meromorphic function in the complex plane and let be a collection of disjoint, bounded, smooth Jordan curves with Here is the total variation of the angle between the tangent to and the real axis. Then there exist absolute constants and a set E of lower logarithmic density c such that

The following result shows that upper bound for the length of in Theorem 12.1 is sharp as soon as the closure of involves a value whose geometric deficiency is greater than zero; for definition of geometric deficient values, see in [3], or [16], Ch. 2, Section 2.2. Theorem 12.2. Let be a meromorphic function in the complex plane, of lower order Suppose that there is complex value in the closure of a continuous curve such that Then there exists a constant depending on only, and a sequence such that

By analogy with the Nevanlinna theory, we say that a given curve non-deficient if

is

and deficient otherwise. Clearly, (12.1) implies an analogue of the Nevanlinna deficiency relation with the theory of Gamma-lines saying, qualitatively, that for any meromorphic function the majority of curves are non-deficient. This is due for majority of as we have, by (12.3), the “smallness” of the corresponding Gamma-lines. Applying this conclusion we arrive at the proximity property of meromorphic functions that permits us to describe geometric locations of of This property permits us to pass to the next stage of investigations in the theory of meromorphic functions, that is to study locations (geometry) of unlike the classical Nevanlinna theory that studies the number of these only. The basic idea here is very simple: Obtaining upper bounds for the length of Gamma-lines of a given curve we obtain a conclusion about the

25

smallness of this length. Reversed, this smallness leads to the conclusion that for “majority” of complex values lying on this the corresponding should be geometrically close. This observation should be true for more general classes of complex functions as well. Therefore, by constructing the theory for Gamma-lines for such classes of functions, we result in a kind of Nevanlinna theory and proximity property in a more the general case than the meromorphic functions only. Since Gamma-lines of are level sets of their applicability is as large as the applicability of level sets. Therefore, investigating Gamma-lines of more general classes of functions should be useful both from the point of view level sets of Re F(z) and from the point of view of the proximity property for functions In particular, if we able to establish the proximity property for functions considered in previous fields, we should be able to make progress in the previous problems as well. Therefore, we pose Problem 12. To study the geometry of Gamma-lines for non-analytic classes of functions Field 13: Gamma-lines of quasiconformal functions and catastrophe sets as tools to investigating of Gamma-lines of general classes of functions Following [17], we now offer an approach to study Gamma-lines and value distribution of general classes of functions. First of all, to study general classes of non-analytic functions in a given domain D we need to get an idea about singularities of the corresponding functions. Roughly speaking, we need to separate those sets where the coefficient of quasiconformity of becomes infinite (catastrophe sets). Then the set may be divided into two subsets, where is either quasiconformal or antiquasiconformal. An essential advantage now is that the topology of on these subsets is the topology of an analytic or an anti-analytic function. Hence, we may apply a number of topological results for analytic functions to investigate on these subsets. Moreover, there is a general method, the tangent variation principle, permitting us to give bounds for the length of level sets of quasiconformal functions. Thus, describing the catastrophe set and then applying the tangent variation principle to the function on these subsets of should give some idea about the geometric behavior of Gamma-lines of the function on whole domain D. For convenience of the reader, we now discuss catastrophe sets and the tangent variation principle in some detail. Catastrophe sets. The idea of the catastrophe set is to determine those sets where a given vector field, a complex function, or a solution of a sys-

26

tem of equations, has “bad” behavior. We believe that this notion is selfinteresting and deserves to be studied independently. Indeed, these sets have important interpretations in applied science. Investigations of several physical phenomena are reduced to considering “smooth” complex functions in a given domain The general ideology here is as follows: these phenomena (such as hydro- and gas dynamics, electromagnetic phenomena, acoustics and so on), where a given physical magnitude is spread more or less uniformly in all directions correspond to functions which are locally quasiconformal, i.e. their coefficient of quasiconformity remains finite in D. If is equal to infinity at a given point or on a given line in D then the behavior of the function as well as of the related physical phenomena there changes essentially. Indeed, the mapping then degenerates in the sense that images of small domains are not any small; respectively, the related physical phenomena are not any more spread uniformly. That is, in these points on these lines, where a certain “catastrophe” takes place, both mathematically and physically. Via the notion of a catastrophe set, we obtain a novel aspect to approach in the general theory of differentiable mappings: By knowing these lines we may extract from D some subsets, where the function is quasiconformal, hence being able to apply the well-developed theory of quasiconformal functions. Thus we come to the following Problem 13.1. What can be said about the points/lines of catastrophe where From the point of view of applications, it would be important to consider solutions of a given elliptic complex differential equation

Indeed, numerous physical phenomena can be described by such equations, see [35]. Therefore, it would be important to describe the sets in terms of F and its coefficients, which may depend on and Having an idea about the points/lines of catastrophe for we may extract from the domain D some subdomains where the solution has “right” behavior. Of course, the lines of catastrophe are included in the boundaries Thus we come to the following Problem 13.2. To describe the lines of catastrophe for solutions of a ceratin class of equations F. Similarly, to describe the lines of catastrophe for functions belonging to a given class Remark 13.1. As another extremal case, it would be interesting to consider the sets where We may call these sets as the con-

27

formal lines. Of course, the conformal lines are remarkable as the function has maximally “regular” behavior there. Problem 13.3. To describe the conformal lines in Problem 13.2.

in the cases mentioned

A more detailed variant of Problem 13.3 would be Problem 13.4. To describe the lines in D, where for solutions of certain classes of equations Similarly, to describe the sets of in D for functions belonging to a given class Remark 13.2. Usually quasiconformal functions are considered with Of course, we may consider more general classes of functions by admitting on certain subsets of D the condition and on their complements the condition Of course, the condition resp. means that mapping preserves (resp. reverses) the local orientation in a neighborhood of the point Tangent variation principle. The standard approach to evaluate assumes estimation of the sum where are the branches of the inverse function of taken on the sheet of the Riemann surface and where, of course, is on the sheet, too. We now give a general method of the “tangent variation principle”. This permits us to estimate the length for an arbitrary meromorphic function in D and an arbitrary curve We remark that all results below remain true with minor changes only for functions. Denote by the angle between the real axis and the tangent to at Similarly, we define and If now is the real axis, then

where denotes the length of brackets in (13.1), we observe that

Denoting by V(D) the expression in

28

where

Thus it follows from (13.1) that

Assuming that

is a smooth Jordan curve, consider the magnitude

where is the angle between the tangent to at and the real axis. Evidently in a sense characterizes the “curvature” of Denoting now we may prove the following result, see [16]. First Fundamental Theorem (Tangent Variation Principle). For any meromorphic function in and for any smooth Jordan curve

Thus, the length can be estimated in terms of and V(D), both indicating the “curvature” (or, what is the same, the tangent variation) of and We now continue to estimate collections of of arbitrary analytic (meromorphic) functions in D. To this end, let be a finite collection of disjoint bounded smooth Jordan curves. Denoting by K different absolute constants and by constants depending on and only, we may prove the Second Fundamental Theorem. Let in the closure of D and let Jordan curves with Then

be a function meromorphic be disjoint bounded smooth

where

Moreover, if then

is an unbounded smooth Jordan curve with

29

All inequalities above remain essentially true for a function. Indeed, we just need to add some new constant multipliers depending on only into the right-hand side of the inequalities above, see [16] for details. Thus, if we consider a smooth complex function in a domain D (in particular, solutions of equations) and we able to get an idea about its catastrophe set, then the investigation of Gamma-lines of can be reduced to investigating on the sets composing where is quasiconformal. Then, clearly, we may apply the above inequalities for quasiconformal functions to describe Gamma-lines of on Consequently, we obtain a description about the Gamma-lines of in D. Field 14: Gamma lines in studying regularities of closeness property type for general classes of functions

In Field 13 above, we described an approach to studying Gamma-lines of general classes of functions. We now explain how similar results can be applied to investigating value distribution for general classes of functions. This can be done quite similarly as for meromorphic functions, see [16]. In this case, it follows from a simple geometric observation (the fundamental inequality) follows that as soon as we have upper bounds for the length of Gamma-lines we immediately obtain consequences in value distribution. However, this geometric observation remains true for larger classes of functions. Therefore, (a) bounds for the length of Gamma-lines and (b) the fundamental inequality for more general classes of functions, we should be able to obtain results in the value distribution of such functions. Thus, we first explain the approach in the case of meromorphic functions, describing types of regularities that can be studied by this approach. For details of what follows, see [16]. 14.1. Preliminaries. Incompleteness of the theory of meromorphic functions. The Nevanlinna value distribution theory and the proximity property of study, respectively, numbers and mutual locations of of meromorphic functions in The main conclusion of the theory and of the property remain true in the unit disk for meromorphic functions of sufficiently fast growth, but not for functions of “slow growth” in general (such as the classes bounded functions, Dirichlet, Blaschke products etc.). The main deficiency relation in the Nevanlinna theory, resp. in the Ahlfors theory qualitatively asserts that for any meromorphic function in the complex plane, there is a set of “good”, i.e. non-deficient, values of such that G “almost coincides” with resp. for any value the counting function is close to the characteristic Hence, for any and the quantities and are

30

asymptotically close. This last phenomenon we shall refer as the Nevanlinna property. As to the functions meromorphic in the unit disk, the Nevanlinna property is true for functions with fast growth, i.e. when

where

is the spherical characteristic function. For the same classes of functions, it is possible to describe mutual arrangement of for different values (see for the proximity property of in [6], [9], [12], [13], [15] and [16]). In addition to the Nevanlinna property, the proximity property states that not only the quantities but the geometric locations of these and as well are close as well. The situation with the Nevanlinna property, resp. the proximity property, is more complicated in the case when is of slow growth, i.e. when

In fact, very few results have been obtained not only for the mutual arrangements, but even for the quantities of For the well-known cases such as for the classes of bounded functions, Dirichlet etc., one can estimate the Blaschke sums for their zeros, while even comparing the number of for different becomes difficult. Indeed, these quantities can be strikingly different. For instance, the functions with slow growth may omit values for “large” sets Therefore, the Nevanlinna deficiency relation is not any more true for functions with slow growth and we have non’t have the Nevanlinna property, resp. the proximity property in the general case. However it turns out, see [16], that if we know the length of Gammalines, we obtain a novel, geometric approach to studying value distribution and mutual locations of for arbitrary meromorphic functions including those with slow growth. The main idea here is very simple. Indeed, if where is a curve in the complex plane, then by estimating the length of we obtain estimates for distances between and This approach permits us to describe the numbers and mutual locations of by using a very simple background accessible even for those who just started to learn complex analysis. Another principal conclusion is that we immediately obtain a kind of the second fundamental theorem for any class of functions in the unit disk,

31

if we just have an upper estimate for the length of functions.

in this class of

14.2. Characteristic functions describing the numbers and the mutual locations of Let be a meromorphic function in and let Moreover, let be a finite collection of pairwise different complex values, let A be the set of all counting multiplicities. As the counting function of the set for the function we define

where

and where is the usual number of ing multiplicities.

in the disk

count-

Let now be a smooth Jordan curve in passing through the given points in this order. We denote by the points of the function and by the set of all in counting multiplicities, where For simplicity, we assume that avoids all points at which We denote by the part of the curve lying between the points and Then the set lying in consist of some curves of three types which, due to the assumption above are completely determined as follows: A) Every curve of type connects an with a Hence, is at most equal to the length of this curve. connects a point with a B) Every curve of type point on the boundary Hence, is at most equal to the length of this curve. or C) Every curve of type has no end points coinciding with Let now A) for all and define

We shall call

be the collection of all pairs

the closeness function of

and

from

32

Observe that at most one of two points of a given curve of type Let points or from B) and let

can be an endpoint be the set of all such a lonely

where

and is the number of from B), counting multiplicities. We now call to be the counting function of lonely and For the close and from à we shall also consider

where

and call

is the number of from Ã,counting multiplicities. We the counting function of close and

14.3. Connection between the closeness function and the counting function of lonely points with the lengths of It is obvious

that

and that and are at most equal to the length of the corresponding curves. Therefore, denoting by the total Euclidean length of the curves in we come to the following Fundamental inequality. For any function and for any curve the set A of all

meromorphic in and on

can

33 be represented as of lonely and

so that for the closeness function and the function we have

The above definitions A), B) and C) remain also true also in the case, when the above assumption does not hold. In this case, the separation of the close pairs and lonely points has to be accomplished in a different way. We are now to state our Approach to studying the value distribution and the closeness property of meromorphic functions by using Gamma-lines: Since (14.4) gives upper bounds for the counting function of lonely and we immediately get conclusions on the value distribution (on the numbers of of an arbitrary meromorphic function in the complex plane as well as in the unit disk as soon as we have upper bounds for Since (14.4) gives upper bounds for the counting function of close and we also find conclusions on the mutual locations of of an arbitrary meromorphic functions in the complex plane as well as in the unit disk as soon as we have upper bounds for It is easy to observe that this approach still works for many other classes of non-analytic functions; at least it is the case while considering functions of classes, say M, whose topological properties are somehow similar to those of of analytic functions. Thus, if we have progress in studying Gamma-lines of functions in a given domain D (which, this time, is not necessarily the complex plane), we may apply the approach above to investigate the value distribution and the closeness property of such functions In turn, to investigate Gamma-lines of these functions, we may apply the approach offered in Field 13 above. Thus, we naturally come to the following task: Problem 14.1. To apply the approaches in Field 13 and in this Field 14 to investigate the value distribution and the closeness property of non-analytic functions. Field 15: Gamma-lines of solutions of systems of partial differential equations It would be interesting in particular to consider Gamma-lines of complex functions where and are solutions of a system of partial differential equations (PDE)

34

Indeed, the theory of Gamma-lines accomplishes similar investigations in the case of meromorphic functions. This means that instead of (15.1), we have to consider the Cauchy–Riemann system. Therefore, we may pose the following Problem 15. To study Gamma-lines and possibly to generalize the theory of Gamma-lines for more general partial differential equations (elliptic, parabolic, hyperbolic, mixed type). Field 16: Gamma-lines of algebraic and algebroid functions

For algebraic functions

that is for solutions of

where P is a polynomial of two complex variables and we may define Gamma-lines similarly as for ordinary meromorphic functions: We call Gamma-lines of the preimages of a given curve However, the situation here is more complicated than in the case of meromorphic functions since is multi-valued. Therefore, we need to decompose the function into finitely many branches where denotes the degree of P with respect to and we need to consider totalities of Gamma-lines, separately for each branch Similarly, we may define Gamma lines for algebroid functions. We may now consider these Gammalines from the point of view of equations. To this end, let us consider the following system of equations:

where is a real function such that the solution of is a smooth curve in The first condition in (16.2) means that we are dealing with an algebraic function The second condition means that we are dealing with a curve in the Thus the solutions of (16.2) are Gamma-lines of where is the smooth curve above. For the particular case when we are dealing with level sets of the function Re that is with solutions of the equations

We first remark that the value distribution theory for algebroid functions corresponding to the Nevanlinna theory has been constructed more than seventy years ago, see e.g. [64]. This implies us to pose the following

35 Problem 16. To construct the theory of Gamma-lines for algebraic and algebroid functions. Field 17: Geometry of tions

of algebraic and algebroid func-

In Field 14 above, we described how to derive conclusions about locations of of complex functions if we know the length of Gamma-lines of these functions. We showed that each time when we have information about the length we get a kind of the proximity property of Now, if we first obtain some conclusions about the Gamma-lines of algebraic and algebroid functions, we can use this approach to study Problem 17. To construct the proximity property for and algebroid functions.

of algebraic

Field 18: Gamma-lines in real algebraic geometry To introduce Gamma-lines in real algebraic geometry, we first consider solutions of a system of equations

with real polynomials and from the point of view of complex analysis by introducing the following complex, in general non-analytic, polynomial

Solutions of (18.1) are then nothing but the zeros of of that is solutions or points in solutions of the following distorted system of equations

Then are

By analogy, we call these solutions of (18.3) as of the equation (18.1). We may now consider a number of problems, similar to those studied in complex analysis, for of solutions of (18.1). In particular, we may consider Gamma-lines for solutions of (18.1) which we define, quite analogously, as the set of those points where Clearly, the set of Gamma lines of (18.1) is the union of all solutions of (18.3) with Now we put

36 Problem 18. To construct the theory of Gamma-lines in real algebraic geometry. In other words, to study Gamma-lines for complex non-analytic polynomials. Let us use for this length the notation taking into account the fact that the level sets of are of We observe that when our curve is the straight line the Gamma-lines becoming ordinary level sets of the polynomial that is the solutions of Therefore, recalling Theorem 2, we have the following inequality:

This result immediately leads to estimates of the length of Gamma-lines of (18.1) for rather large classes of curves which are solutions of the equations where F is a real polynomials. Clearly, our curves then are connected components of the algebraic curves defined by F = 0. Indeed, for these the of (18.1) are solutions of the equations Hence, we again deal with level sets of polynomials. For each such function there is a certain finite set of pairs of indices such that

where the coefficients mial

are real constants. Then the degree of the polynowith respect to is, of course, equal to

and, respectively, with respect to

equal to

Thus, denoting

and applying Theorem 1.2, we obtain the following Theorem 18.1 ([25]). For an arbitrary equation (18.1), an arbitrary (algebraic) curve defined by the equations (18.1) and an arbitrary domain D with piecewise smooth boundary, it is true that

37

Field 19: Gamma-lines in studying the influence of geometry in complex interpolations problems 19.1. Interplay between the Gamma-lines and interpolation by an analytic function with a certain geometry of interpolating points. It has been expressed repeatedly during the last decades that interpolation problems for different classes of analytic functions in a given domain D strongly depend on the geometric location of the interpolation points (in particular for classes of functions defined by double integrals). However, it seems that we don’t yet have any clear results presenting this dependence for at least one class of functions. This reflects the common situation in complex analysis that geometric locations of have been studied very little, although quantities of have been described for numerous classes of complex functions. It turns out that the proximity property of of meromorphic functions and can be applied to study interpolations by complex functions with a prescribed geometry of interpolating points. Our approach (to be demonstrated below) is, in short, as follows. Using Gamma-lines we shall establish a renewed version of the proximity property dealing with an arbitrary analytic function in a given domain D and a (finite or infinite) set Z of points in a domain D with lying on a given curve 4 The renewed proximity property describes an interplay between the locations of Z and the growth of This property is also true for an (interpolating) function taking the values at the interpolating points belonging to Z. Thus we immediately get a general result describing the interplay between the geometry of Z and the growth of this interpolating function Indeed, we believe that this approach deserves to be tested for other classes of functions and interpolating problems as well. So, we pose Problem 19. To study the interplay between the geometry of interpolating points Z and the growth of the meromorphic interpolating function Similarly, it would be interesting to consider certain classes of non-analytic functions such as quasimeromorphic functions or complex non-analytic polynomials. Moreover, at least for applications, it might deserve to consider the case when is a solution of a given boundary value problem. 19.2. Some simple results related to interpolation by analytic functions. Following [21], we first define a geometric characteristic of a given pointwise set Z in D, where D is a given domain with piecewise 4

As a particular case this set Z may coincide with the set of considered in the Nevanlinna theory.

of

38 smooth boundary Suppose that there is a set that consists of piecewise smooth, non-closed Jordan curves with both ends in Let be the totality of all such sets containing Z. Denoting by the length of a curve we call the family of shortest curves containing Z if

Let now be an analytic function in a domain D, be a given set in D with belonging to the real axis Then the totality of all sets exists since then belongs to the totality of level sets of function The next result now immediately follows from the tangent variation principle, showing that for an arbitrary analytic (meromorphic, quasimeromorphic) function given in an arbitrary domain D, the length of these shortest curves can be estimated in terms of the length of the boundary and of

This integral arises in numerous investigations in complex analysis. In this paper, we shall consider it as a characteristics for the “power” of the function Simplified tangent variation principle. For any function analytic in the closure of D and for an arbitrary set the inequality

holds. We now proceed to show that this corollary of the tangent variation principle is, in fact, a general interpolation result. Indeed, the corollary can be rephrased as follows: Theorem 19.1. Let be a solution in a given domain D of an interpolation problem with a set Z of interpolating points and with real values of Then (19.3) holds. From this result some immediate consequences follow, often arising in applications. These corollaries determine the power of an interpolation problem in terms of the geometry of interpolating points. To offer an example, let be the set of interpolating points coinciding with all integer complex points lying in a closed rectangular domain D* with sides parallel to the coordinate axis. By we denote a rectangular domain with sides parallel to the axis and containing the points

39 Corollary 19.1. Let be an analytic solution of the interpolation problem with the above set of interpolating points whose are supposed to be real. Then

Applying this corollary, we get the following result related to interpolations by entire functions: Theorem 19.2. Let be an entire function accepting real values at integer complex points. Then the lower order of is at least equal to 1. Another simple result relates to the polynomial case: Theorem 19.3. Let be a polynomial of degree be a set in D with If is a solution of an interpolation problem accepting real values at this set Z, then

In other words, a polynomial problem accepting real values lying in D only, if

can be a solution of the interpolation at a given set of interpolation points

Similarly we have Corollary 19.2. Let be a polynomial solution of the interpolation problem in Corollary 19.1. Then

Field 20: Gamma-lines and interpolation problems in algebraic geometry In the above section, we saw that the simplified tangent variation principle immediately results in Theorem 19.1 which describes the interplay between the growth of an interpolating analytic function in D and the geometry of an interpolating set Z, provided that whenever On the other hand, see Field 18, for a complex non-analytic polynomial

40

and an algebraic curve defined by

in (18.5) the

following inequality

holds, see (18.7). Of course, this is an analogue of the tangent variation principle for these polynomials or, what is the same, for the solutions of an equation (with

in real algebraic geometry. Now, similarly as we derived Theorem 19.1, we get the following interpolation result in real algebraic geometry from (18.7): Theorem 20.1. Let Z be a given set of pairs in a domain D with proper shortest curves involving Z. Suppose that for given poly-

nomials

and

with

Then

Theorem 20.1 has a clear meaning: If the geometry of Z is somehow complicated (so that the quantity is large enough) then due to the definition of see (18.6), we conclude that at least some of the quantities should be large as well. Now we want to investigate Problem 20. What can be said about the interpolation problem above for non-algebraic curves Field 21: Transfer of concepts and results of complex analysis in differentia l geometry

We now introduce the notion of in differential geometry. Let M be a smooth surface in whose Gaussian mapping G transfers M onto a set G(M) on the Riemann sphere S. 5 5 The Gaussian mapping G maps each point of M onto the sphere S with radius equal to 1 such that the normal to M at the point and the normal to the sphere at the Gaussian image coincide. In what follows, we consider on the Riemann sphere S (defined as a compressed sphere of radius 1/2).

41

For “good” surfaces the set G(M) consists of several sheets over some schlicht domains on the sphere S. We call of M all those points on M where the normal to M coincides with the normal to the Riemann sphere at the point whose stereographic projection equals to the complex value The direction of the normal on the Riemann sphere corresponding to the is called the

Let us now introduce a complex characteristic function of the surface as a function which transfers into this complex value Thus any of the complex characteristic function corresponds to a point on the surface M which has an normal. Therefore, any result related to

42

the distribution of of this complex function can be interpreted in terms of the distribution of the on the surface M. Hence, we may think about transferring problems considered in complex analysis into differential geometry. For instance, in analogy with complex analysis, we may consider Uniqueness problems in differential geometry: Let be a family of surfaces and let for a given set the condition imply that the point is Clearly, if all points of D are then our surface is just a part of the plane with normals. Now we may ask how “large” should the set Z be to conclude that the surface with is a part of a planar surface?

We may also consider Interpolation problems in differential geometry: Suppose that is a family of surfaces is a given set in D, and is a given set in the complex plane. Can we then construct a surface with an normal at the points To consider problems of this type we first need to determine the complex characteristic function for a given surface or more generally, to determine for a family of surfaces Thus we come to the following Problem 21. To determine the complex characteristic function for a given surface or more generally, to determine for a family of surfaces Then, we may consider such problems in differential

43 geometry (similarly as to corresponding problems in complex analysis) by reducing the situation to investigating the complex functions Field 22: Gamma-lines and proximity property in differential geometry with emphasis on minimal surfaces As a particular case of the previous field we propose to consider Gammalines in differential geometry for some classes of smooth surfaces M in involving minimal surfaces. For minimal surfaces, see [59], Nevanlinna value distribution theory has been established long ago [34], [43], [44] so that what we can offer should give a further development of this theory in the same way as the theory of Gamma-lines of meromorphic functions offers additions to the Nevanlinna theory. We may also give another geometric version of the value distribution theory of minimal surfaces, based on the Ahlfors theory of covering surfaces, while papers cited above construct a Nevanlinna type theory only. Our version seems to be more convenient to considering problems of Gammalines type. Let M be a smooth surface in whose Gaussian mapping G transfers M into a covering surface G(M) over the Riemann sphere S, see the Ahlfors theory of covering surfaces [2], [58], Ch. 13. In [3], a new type of geometric deficient values were introduced and an analogue of the first fundamental theorem has been established to describe covering of values by the covering surface G(M). The Ahlfors theory itself describes covering of curves or domains but not covering of distinct, complex values It turned out that thanks to these geometric deficient values, the Ahlfors theory can be immediately transferred into a theory of minimal surfaces. More generally, a version of the Ahlfors theory remains valid for families of surfaces M involving minimal surfaces, see [3]. Thus, we get a geometric version of main theorems of the value distribution theory, describing the quantities of in such a family surfaces M. We now proceed to consider location properties of of the surfaces M, starting with Gamma-lines of M. These can be naturally defined as those sets on the surface M whose G-images lie on a curve on the Riemann sphere S. Hence, it is pertinent to pose the following problems: Problem 22.1. What can be said about Gamma-lines of the surfaces M described above? If we find information on Gamma-lines, then, similarly as to the standard situation of meromorphic functions, we may consider applications of these results to describe geometric locations of of M.

44 Problem 22.2. What can be said about geometric locations of M? What about the proximity property for these surfaces?

of

Finally, let us intersect M with a plane in . Any similar intersection is clearly a kind of level sets of M and we naturally come to Problem 22.3. What can be said about the geometry of these intersections? Field 23: Differential level sets of real functions generalizing the concept of level sets, their connection with partial differential equations and with boundary value problems The problems to be posed in this section were in fact posed by the author before, see the collection [33] of open problems. So, this section actually serves to consider these problems in more detail. We also mentioned above that sometimes to get an idea about the distribution of level sets for a solution of a given partial differential equation is more important than to find the solution itself. Indeed, by describing the level sets of the solutions we get a visual picture of the solution. In what follows, we denote by the family of solutions of (23.1). However, sometimes, it may be of great importance in applied sciences to consider functions in a larger class of functions instead of (or together with) considering actual solutions in Note, that for any function we may consider the collection of level sets of called differential level sets of However, for solutions of a partial differential equation (23.1), equals to the domain D under consideration. On the other hand, we may observe an “inverse” similarity with the boundary value problems related to the solutions in a domain D, of an equation which satisfies a condition of type (23.1) on the boundary of D. Thus, what we consider now is somehow inverse with respect to the boundary value problems: We know a function and we want to study the level sets In boundary problems, we want to find or to describe its solution under the condition that a part of the set for this solution coincides with the boundary of D. Thus, we get another interesting type of curves similar to level sets, establishing another connection with partial differential equations. One of the main ideas of our proposal is as follows: Instead of the usual idea of finding solutions of a partial differential equation (23.1), we propose to study for larger classes of functions If these

45

“degenerate”, that is when they “fill” a subdomain then, clearly, this function is a solution of (23.1) in In diverse practical situations, studying these type of curves, or their interplay, can be interesting and important. Such an interplay leads, mathematically, to wide generalizations or new viewpoints as well as to new types of problems in complex analysis, differential equations, applied mathematics and their interplay. We feel that this is just a starting point for interesting investigations. Of course, to avoid unnecessary complications, we may assume at the beginning of such studies, that all functions, equations, or curves we consider are smooth enough. As a clear consequence, degenerating may “fill” domains only. Problem 23.1. Let M be a given class of functions in D. What can be said about the geometry, in particular about the length, of of for expressions of type (23.1), about their interplay with solutions of the partial differential equation (23.1) and about the boundary value problems associated with (23.1)? Remark 23.1.

generalize level sets) Indeed, if we take

we are dealing with the level sets of Remark 23.2. and partial differential equations) The topic of partial differential equations can be considered as a limiting case of corresponding to the following situation: If its “fill” a domain D, then satisfies the equation (23.1). On the other hand, absence of degeneration means that M does not involve a solution of the equation (23.1). Therefore, presence of degeneration means that we are dealing with a solution of the partial differential equation (23.1) considered in a domain Thus, to use this unusual approach to studying partial differential equations, we first need to consider the following Problem 23.2. Is there any degeneration in the above sense in a given of coinciding with class M? In other words, are there some subsets a subdomain of D? We remark that the sets in D are called despite of the fact that may consist of some curves and domains provided that the functions D and are smooth enough. Denote now by the length of non-degenerating parts of i.e. is the total length of all curves plus the total length of all boundaries of It is then natural to pose

46

Problem 23.3. To find bounds for the length in a given class M.

for functions

Remark 23.3. (Connecting the problems above with the Nevanlinna theory and Gamma-lines) Let us consider the following particular cases: Let M be the class of real parts of entire functions (consequently, is a harmonic function in the classical sense) and Then, we are dealing with the length of where is the straight line The following case is likely to be more interesting, while involving derivatives. For the same class M, let R be a linear combination

where the coefficients are real constants. Then this combination is the real part of an entire function and the above problems reduce to studying the length of for the entire function with But the length of for Re Q have been studied in the theory of Gamma-lines. Moreover, we have analogues with the Nevanlinna main theorems and the deficiency relation for these This leads us to pose the following Problem 23.4. To find an analogue of the deficiency relation for of other types of classes M and other conditions of type R = 0. Field 24: Random differential equations

as a geometric point of view to partial

In this section, we shall consider a notion appearing everywhere in applications (mechanics, physics, chemistry, biology) and which, consequently, deserves to be studied mathematically. Hence, let us consider a family M of function defined in a domain D. This class may be the class of harmonic functions, periodic functions and so on. In applications, it often happens that we have to consider the behavior of such a function on a random set, in particular on a random curve, in D. Very often such a situation may imply the case that on these random curves, a differential equation of type (23.1) holds. For instance, some observation data may have been collected about a function along certain random curves, in order to determine that an expression of type (23.1) gives a good approximation for the phenomenon under investigation. Obviously, mathematically, we are dealing with a part of of a certain function Then, an immediate important question arises: Is it true that the data observed along these lines remains true for the whole domain under investigation? To consider this situation from the point of view of mathematics we need a concept of random lines (or random curves). Mathematically, the

47 above problem is as follows: Whether a function which satisfies an equality (23.1) (according to observations) on some randomly taken curves satisfies (23.1) in the whole domain D? In fact, such randomly taken curves simply form a “random” subset of of R in D. Therefore, we simply come to the following Definition of random called a set of random

Any set of R.

of level curves of R is

Let us denote by lengths of all these lines belonging to Note that the result of calculation may give . This inequality is possible a priori. Indeed, if we consider curves inside of the degenerated parts see the definition of in the preceding section, then their contribution in is positive while In terms of the length of and of random we may offer an approach to studying partial differential equations. This approach is similar to that of the commonly used approximation of solutions. Suppose a solution of (23.1) belong to M, and consider a sequence respectively the sets Clearly, when these lines are extended as so that in any subdomain of D the set is nonempty starting with then tends to a solution On the other hand, by intuition, we may conclude that is a solution of an equation (23.1), or that is a good approximation of the solution. Suppose now that for a function we proved an inequality

If as the random observations, we get for a given function

then, clearly, we conclude that equation (23.1) in a subdomain

is a solution of the partial differential Thus we come to the following

Approach to study solutions of a partial differential equation (23.1). We just need to get estimates of type (24.1) of the length of for see Problem 23.1. Then calculate the length of a random set for a given As soon as we have for the length estimates of type (24.2), we may decide that has to be a solution of (23.1) in a subdomain Problems 24.1. Apply this approach for different classes of functions M and for different classes of equalities (23.1).

48 Field 25: Differential differential Gamma-lines, and their connections with different topics in pure and applied mathematics For the complex case, we may similarly consider a class of complex functions in D. Respectively, we may consider an equality of type or else to consider a system of equalities

where C,

and

are supposed to be real functions.

Remark 25.1 (generalizations of called C-lines below, of a function of of functions Remark 25.2 (generalizations of (25.2) may be we called differential tion of the classical concept of in the simplest particular case when

the set

coincides with the set of

Differential Gamma-lines (also are generalizations of the concept Solutions of a system of these are a generalizaof complex functions Indeed,

of

It is easy to see that in a general non-degenerating case, the set consists of some curves, while the set consists of some points. Remark 25.3 (degenerating differential and differential Gammalines are solutions of complex differential equations). Clearly, if the set involves a subdomain or coincides with D, then the function is a solution of the system of partial differential equations (25.2). In the particular case when satisfies

where P is a polynomial, these degenerating solutions coincide with the solutions of an algebraic (complex) differential equation

Similarly, if the set involves a subdomain or coincides with D, then the function is a solution of the partial differential equation (25.1). Again, in the particular case, when

49 these degenerating solutions coincide with the solutions of As complex functions may represent very different applied such as velocity, force and so on, the above observation leads to the following problems: Problems 25.1. To study the set a complex function in a given class

that is differential for different

of

In particular, it would be interesting to obtain results similar to the classical theory of analytic functions related to perhaps even some results related to the classical Nevanlinna and Ahlfors theories. Problems 25.2. To study differential Gamma-lines (C-lines) of complex a function in a given class for different C. Investigation of C-lines seems much more valuable in applications that of Gamma-lines since, unlike of Gamma-lines, C-lines take into account presents of derivatives. From mathematical points of view it would be highly interesting to obtain analogs of main results related to Gamma-lines. Finally, as a new type of problem, it would be interesting to study degeneration of differential and differential Gamma-lines: Problems 25.3. Consider a function in a given class together with (25.1). How large or “powerful” should be the set be to conclude that is a solution of (25.1) in a subdomain of D. Similarly, considering a function in a given class and a system (25.2), how large or “powerful” should the set be to conclude that is a solution of (25.2). We now add here a short survey of similar problems: A particular case interesting in applied mathematics. From the point of view applications, investigating of generalized seems to be even more promising than in the classical case. The matter is that, unlike to the classical case of of a complex function have important interpretations in very different applied situations. For instance, let us consider the following situation: Let be an analytic function and

This system of equations is studied in the free boundary value theory, being of crucial value in several applied problems. Consider now a more general system

50 As to solutions of (25.8), we already have some analogues of the Nevanlinna value distribution theory, see Ch. 5 in [16], while considering the value distribution of the sets In particular, this gives a quantitative description of the solutions of (25.7). A similar, more general problem. In practice, we may often meet situations, where it suffices to consider an inequality instead of one of the equations above. For instance, it may be the case that we should have, say, to consider a system of type

Similarly, we may have to consider

or even more generally,

Such situations are essentially different to (25.2), despite of their apparent similarity. In fact, we are dealing just with a subset of the level sets In [30], we offer a general method to obtain upper bounds for the length of these level sets for large classes of real functions considered above. If where is a meromorphic function in the complex plane, we are able to express these upper bounds in terms of the classical characteristic function. Hence, we again come to some new problems similar to those in the value distribution theory. Connection of differential Gamma-lines with boundary value problems of elliptic complex differential equations. Boundary value problems for analytic functions in a domain D or, more generally, for solutions of elliptic complex differential equations form one of the most applicable parts of complex analysis. The boundary conditions permit numerous interesting interpretations in applied science. From the point of view applications, boundary value problems answer to the following type of questions: Is it true that a given phenomenon described by some condition remains true up to the boundary The last problem actually leads to C-lines related to (25.1). Thus, differential Gamma-lines (C-lines) actually are connected with certain types of boundary value problems. On the other hand, for we already have

51 some results similar to the main results in the Nevanlinna theory. This prompts that for C-lines we may expect to obtain some results similar to the Nevanlinna theory. Hence, we may hope to obtain some new types of results in such problems related with boundary value problems. Indeed, a similar case has been already considered in [19]. Namely, we there consider a Riemann-Hilbert type condition with certain restrictions imposed upon the functions giving estimates for the length of C-lines. Moreover we also consider systems of finitely many conditions in a similar way as in the Nevanlinna theory and its analogues related to We then obtain some results, related to similar to the second fundamental theorem of Nevanlinna and the deficiency relation. Connection of differential level sets with complex differential equations and a stability phenomenon. As already mentioned above, solutions of complex differential equations formally are degenerations of differential since if is a solution of (25.4) then of the system (25.3) “fill” all of the domain D. We now raise a new type of approach related to complex differential equations. To this end, let us consider analytic solutions of (25.4) or, what is the same, solutions of the system (25.3). Then these solutions actually are completely determined by one of the equalities in (25.3); suppose these solutions are determined by

We now define the length of non-degenerating Let us also consider, for given analytic function in D, the random length of which we define as to above: We take random curves corresponding to the condition measuring their length. We now apply the idea above related to the application of random lengths. Suppose we know that for given class of analytic functions and a given the length of in domains satisfies where If for a given and a given sequence we have

where is an arbitrary function tending to when then we conclude that is a solution equation (25.3) and consequently (25.4) as well. We now give an example demonstrating this approach. It is known that if is a linear differential equation with polynomial coefficients, then any entire solution of is of a finite order and of normal type, so that Therefore, using

52 standard computations of the value distribution theory, we obtain that for a function we have Consequently, we obtain for the length of level sets the following inequality: where the constant see [16], Ch. 4. Thus, arguing as above we obtain the following Theorem 25.1. Let be a linear differential equation with polynomial coefficients. Then any entire function satisfying

is a solution of this equation. Problem 25.4. Apply the above approach for different classes of complex differential equations. Note that the degeneration here is of another type than before. Indeed, here points “fill” the domain D, while this was made by (or Clines) above. It would be pertinent to show connections of this phenomenon with the recently established stability phenomena, see [20], [22], [24]. This phenomenon shows that in some cases to get a conclusion related to a meromorphic solution of a complex differential equation (25.4), it is sufficient to show that satisfies (25.4) on a small subset of the complex plane only. As an example, we considered in [22] the classical Gold’berg result asserting that any meromorphic solution of a first order algebraic differential equation is of finite Nevanlinna order. Indeed, we proved the following Theorem 25.2. If satisfies on the set of of where are arbitrary distinct complex values, then is of finite Nevanlinna order. This result gives a visible example of the stability phenomenon. Of course, this phenomenon may be considered for several other classes of functions and differential equations We then say that given in a domain D, satisfies a stability phenomenon with respect to the equation P and a set if for all implies for all Similar phenomena can obviously be considered for partial differential equations, for systems of real equations with complex solutions etc. Respectively, we pose the following Problem 25.5. Can we conclude that for a given function (complex, or real of two variables) in D satisfying a given equality P = 0 (or a system of equalities) on given subset of D is a solution of the equation P = 0 for all values or

53 Field 26: Generalizations in the theory of quasiregular functions, generalized value distribution theory and complex differential equations with quasimeromorphic solutions In this section, we consider some of the problems posed above from the point of view of quasiregular mappings. We first remark that any generalized class of analytic functions (in particular any solution of an elliptic complex differential equation) may be represented in an arbitrary neighborhood of any point as a solution of an equation

with a given function Remark 26.1. The solutions of (26.1) immediately generalize quasiconformal functions which are solutions of (26.1) with In the general case, that is without the restriction solutions of (26.1) are more complicated: It is enough to mention that if then the solutions of (26.1) have a similar topology as that of analytic functions while this is not quite so for solutions (26.1) with an arbitrary Obviously, we may apply the approaches described above, and in particular the catastrophe set approach, to investigate complex functions satisfying (26.1). The essential difference is that we now have to describe the behavior of the solution in terms of Problem 26.1. To study as well as differential (numbers and locations) for solutions of (26.1) in the general case in terms of Similarly, to study Gamma-lines as well as differential Gamma-lines for solutions of (26.1) in the general case in terms of Remark 26.2. It would be interesting to study the geometry (locations) of the of these functions: This could be considered as a far-reaching progress of the original theory of quasiregular functions, which restricts to describing the quantities of only. Field 27: Complex differential equations with generalized quasiregular solutions We are coming to a new topic in the theory of complex differential equations by considering K-quasiconformal solutions of

or, more generally, solutions that satisfy (26.1) where is not necessarily < 1. This means that we now consider generalized analytic solutions

54 of complex differential equations. Note that solutions of ordinary first order complex differential equations may be determined in a similar way as solutions of where satisfies Recently, some results in this direction have been obtained in [27] while considering differential equations

with some algebraic restrictions imposed upon F, with quasiconformal solutions. For these solutions, an analogue of the classical Gold’berg result has been proved. Consider now the case when the coefficients of are rational functions, either complex rational, or real rational functions of two variables. If we suppose that a solution satisfies a particular type of equation (26.1) such as

or then the situation with (27.1) becomes similar to the standard case. Indeed, we may then substitute in (27.1) by (or by dealing then with a comparatively simple equation of type (27.2). Problem 27. To study partial differential equations (27.1) satisfying (26.1). In particular, to study value distribution, proximity property of Gamma-lines of the solutions. The first step into this direction might be to consider these equations (27.1) and with solutions satisfying either (27.3) or (27.4). Several problems related to analytic or meromorphic solutions of ordinary differential equations might be considered for generalized analytic solutions. We believe that further investigations in this field might be beneficial. Field 28: Generalizations of analytic functions associated with a given polynomial In connection with Problem 27, we observe call attention to a particular case where instead of (27.3) or (27.4), we consider functions satisfying

55 or This means that we are then dealing with “almost” analytic functions. In fact, when then if satisfies (28.1), and if satisfies (28.2), then we have provided keeps off of some possible curves, where is finite. Thus, the functions satisfying (28.1) and (28.2) are quasiconformal functions (but not necessarily K-quasiconformal). This should, in turn, imply “almost analytical” properties in a neighborhood of infinity. Therefore, we pose Problem 28.1. To study these “almost analytical” properties in terms of (or The next challenging question is now Problem 28.2. What can be said about “almost analytical” properties of solutions of

or of

in terms of

or

Field 29: Topological properties of multi-valued functions and multi-valued solutions of some classes of complex differential equations We first remark that this section is essentially based on [26]. Considering differential equations, or systems of differential equations from a global point of view it is natural to be interested in the description of global properties of solutions in maximally large domains of definition. There are plenty of investigations of single-valued solutions, in particular of entire or meromorphic solutions, see for instance the monograph [52]. On the other hand, multi-valued solutions, which we often come across, see for instance [46], have been studied very little. To mention a few examples of such studies, recall first two papers by Poincaré considering algebraic solutions of some particular classes of first order algebraic differential equations, see [60]–[61], and an recent investigation [40] of algebraic solutions of the (second order) sixth Painlevé equation by Dubrovin and Mazzocco. Concerning algebroid solutions we only know a few references, see for instance [51] who considered algebroid solutions of some particular classes of algebraic differential equations. We also remark that the multi-valued

56 character of solutions plays an important role in some physical problems, see [1], [39]. Speaking about multi-valued functions (more general than algebraic or algebroid functions), it is indeed the case that there does not exist any theory of Nevanlinna type, as it is the case for meromorphic or algebroid functions. However, the concept of islands from the Ahlfors theory of covering surfaces could perhaps successfully applied to considering general multivalued functions. To recall the Ahlfors theory, let W be a meromorphic function in a simple connected domain and consider the image W(G) which is a covering surface A connected part of that lies over a domain D is called an island over D if the projection of the boundary of this part on D does not intersect with D. Otherwise it is called a peninsula. The multiplicity of an island is the number of its sheets. The island is called simple if it is of multiplicity 1. The number of simple islands over D will be denoted by In the case of meromorphic functions in the Ahlfors theory studies the interplay between the classical Ahlfors characteristic function and the number of simple islands of We can, speaking a little bit qualitatively, present the main conclusion of the Ahlfors theory as follows: If we take different domains then for at least one of them, say for we have

when where E is a small set of the finite logarithmic measure. Thus the concept of the number of simple islands can in some extent substitute the concept of the characteristic for meromorphic functions. Therefore, we pose the next Problem 29. To study multi-valued functions (and multi-valued solutions of equations) in terms of the number of islands. Before, we continue, we offer a result into this direction: Suppose is a solution, in a simple connected domain G, of a differential equation of type

where are analytic functions in the variables and that can be multi-valued in both of their variables. We only pose the restriction that for a given complex value there is a constant M(G) such that

57 The solution is permitted to be, in general, a multi-valued analytic function that may be considered as the totality of some branches in simply connected domains When is an algebraic or algebroid function, then appear as usual simply connected domains where are curves connecting critical points of with infinity. From now on, we consider such a branch in a subdomain For simplicity, we will write just and G and we also consider the domain D in the definition of islands to be just a disks Remark 29.1 (Interplay between the islands and Denoting by the islands over we observe that the of whose images belong to may be considered as good as the inverse function to has a good behavior in a neighborhood is then a one-to-one mapping in the of This means that when we obtain bounds for the number of islands (see Theorem 1 below), we simultaneously obtain bounds for the number of such good and so we are considering the global behavior and the value distribution of the solutions. Denoting now by the diameter of G, we prove Theorem 29.1. Let be a branch a solution of (29.1) in a domain G, and suppose that the coefficients satisfy (29.2). Then, for any disk there is a constant C depending on M(G), and only such that

More precisely, we may prove that

Although Theorem 29.1 relates to general classes of multivalued solutions, it seems to be useful even in the standard class of meromorphic functions in the complex plane. For instance, from Theorem 29.1 we may derive the following classical theorem of Gol’dberg: Theorem 29.2 ([45]). Any meromorphic solution of the first order equation algebraic differential equation is of finite Nevanlinna order. In further investigations, it would be pertinent to combine metric topological methods of the Ahlfors theory, the geometric and analytic methods of Gamma-lines and the topological approaches developed in the singularity theory, see for instance [53] and [54].

58 Field 30: Global multi-valued functions (and solutions of differential equations) from the point of view value distribution and the closeness property Clearly, once we can study islands of multi-valued functions, and more aspects of these functions can be described in terms of these islands, the more useful is to investigate these islands. For example, if we are able to describe the value distribution, the closeness property, the Gamma-lines in terms of then we obtain a lot of additional information on different aspects of the global behavior of the solutions. Hence, we next state Problem 30. To study and Gamma-lines of different classes of multi-valued functions in terms of its islands. As already mentioned above, there are quite few investigations of algebraic and algebroid solutions of differential equations. Indeed, the value distribution theory of algebroid functions, similar to the Nevanlinna theory, was constructed around 1930, see e.g. [64]. If, in addition, we investigate the proximity property and Gamma-lines of these functions, this seems to give pertinent additions to our knowledge of these functions that possibly may have applications in the study of the global behavior of such solutionsof differential equations. Field 31: An approach to reducing investigations of solutions of differential equations to a new type of problems in value distribution theory Consider a differential equation

where are analytic functions in the variables and admitting a transcendental meromorphic solutions Suppose first that the coefficients in are small functions with respect to in the usual Nevanlinna theory sense. Hence, in particular, this covers the case of rational solutions, if is transcendental. Denoting of functions F by we now consider the equation (31.1) at the of the derivative that is at the points We see that any of of the derivative is a zero of the function

Thus we see that investigating the value distribution of solutions of (31.1) is connected with studies of a new type of value distribution of meromorphic

59 functions, where instead of the ordinary we should study solutions of the equations

(i.e., solutions of

Of course, this is connected with the small functions topic in value distribution theory, directed to studying solutions of where is a small function in the sense of Indeed, we supposed above that the coefficients in (31.1) are small functions with respect to This means that is just a general form of The Nevanlinna second fundamental theorem has been generalized to this small functions case, see [65] by considering instead Observe now that if we consider this kind value distribution for a given meromorphic function then denoting by the number of solutions of (31.3) in we expect that in the general case. i.e. for the majority of values the quantities should be much smaller than or However, if is a solution of (31.1) then for the majority of values according to the Nevanlinna theory, the quantities should be close to the quantities or at least in some sense. Therefore, it is natural to pose the following Problem 31. To study the value distribution of the solutions of (31.3) and their interplay with the solutions of complex differential equations (31.1). Of course, all this may also be rephrased for multi-valued functions as well. Field 32: Gamma-lines of polynomials and the proximity property of of polynomials It is well known that the first main theorem in the value distribution theory of meromorphic functions in the complex plane is, qualitatively, quite similar to the main theorem of algebra. Indeed, the Nevanlinna theory states that the quantities and are close to the characteristic function for the majority of complex values and and for most of the values On the other hand, by the main theorem of algebra for polynomials of degree the quantities and are equal to for all large enough. Thus, the Nevanlinna characteristic function and the degree play a similar role for transcendental meromorphic functions and for polynomials, respectively. Hence, value distribution theory is well established (even in a more comprehensive way) in the polynomial case. But concerning the mutual locations of (the proximity property) and the Gamma-lines of polynomials, it seems we have a somewhat curious situation presently: These subjects have been studied more extensively

60 for meromorphic functions than for polynomials. Therefore, we invite the reader to study Problem 32. To study the closeness property of for polynomials.

and Gamma-lines

For the convenience of the reader, we shows how some methods developed for meromorphic functions may can be easily applied in the case of polynomials. In fact, we shall shortly consider Gamma-lines of polynomials and a problem by Erdös-Herzog-Piranian. Namely, Erdös, Herzog and Piranian (1958) proposed some problems [41] related to the length of the set for monic polynomials In particular,they conjectured that In the light of Gammalines, the set is a special case of of Denoting by the length of a of and by we may rewrite the above conjecture in the form

The first essential progress was made by Pommerenke in 1961 [63] who showed that while Borwein proved in 1995 [36] that

Thus, Borwein gave the correct rate of growth in this long standing open problem. The best estimate up to now for the constant in (32.2) has been obtained by Eremenko and Hayman in [42] who showed that the constant is at most 9.173. We now illustrate how the technique developed to studying Gammalines of meromorphic functions can be applied in the polynomial case. Instead of we proceed to consider much larger classes of smooth Jordan curves lying in the closure of the unit disk. We only suppose that the absolute integral curvature of is finite. 6 We now use the notation only for this type of curves and show that for these curves the following result is true: Theorem 32.1. For any curve

and any polynomial P,

where Remark. Our method applied carefully in the particular case of gives in (32.2) a better constant than obtained in [36], but not as good as that one obtained in [42]. 6

This means that point

where

denotes the curvature of

at the

61

Our proof for Theorem 32.1 follows almost immediately from the following inequality in [7], see also p. 20 in [16]: For any meromorphic function in the closure of a domain D with the piecewise smooth boundary and for any curve it is true that

where is the length of a of in D, and is the length of Now, it is known that for a monic polynomial P of degree the set is contained in the union of some 7 disks the sum of whose radii is Note that all our Gamma-lines lie in the union of similar disks taken for M = 1. With this M, the union consists, clearly, of some closed non-overlapping domains with total length of the boundary Applying (32.4) to P in these domains we have

where and where are the zeros of the derivative so that taking into account that the double integral here is we obtain (32.3). We may apply a similar approach, see [25], to study the length of and, more generally, to study Gamma-lines, provided is an algebraic function. Appendix: Characteristic sets and deficiencies describing geometric locations of of meromorphic functions In this appendix, a new characteristic (characteristic set) for meromorphic functions will be introduced that describes the location of in a similar way as the Nevanlinna characteristic function describes their quantity. Qualitatively speaking, we show that geometric locations of for almost all values can be described if we know the locations of for at least one “good” value Introduction. A substantial part of the research program described above deals with the geometric theory of meromorphic functions, describing, in particular, geometric locations of of an arbitrary meromorphic 7

This lemma is due to Cartan, see [55], p. 19; Pommerenke [62] improved the constant 2e to 2.59.

62

function in the complex plane. We first offer a short survey of this theory in the present appendix, for the convenience of the reader. These recent results include the proximity property of the principle of partitioning and the comparability property of for meromorphic functions. We suppose that the reader is familiar with basic concepts and notations of these theories. By the main conclusions of the Nevanlinna theory, a clear majority of values are “good” values for a function in the sense that is not deficient in the Nevanlinna, resp. Valiron meaning. For any “good” value it is true that as Thus, if and are both “good” values we immediately observe that a certain closeness between the number of and holds since as During the last two decades, some new properties were discovered such as the proximity property, the principle of partitioning, and the comparability property, see [8], [9], [12]–[15], [31] and [32], which describe geometric locations of of arbitrary meromorphic functions in the complex plane and its subdomains. Moreover, these geometric results contain the main conclusions of the classical value distribution theory. Accumulators of and the proximity property of We first consider these general properties from a new point of view of characteristic sets that describes geometric locations of in the disks We show that for any outside of an exceptional set of finite logarithmic measure, there are finitely many non-overlapping domains as satisfying the following properties: (a)for any collection of pairwise different complex values each contains exactly one simple for all indices except perhaps for exceptional indices such that (b) for these exceptional indices, it is true that

(c) Moreover, the diameters of the domains are small in the sense that for an arbitrary monotone increasing function as we have

The domains satisfying the above properties we are called proper small univalent accumulators.

63 Comments to the above definitions. Denoting by the number of simple in we observe that

Therefore, taking into account the limiting property immediately deduce the following inequality:

we

This improves the main conclusion of the Ahlfors theory of covering surfaces

describing the number of simple of in see [2] and [58], Ch. 13. Thus, the domains accumulate, in some sense, simple and they may be called proper since they accumulate not less that simple The main conclusion of the value distribution theory ensures that there is the same amount of simple in the disk Hence, we may now state the following result reflecting the proximity property of of meromorphic functions: Theorem A.1. For an arbitrary meromorphic function in the complex plane, for an arbitrary monotone increasing function as and for an arbitrary outside of a possible exceptional set of finite logarithmic measure, there exist a collection of approximately many proper, small, univalent accumulators in Remark A.1. The proximity property of from Theorem A.1: Due to the small diameter lying in each of the domains other. In fact, if we denote the resp. resp. then we have

immediately follows all simple should be close to each in by

Remark A.2. The main inequality (A.4) of the Ahlfors theory also follows from Theorem A.1. This, in turn, means that most of the simple

64

of

are involved in the closeness property, reflected

by Theorem A.1. Denoting by the number of those indices for which the domains don’t involve an we may state Theorem A.1 as follows: For an arbitrary finite collection all accumulators except perhaps many exceptions contain exactly one simple and

Due to (a), this means that there are at most accumulators which don’t contain a simple Qualitatively, this means that each of the accumulators involves, in the average, not less than q – 4 (and clearly not more that q) simple and that all these points are close to each other, due to the smallness of Hence, simple of mainly lie in the accumulators This allows us to consider an extended complex value as a deficient value with deficiency if the number of simple lying in all domains is less than so that we have

Then it follows from Theorem A.1 that we have Theorem A.2. (Deficiency relation for geometric locations of For any value where is at most a countable set,

For exceptional values a such that

we have

This deficiency relation reveals a phenomenon, the particular appearances of which may be observed in several investigations of the locations of carried through during the last century, including the classical results related to Julia rays. For short surveys on these results, see [66] and [9].

Proximity property and the Littlewood property. We say, qualitatively, that a set is a Nevanlinna set for a meromorphic function

65

if for most extended complex values a majority of simple of is contained in G (clearly, we may have different meanings for the words ‘most’ and ‘majority’). Littlewood conjectured the following marvelous property [57], proved by Lewis and Wu, see [56]: For any entire function of order there exist a Nevanlinna set G in the complex plane which is comparatively small so that

where S(X) denotes the area of a set We say that a Nevanlinna set G satisfies the Littlewood property is (A.8) holds. Theorem A.1 now shows that for most complex values a majority of simple of is contained in the accumulators Thus, their union is a Nevanlinna set. The next result then shows that the ratio

plays a crucial role in the theory of the locations of used the notation

Here we have

In terms of we may precisely determine whether a given meromorphic function has the Littlewood property. Moreover, in terms of we may also find a better estimate for the diameters of the accumulators Theorem A.3. For an arbitrary meromorphic function in the complex plane, for an arbitrary monotone increasing function as and for an arbitrary outside of a possible exceptional set of finite logarithmic measure, there exists a collection of approximately many proper, univalent accumulators in For the diameters of it is true that

It is now easy to show that Theorem A.3 implies Theorem A.1. Indeed, the inequality (A.2) follows from (A.9). This means that the accumulators in Theorem A.3 must be small. Suppose now that we have

66

Then, from (A.9) and (A.10) and taking into account that we readily obtain

Recalling that may have as small growth as we please, we may choose appropriately to obtain

Therefore, Theorem A.3 immediately implies Theorem A.4. Any meromorphic function the Littlewood property.

satisfying (A.10) possesses

Remark A.3. Theorem A.3 for meromorphic functions satisfying (A.10) deals, in fact, with an improved version of the Littlewood property, since in addition to this property (treating the area of the accumulators Theorem A.3 also offers information about the geometry of these accumulators by describing their numbers, diameters and univalence. Remark A.4. It is easy to show that

On the other hand, due to the inequality (A.2), we have

for an arbitrary meromorphic function. Improving this last estimate slightly results in the Littlewood property. In particular, such an improvement is possible, if is of infinite lower order This implies Theorem A. 5. Any meromorphic function sesses the Littlewood property.

of infinite lower order pos-

Of course, Theorem A.5 offers an addition to the result by Lewis and Wu [56] cited above. Conjecture. Any composite function with transcendental meromorphic and entire transcendental possesses the Littlewood property. The following result offers useful additional information to Theorem A.1 by considering distances between simple in The result

67

below reflects the comparability property, stating that for arbitrary distinct points, the distances and are comparable. Concerning this concept, we say that two quantities X, Y are comparable by denoted this by if the double inequality holds. Recalling Theorem A.1 and the subsequent Remark A.1, we have Theorem A.6. For arbitrary distances it is true that

and

Since (A.11) holds for arbitrary

we obtain the

Comparability property for distances: For arbitrary distances and between of a meromorphic function, the double inequality

holds. We also get, from (A.11) and (A.12), the following Comparability property for derivatives: For arbitrary distances and the double inequality

holds. For a detailed description of the above properties and their connections to other problems, see [12], [15], [31], [32]. From Theorem A.1 and Theorem A.6, it is not difficult to derive a Picard type result for locations of This result shows that there is at most four exceptional values for which the geometric behavior of is essentially different than the geometric behavior of of the normal values In fact, by (A.1), we conclude that for six arbitrary distinct values approximately accumulators contain not less that simple Taking into account that each of domains can involve not more than one simple we conclude that some of accumulators should involve at least two simple with different indices Consequently, for these two simple (A.11) must hold. Thus, denoting by the set of all simple of lying in we come to the following five point theorem:

68

Theorem A.7. For an arbitrary meromorphic function five pairwise different complex values at most, say

there may exist such that

for all indices Remark A.4. Observe that this result has to be interpreted as follows. We should not consider as exceptional all five values above but only four of them. In fact, one of these values, say may well be a normal value and Theorem A.7 just means that for only, the geometry of may essentially differ from the geometry of The sharpness of Theorem A.7 is immediate: The Weierstraß double periodic function has no simple for four values, say Therefore, there is nothing to discuss by taking any five values with the above However, if we now add a sixth value then for the simple with moduli large enough, we can indicate a simple such that

Characteristic sets of locations of Due to Remark A.4, we observe that to get a comprehensive knowledge of the geometric locations of it suffices to describe (a) the geometric locations and (b) the diameters of the accumulators In turn, to describe the locations of it is enough to indicate the location of one point only, say in each of In fact, by (A.2), the diameters are small. Moreover, to describe the diameters it is enough to find real positive values, say such that Thus, we arrive at the following main problem for the locations of To find a characteristic set of pairs that describes (in the above meaning) the locations and diameters of the accumulators After having found Theorem A.1 then offers a detailed knowledge of the locations of simple for an arbitrary collection of values Clearly, the characteristic set is not unique, as may be seen by distorting slightly, or adding/removing some pairs However, complete determination of for a given meromorphic function can appear to be a difficult problem, as compared with determining the

69

usual characteristic function in the Nevanlinna theory. Therefore, it may be useful, at least, to find some characteristic subsets where and determine the locations and diameters of the corresponding accumulators The following result gives an approximate idea about how to proceed: Theorem A.8. For arbitrary distinct values and an arbitrary the set of all simple of in contains a subset such that the set is a characteristic subset and for Remark A.5. Observe that if q is large enough and is small enough, then can be as close to 1 as we please. This means that for such q and the set of pairs in Theorem A.8 involves almost all possible characteristic pairs, as the total number of all characteristic pairs equal to and the quantity are close in this case. In a parallel way as in the classical Nevanlinna value distribution theory, the main result of the theory of geometric locations of of meromorphic functions implies, essentially, that whenever we know locations of and the quantities for at least one “good” value then we may describe geometric locations of an arbitrary collection of To obtain such a result, we need to show that in the collection of characteristic pairs the set of essentially coincides with the and the set with the quantities as soon as is a “good” value for Considering geometric locations, we define “good” values using the Ahlfors theory of covering surfaces. To this end, let D be a domain on the Riemann sphere. For simplicity, we may assume D to be a small disk centered at The multiplicity of an island, i.e. a connected part of that lies over D, is the number of its sheets, and the order of an island is its multiplicity minus 1. The islands of multiplicity one are called simple islands. Recall that a simple island of over D is a univalent domain of this surface whose projection onto the plane coincides with D. Denote by the simple islands of lying over D. Clearly, is then a one-to-one mapping in This means that has a “good” behavior in a neighborhood of each point and function has a “good” behavior in neighborhood of each point Obviously, the domain is one of the simple islands We now give the following Definition A.1. For a given we say that is a “good” for if Of course, we call the collection of all such as the set of “good”

70 Now, it is clear that we can consider a value as a “good” value of if admits a large number of “good” so that the quantities are small. 8 We may now state the following theorem, which is, in some sense, the main lemma here: Theorem A.9. For every D, of the set

and of “good”

we may choose a subset such that the set

is a characteristic subset and that Theorem A.9 plays a crucial role in determining characteristic subsets and therefore in the whole theory of locations of for the following reason: For an arbitrary and an arbitrary collection we may take some of their neighborhoods with non-intersecting boundaries to apply the second fundamental theorem in the Ahlfors theory for these domains. Then we conclude that at least for one of these domains, say for the inequality holds for is a constant depending on Therefore, due to where Theorem A.9, we can construct a characteristic subset

such that

Finally, arguing as in Remark A.5, we obtain that whenever the number q is large enough and is small enough, small then is close to 1. Hence, such a characteristic subset “almost” exhausts a “complete” characteristic set This means that when the number q is large enough, we can always choose a “good” value, say in a given collection of After having corresponding found the corresponding “good” we may can describe the locations of the accumulators. On the other hand, since we may take q as large as we please, we may add new domains while having large enough. Choosing then domains to satisfy we may “exhaust” as completely as we please. Respectively, having the corresponding values we can determine the closeness between in as completely as we please by the inequalities (A.11), (A.12)). 8

Observe that by the Ahlfors theory, there are always domains D with

and that for large

71

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ON LEVEL SETS OF QUASICONFORMAL MAPPINGS

G. A. SUKIASYAN Institute of Matematics of National Academy of Sciences of Armenia 24-b Marshal Bagramian ave., Yerevan 375019, Armenia, [email protected] Abstract. In the present article some analogs and generalizations of the tangent variation principle are given for quasiconformal and continuously differentiable mappings. Mathematics Subject Classification 2000: 30C70, 26B15. Key words and phrases: quasiconformal mappings, tangent variation principle, level sets

1. Introduction and statements of the results Let the

be a complex function in a domain and be a curve in The pre-images are called of the function see [6]. Denote by the total length of the preimages (that is of Gamma-lines) for a given mapping and a curve One of the most general results in the geometric theory of analytic functions is the length-area principle of L. Ahlfors [1]. This principle has numerous applications in the theory of univalent functions, circumstantially or areally mean functions in the unit disk, quasiconformal mappings (see, e.g., the books by W. Hayman [8] and by J. Lelong-Ferrand [9]). The principle gives above bounds for integrals of where is the circumference G.A. Barsegian revealed in [3], [4] a tangent variation principle that permits to give upper bounds of for an arbitrary meromorphic function in D and for large classes of curves Moreover, it turned out that for meromorphic functions in the complex plane, main results related to Gamma-lines are analogous to main results in the Nevanlinna theory and

© 2004 Kluwer Academic Publishers

76

that there is an analogous deficiency relation for Gamma-lines. Moreover, these results lead to a new proximity property that describes the geometry of of meromorphic functions instead of their cardinality in the Nevanlinna theory. These results also have other applications, say, in problems related to Gelfond quantities, asymptotic behavior, representations of Riemann surfaces, etc. Following the results in [3], [4], W. Hayman and J.M.G. Wu [8] studied Gamma-lines for the class of univalent functions and particular classes of namely for straight lines and circles. These results have been continued by many successors, see [6] for more details. Generalizing the theory of Gamma-lines for quasiconformal mappings is a natural problem. G.A. Barsegian, see [3]–[6], first defined the classes of angular-quasiconformal mappings for which the main results of Theory of are extendable. In the present article, some analogs and generalizations of the tangent variation principle will be established for quasiconformal mappings in a usual sense and moreover for continuously differentiable mappings. Denote by resp. the class of continuously differentiable, resp. twice continuously differentiable, complex-valued mappings defined in a domain D of the complex plane with positive Jacobian at each point of D. Let be the Jacobian of a mapping at and be the formal derivatives (in these notations Suppose that In the theory of quasiconformal mappings, the magnitude

called the characteristic of the mapping at plays an important role. It has a simple geometrical meaning: any infinitely small circle with the center at is mapped by onto an infinitely small ellipse for which is the ratio of the largest semi-axis to the smallest one. The following classical definition of Q-quasiconformality given by Grötzsch is very convenient to establish various relations. Definition 1 A mapping the inequality

for which at each point

holds, where Q is a finite number, is called Q-quasiconformal in D. Obviously, a 1-quasiconformal mapping is conformal. In the subsequent theorems, is a bounded domain with a rectifiable boundary and is the length of the boundary of D. As a charac-

77 teristic of the curvature of

we consider the magnitude

where is the angle between the tangent to at and the real axis. The following result is due to G.A. Barsegian, see [3]–[6]. Tangent Variation Principle. For any meromorphic function D and for any smooth Jordan curve with the inequality

holds, where and only and is the area element.

in

are constants depending on

Tangent Variation Principle permits us to estimate the lengths for different classes of functions and curve The last integral frequently encountered in the theory of complex functions, in particular, for is a characteristic for some known quantities and, in its turn, may be estimated sufficiently effectively in terms of the classical 1 characteristics and The next theorem is a generalization of the Principle for quasiconformal mappings. Theorem 1 For any Q-quasiconformal mapping any smooth Jordan curve with we have

where

and

are constants depending on

and for

and Q only.

Taking into account that the mapping by a meromorphic function is conformal, except for multiple points and poles not affecting on the quantities of the inequality (1.2), we obtain the Tangent Variation Principle by 1

Definitions and results of the value distribution theory are assumed to be known, see [10].

78 Theorem 1. Therefore, this theorem may be interpreted as Tangent Variation Principle for quasiconformal mappings. In the above theorems, we only require that for the curve the magnitude is bounded. It should be expected that any additional information about (for instance, the analytical aspect of the curve) should somehow be reflected in the eventual results. Moreover, the use of methods of differential geometry in the proofs allows us to improve the estimates considerably in some cases. This idea is accomplished in the next theorems. Definition 2 The set the equation

of all points

of an open set

satisfying

where F is a smooth function on U, is called a smooth (or regular) curve in U, if grad at each point Theorem 2 Let be a smooth curve (in the sense of the above definition) with a twice continuously differentiable function F in and Then for any mapping we have

where is the characteristic of the mapping Furthermore, if is Q-quasiconformal then

at

By an example, we shall show advantages of the estimates (1.3) and (1.4). For simplicity, consider a function meromorphic in D and an ellipse in the with the focuses at and and with the eccentricity e. Such an ellipse can be given by the equation

79 Having done the appropriate calculations (see Section 2 below), we get

For a circle

we now get

The inequality (1.6) compared with the inequality (1.1) is more sharp for a circle This may be easily seen by looking at the function Theorem 3 Suppose that is a smooth Jordan curve and F be a twice continuously differentiable function in such that grad for any except possibly at isolated points. Then for any mapping we have

80 For a Q-quasiconformal mapping

We next consider the special case of a straight line. Given a straight line by the equation then from (1.7) we infer that

and, respectively, for meromorphic functions

We may also consider curves consisting of several separate parts, which possibility is not included in the Principle. For instance, for a equilateral

81

hyperbola

as in the previous case, we get

and, respectively, for meromorphic functions

For Q-quasiconformal mappings in the inequalities (1.9) and (1.10) the magnitude can be replaced by Q. Remark 1 In Theorem 2 and Theorem 3 above, the existence of isolated singular points (for instance, points at which is not differentiable or grad is admitted, by the fact that they do not affect on the quantities of the corresponding inequalities. Remark 2 If the domain of definition of is an open set everywhere in the inequalities above, D is to be replaced by

then

2. Proofs

We first prove a lemma which shows that the magnitude is a measure of angular distortion under continuously differentiable mappings. Lemma. that

Let

and be an arbitrary point of D. Suppose are smooth curves intersecting at the point and let

82 be the resp. and resp. and the real axis.

2

of the curves and respectively. Denote by and the angles between the tangents to the curves and at the point resp. and the positive direction of Then the inequality

holds. Moreover, if

and if

or

or

then

then

Proof. In a sufficiently small neighborhood of the point the mapping can be replaced, approximately, by the linear transformation of the plane

where

and

Let then be smooth curves, intersecting at the point represent these curves in the form

We

Since then by means of the transformation (2.4) the straight lines (2.5) correspond to

2

The direction of the tangent to a curve at a point and, respectively, the angle between the tangent and the positive direction of the real axis is determined in accordance with the direction of the curve

83 where, as

for The directions of the straight lines (2.6) determined by the vectors correspond to the directions of the straight lines (2.5) determined by the vectors Note that the transformation (2.4), i.e. the differential of the mapping at with the condition carries with regard to directions the tangents to the curves and at onto the tangents to the curves and (the of and at and and are the angles between these tangents and the positive direction of the real axis. We now compute

and where Taking into account that

we get from (2.7) that

If we take

or

in (2.7), then

84

whence taking into account that

then

The inequality (2.2) now follows from (2.9) and (2.10). The inequality (2.3) can be proved in the same way as (2.2). Remark 3 For Q-quasiconformal mappings the magnitude replaced by Q in the inequalities (2.1)–(2.3).

can be

Proof of Theorem 1. We make use of the Barsegian method to prove the Tangent Variation Principle. Let be a smooth Jordan curve. Let us represent the lines as the union Here is the totality of the arcs from in each point of which the smaller angle between the tangent and the real axis is less or equal to Similarly, is the totality of the arcs from in each point of which the smaller angle between the tangent and the imaginary axis is less than If we denote by and the total lengths of the arcs and respectively, then

Let

be the number of points be the smaller angle between the tangent to the arc from at the point and the real axis (see Fig. 1). By virtue of the definition of we have and where is the length element of an arc from By using these notations we get

where

be the set

and

85

Similarly, if we denote by number of points the tangent to the arc from axis, then we get

where Since we have

and and

the set by the by the smaller angle between at the point and the imaginary

then by virtue of (2.11)–(2.13)

For a given we denote by the totality of intervals making up by the interval lying between the points and and by the number of points We first prove the theorem for a curve with arcsin Let be a straight line joining the points and Denote by resp. the smaller angle between the tangent to the curves and at the point resp. by resp. the smaller angle between the tangent to the curve at the

86

point

resp. and the line and by resp. the smaller angle between to the tangent to the curve at the point resp. and the line (see Fig. 2a and Fig. 2b).

Applying the inequality (2.1) of Lemma with have

replaced by Q, we

Obviously there is a point such that the tangent to the curve at the point is parallel to the line and the angle between the tangent and the real axis is equal to arg Therefore,

Note that either see Fig. 2a, or see Fig. 2b. The same is true for Consequently, in any case,

and

87 and by applying (2.15) and (2.16), we obtain

Summing up over all results in

and

which is true in the case of

as well. Consequently,

Moreover, from geometric constructions, it follows that

88 where we have

is the length of the boundary of D. Thus, from (2.17) and (2.18),

and similarly

The last two inequalities, together with (2.14), imply the following inequality

which is the assertion of Theorem 1 in the case We now consider the case arcsin Beginning from an end of we successively mark on it disjoint, arcs with Either we completely exhaust the curve by a finite number arcs in a such way or an arc remains with In both cases the number of the arcs is estimated by where is the integral part of Applying the inequality (2.19) for each arc taking into account that we get the inequality (1.2) with

So, Theorem 1 is completely proved. Proof of Theorem 2. Under the notations of the previous proof, we consider the function on the interval Recall that this is the interval lying between the points and As then by the Rolle theorem there is a point not necessarily the same as in Theorem 1, such that

89

tors

and

Here means the scalar product of vecSo, applying Lemma we get

Further, exactly as in the previous proof, the inequalities

resp.

follow. From (2.14) and the last two inequalities (1.3) then follows. Replacing everywhere by Q, we obtain (1.4), proving Theorem 2. Consider now the ellipse

Except for the focuses, the function

90 is twice continuously differentiable in We exclude the pre-images of the focuses from the domain of definition. It is easy to show that

Compute next the scalar product

Substituting these relations into (1.3), we have

91 Taking into account that for any meromorphic function and (2.21) implies that

If is Q-quasiconformal then we need to apply the inequality (1.4), replacing by Q everywhere in (2.21). Proof of Theorem 3. It is pertinent to mention again that on isolated singularity of the function F does not affect on the proof of Theorems 2 and 3. We can first exclude the pre-images of these points from the domain of definition and then carry out the proof for a remaining part. Hence, the inequality (2.20) can be replaced by the more simple inequality

To complete the proof, it suffices to repeat the proof of Theorem 2, word by word. Acknowledgements The author thanks G.A. Barsegian for valuable discussions of the results.

92

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Ahlfors L., Untersuchungen zur Theorie der konformen Abbildungen und der ganzen Funktionen, Acta Soc. Sci. Fenn. 1, n. 9 (1930), 1–40. Ahlfors L., Zur Theorie der Uberlagerungsflachen, Acta Math. 65 (1935), 157–194. Barsegian G.A., New results in the theory of meromorphic functions, Dokl. Acad. Nauk SSSR 238 (1978), no. 4, 777–780. (in Russian, translated in Soviet Math. Dokl.) Barsegian G.A., Geometry of meromorphic functions, Mat. Sb. (N.S.) 114(156) (1981), n. 2, 179–225, 335. (in Russian, translated in Math. USSR Sbornic) Barsegian G.A., The tangent variation principle in complex analysis, Izv. Nats. Akad. Nauk Armenii Mat. 27 (1992), n. 3, 39–65. (in Russian, translated in J. Contemp. Math. Anal. 27 (1992), no. 3, 34–56) Barsegian G.A., Gamma-lines: on the geometry of real and complex functions, Taylor and Francis, London, New York, 2002. Hayman W., Multivalent functions, Cambridge University Press, Cambridge, 1958. Hayman W. and Wu J.M.G., Level sets of univalent functions, Comment. Math. Helv. 56 (1981), 366–403. Lelong-Ferrand J., Représentation conforme et transformations à intégrale de Dirichlet bornée, Gauthier-Villars, Paris, 1955. Nevanlinna R., Eindeutige analitische Funktionen, Springer, Berlin, 1936.

ON THE UNINTEGRATED NEVANLINNA FUNDAMENTAL INEQUALITY FOR MEROMORPHIC FUNCTIONS OF SLOW GROWTH

ANGEL ALONSO E.U.I.T.Agrícolas. Ciudad Universitaria, 28040 Madrid, Spain [email protected] ARTURO FERNÁNDEZ Facultad de Ciencias, U.N.E.D. Ciudad Universitaria, 28040 Madrid, Spain [email protected] AND JAVIER PÉREZ Facultad de Ciencias, U.N.E.D. Ciudad Universitaria, 28040 Madrid, Spain

Abstract. J. Miles proved that for a meromorphic function and values the inequality holds for some constant K, for all large in a set of positive lower logarithmic density. This inequality is in some sense stronger than the unintegrated Nevanlinna fundamental inequality However, it remains the question about the size of the constant K. In this work, the above mentioned inequality will be considered for functions of slow and regular growth, observing that in this case, which is a natural extension of the rational functions class, the constant K can be considerably reduced in relation with the numerical values suggested by Miles. We make use of a result of Barsegian which follows from some beautiful considerations around the main theorems of the Ahlfors theory of covering surfaces. Mathematics Subject Classification 2000: 30F10. Key words and phrases: Meromorphic function, value distribution, characteristic function.

© 2004 Kluwer Academic Publishers

94 1. Introduction

If is a nonconstant meromorphic function in and are distinct elements of the Riemann sphere, then the fundamental inequality in the Nevanlinna second main theorem states that

with and measure, so that we also obtain

outside an exceptional set of finite

Taking into account that

by the first main theorem, we conclude from (1) and (2) after division by

or

The functions

can be defined by

so that

Therefore N and T, the usual Nevanlinna functions, can be considered as the integrated magnitudes of and A, the simple counting function and the mean covering number, respectively. Conversely, and A are the differentiated N and T. Ahlfors [1], see also Nevanlinna [11], p. 349, proved the unintegrated version of (3) i.e.

95

and Miles [9] obtained the boundedness by an absolute constant of the larger magnitude

in a set of positive logarithmic measure. More precisely, he proved

when E is a set of positive logarithmic measure. On the one hand (5) is stronger than (4) in the sense that it yields an estimate for the sum of absolute values. On the other hand, (5) is valid in a smaller set, namely a set of positive logarithmic measure, whereas (4) is valid for all outside a set of finite measure. In this paper we shall pay attention to the Miles inequality, obtaining a quantitative estimate for K for slowly growing functions, improving notably the Miles estimate for such functions. 2. Preliminary facts and remarks

The following lemma is due to Miles [9], who used previous ideas of W. Fuchs. The lemma plays a central role in the Miles inequality (5) and in all further considerations in this work. Lemma A There exist an absolute constant and such that if is a nonconstant meromorphic function in C, then there exists a set with lower logarithmic density at least C such that

for all sufficiently large The constant obtained by Miles is Miles makes use of Lemma A to prove (5), obtaining for the constant K the estimate

However, G. Barsegian [3] makes use of original and powerful ideas of strong geometric nature to prove the following Theorem A The constant K in (5) satisfies

96

where is the same constant as in (6), and previously given positive number.

is an arbitrarily small

From Theorem A, it becomes clear the interest to obtain precise estimates for the constant of Lemma A in order to improve the estimates of K in the Miles inequality. 3. Statement of results

Next we present some results to improve the estimates given by Miles for the constant while considering functions of slow growth. Theorem 1

where

For

rational, it holds for all sufficiently large

is an arbitrarily small previously given positive number.

Theorem 1 yields in Lemma A for rational. We recall that Miles gives in this case. Rational functions are characterized by the condition of slow regular growth: We also present some improvement for the estimate of subject to a weaker condition: Theorem 2 For a meromorphic function

satisfying

we can take Further, for meromorphic functions satisfying

we can improve (10) to

for functions

97 4. Proof of Theorem 1 To prove Theorem 1, we recall the expression for

and let

So, we may write

where rational functions. We also have

as

where is a polynomial of degree strictly less than polynomial of degree strictly less than and therefore are rational functions tending to zero as From (13), (14) and (15), we obtain

Here we have

and

are

is a

98 clearly tends also uniformly to zero as From (17), we obtain

where again uniformly as From (16) and (18) we conclude

with

uniformly as

From (19), we obtain

To evaluate the left hand side of (8), we again make use of (14) and (15) to obtain

Arguing with

so that

instead of

we get instead of (21)

99

Finally, (20) and (23) yield the assertion

5. Proof of Theorem 2

We shall follow the steps of the Miles proof of Lemma A and try to improve his estimates for in our particular case of functions of slow growth. We shall set for Then we shall consider, similarly to Miles, the differentiated Poisson-Jensen formula applied to We obtain for see Hayman [6], p. 22,

First of all, we shall consider the integral on the right hand side of (24) following the ideas in [6], p. 23. We have

Therefore, the integral can be estimated as follows:

since On the other hand, we have

and so

100

Therefore, summing up for all poles

in

we get

and so the right hand side tends to zero as A similar result holds for the term in (24) corresponding to the zeros Thus, we conclude from (24), (25) and (26) that

where

The convergence is uniform on bounded sets containing neither a zero nor a pole of Therefore, after multiplying by we obtain

Taking into account

we obtain from (28)

Now, by the standard estimate

and making use of the estimate in the logarithmic derivative lemma obtained by S. Lang, see [8] p. 48,

101

we conclude from (30), (31) and (32) that

Next, we shall apply, following Miles, the following growth lemma due to Hayman: Lemma. Let and

holds for all

and

be positive and nondecreasing for for Then

in a set having lower density

Noting that we may apply this lemma to Then it follows from the lemma that

on a set of values of having lower density at least Introducing the change of variable we get

so that (34) yields

on a set of values of logarithmic density at least From (33) and (35), we conclude

on a set of If satisfies for some C,

of lower logarithmic density at least

then

102

whence and since we can take and therefore also wish, we get from (36) and (37),

so close to 1 as we

on a set of r-values of lower logarithmic density at least we have proved (10). Now, to prove (12) we assume

that is,

so that

In this case, we can slightly refine the estimate (34) in Hayman’s Lemma. In fact, setting again we get in this case

Therefore and

so that

Thus

Together with (33), this yields

Again, since

with

arbitrarily small, recalling

103

so that

we conclude that

From (39) and (40) we obtain (12). We remark that for small (12) yields a perceptible improvement with respect to (10). For instance for we can take

instead of 6. Example

A well known example of entire functions of regular growth are the so-called Lindelöf functions, see [12], p. 18, namely the infinite products

where We have the asymptotic expansion

so that

Thus, the Lindelöf functions are an example for which (10) is applicable, i.e. we can take Acknowledgements A. Fernández is partially supported by the Grant BFM2002-04801 and J. Pérez by the Grant BFM2002-00141.

104 References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Ahlfors L. V., Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157– 194. Barsegian G. A., A geometric approach to the problem of ramification of Riemann surfaces, Soviet Math. Dokl. 18, 6 (1977). Barsegian G. A., On the geometric structure of the image of a disk under mappings by meromorphic functions, Math. USSR Sbornik. 34,5 (1978). Barsegian G. A., Gamma-lines: on the geometry of real and complex functions, Taylor and Francis, London, New York, 2002. Hayman W. K., An inequality for real positive functions, Proc. Cambridge Philos. Soc. 48 (1952), 93–105. Hayman W. K., Meromorphic functions, Clarendon Press, Oxford, 1964. Lang S., The error term in Nevanlinna theory, Duke Math. J. 56 (1988), 193–218. Lang S. and Cherry W., Topics in Nevanlinna theory, Lectures Notes in Mathematics, 1433, Springer-Verlag, Berlin, 1990. Miles J., On the counting functions for the of a meromorphic function, Trans. Amer. Math. Soc. 147 (1970), 203–222. Miles J., Bounds on the ratio for meromorphic functions, Trans. Amer. Math. Soc. 162 (1971), 383–393. Nevanlinna R., Analytic functions, Springer-Verlag, New York-Berlin, 1970. Nevanlinna R., Le théorème de Picard-Borel et la théorie des fonctions méromorphes, Chelsea Publishing Co., New York, 1974.

ON SOME NEW CONCEPT OF EXCEPTIONAL VALUES

G. A. BARSEGIAN Institute of Mathematics of National Academy of Sciences of Armenia 24-b Marshal Bagramian ave., Yerevan 375019, Armenia, [email protected] AND

C.C. YANG Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China, [email protected]

Abstract. We introduce a concept of jumping for functions meromorphic in the complex plane. This concept generalizes the classical concept of multiple points. By making use of the new concept, we are able to generalize some main conclusions of the Nevanlinna value distribution theory related to multiple In particular, it turns out that not only these multiple are exceptional in the sense of deficiency but also those where is sufficiently small. Mathematics Subject Classification 2000: 30D30, 30D35. Key words and phrases: level sets, Gamma-lines, value distribution, multiple values.

1. Introduction and results In this paper, we introduce a concept of jumping for function meromorphic in the complex plane, generalizing the classical concept of multiple points. To this end, we apply the estimates of the derivatives of an arbitrary function meromorphic in the complex plane and its inverse function obtained in our previous articles [7], [8]. Indeed, making use this new concept, we are able to generalize the main conclusions of the Nevan-

© 2004 Kluwer Academic Publishers

106

linna theory related to multiple points. More precisely, we show that not only multiple are exceptional in the Nevanlinna theory but also those at which is small enough. We come to this conclusion by showing that the number of jumping is small in a similar sense as the number of multiple points of is small in the Nevanlinna theory. Thus, we come to a new type of exceptional including the classical concept of exceptional multiple Let be an of and let

be the Taylor coefficient of about Of course, for a multiple point, we have while the concept of a jumping to be defined below includes other phenomena such that is very small, or that some is very big while remains comparatively small. We also consider another type of generalization of the concept of multiple by looking, for arbitrary values and at the points for which simultaneously and Of course, for a usual multiple point, It appears that for with lower order greater than two, such points are exceptional as well. A similar phenomenon for inverse meromorphic derivatives has been introduced in [8]. Denote by the number those in of counting multiplicity, for which and denote, as in the Nevanlinna theory,

To define our new concept of jumping consider those in of function for which the Taylor series (1.1) at has at least one jumping coefficient in the sense that

with

These

will now be called jumping is also a jumping Denote by the number of jumping counting multiplicity, and

Clearly, any multiple in

107

Since multiple

are jumping points, we trivially get

Using the notation in (1.5), we may rewrite the main conclusion the Nevanlinna theory concerning multiple as follows: Given distinct complex values we have

From the inequality (1.6) it follows that the multiple points of are exceptional in the sense that there are at most countably many of them. The following result now generalizes (1.6) is at most countable as well: Theorem 1. Suppose is a meromorphic function in are distinct complex values. Then

and

From the proof of this theorem, it immediately follows an interesting result related to previous investigations in algebraic complex differential equations. Indeed, it was shown, see [2], [3], that any meromorphic solution of in the complex plane is of finite order, provided that where is the maximum of all weights taken for all monomials of the polynomial P. For some related developments, we refer the reader to [6], [5]. Corollary 1. Let be a meromorphic function in and let be distinct complex numbers, Suppose that where M, are finite constants for all in the collection all of in Then the order of is at most equal to Remark. In particular, if is bounded in the collection of all points, then Therefore, instead of Q, we may take or the ratio and the conclusion remains valid. We next consider two other types of exceptional values. We first show that for of lower order greater than two the points for which simultaneously and are exceptional as well. Secondly, we

108

show that a similar phenomenon occurs for a new type of meromorphic functions introduced in [8], associated with and its inverse function More specifically, for any point with we define to be the value of branches of the function for which and denote by the derivative of with respect to We call the composition of and inverse meromorphic derivatives, see [8]; they are, indeed, meromorphic functions with respect to variable This is easy to verify: If is an ordinary point, then, clearly, the function is single valued in a small neighborhood of On the other hand, if is a multiple point with multiplicity then can be represented as and has a representation of the form so that is single valued in a small neighborhood of and is a pole with multiplicity for Thus the composition of and is meromorphic and is just equal to Denote by the number of those in counting multiplicity, of our function for which As usual, denote

Similarly, we use the notations and correspondingly for of Theorem 2. Let be a meromorphic function of lower order R be an arbitrary positive constant. Then we have: (1) For any where D is a countable set,

instead and

and

(2) For any integer

and

and any

where

is a countable set,

109

The inequalities (1.8) and (1.9) are similar to the usual deficiency relation:

which shows that the number of is large for the majority of values of Theorem 2 shows that if we consider those only for which additional restrictions such as are valid, then the number of such is small in general. We now apply Theorem 2 into the Nevanlinna theory and into periodic functions. The following corollary from Theorem 2 generalizes the inequality (1.6), provided Corollary 2. Let be a meromorphic function of lower order be distinct complex numbers and for and be a set of distinct complex numbers. Then

and

This corollary immediately follows from the assertions (1.8) and (1.9) by taking into account that, for an R appropriately chosen,

and, for

110

Remark. The Nevanlinna inequality (1.6) follows from (1.10) by taking and Theorem 2 is not true with The case remains open. We next consider a periodic function with period from the value distribution theory point of view by examining the behavior of in the set of Then, for any we have and Note that most of well known periodic functions have the following property: Moduli of their derivatives are bounded in the set by a constant depending on only. We call such periodic functions simple. For instance, for or for the Weierstrass doubly periodic function their derivatives take a finite number of values only in for any value Consequently, they are simple periodic functions. We may also consider the concept of simple periodic values for an arbitrary meromorphic function By this we mean those values a for which with some complex constant whenever and With these definitions we deduce from Theorem 2 the following: Corollary 3. Let be a meromorphic function, be a set of distinct complex values and let be bounded by a constant M in the collection of of Then Consequently, any meromorphic function with five simple complex periodic values is of order and so is a simple periodic function. Remark. In particular, any doubly periodic function is of order 2. Proofs Remark. We first explain, qualitatively, how Theorem 1 can be derived from the results in [7], [8]. These results expand the main conclusions of the Nevanlinna value distribution theory and the Ahlfors theory of covering surfaces. By these classical theories, the majority of are simple for the majority of values Our results extend this conclusion by establishing several “good” properties for such simple In particular, we show that, in the general case, the Taylor expansion considered at these simple has no jumping coefficients. Together with the Nevanlinna– Ahlfors conclusion, this yields that for most of the Taylor expansion has no jumping coefficients. In other words, the number of admitting jumping coefficients is “small” and, consequently, jumping are exceptional. Before proceeding to explain this in detail, we recall the following

111 Lemma 1 ([8], Theorem 2). Let be a meromorphic function in be a monotone increasing function and be distinct values. Suppose U and are integers are constants and is the disk Then there exist pairwise disjoint simply connected domains in the disks where E is a set of finite logarithmic measure, such that the following assertions are valid: (1) The function is univalent in any of the domains Consequently, the inverse function is a one-to-one mapping in any of the domains considered on the covering surface and thus determines the branches of the inverse function for which (2) The collection of all simple in form an Ahlfors set of simple in the disks and the set is a totality of accumulations which, in turn, are totalities of close and simple of in In other words,

where and

is the number of all simple

of

in

where is the diameter of the set Consequently, all points of in are close. (3) For the number of the domains

(4) If, for a given then we have has a preimage with

the domain in Moreover,

involves an (hence any value and for an arbitrary value

112

and

(5) For any

Consequently, for any

(by (4), all these points (6) For any integer

and exist). and for any

with

where

and

(7) For any integer

and

and for any

with

113

Comments on Lemma 1. The inequality (2.1) improves the main conclusion of the Ahlfors theory related to simple The assertions (1) – (3) reflect the proximity property of of meromorphic functions, describing the locations of (in addition to the classical conclusions in the Nevanlinna and Ahlfors theories which describe the numbers of only). The inequality (2.2) generalizes previous similar estimates established in [1] for the case The assertion (2.4) clearly reflects the comparability of the related quantities. Remember that due to the reasons given in [7], the comparability as well as the estimates in (5) and (6) concern to the majority of The inequalities (2.5)– (2.7) generalize earlier similar estimates established in [4] for the case The inequalities (2.3), (2,8)-(2.10) seem to be new results. The sharpness of all conclusions in Lemma 1 can be shown by examining the Weierstrass doubly periodic function. To complete the proof of Theorem 1, we also need the following upper bounds for the spherical length of due to Miles [10] and [9]. Lemma 2. Let satisfying

If

be a meromorphic function in such that

Then there is a set

is a meromorphic function in the unit disk and

then there is a sequence of values

such that

Proof of Theorem 1. Recall first how the Ahlfors sets of simple were obtained, see [8], Theorem 2(2) or (2.1) in Lemma 1 above. As for the proof of (2.1), see the Generalized Theorem 2 in [7] and Section 3 of [7]. In fact, we there proved the following inequality which immediately leads to the inequality (2.1):

114

where is a monotone increasing function tending to infinity as The function depends on in Lemma 1 and it can be chosen to tend to infinity as slowly as we please. Consequently,

Then, as a result of Lemma 2, we have

where

It is known [11] that

and, for

transcendental,

Therefore, by the second fundamental theorem, (2.14) and (2.15), we obtain

as Due to (2.6) in Lemma 1, applied for and for the values we obtain that all in Theorem 1 are not jumping as soon as the moduli of these are greater than Consequently, for the number of jumping of

where means the number of simple fore, applying the inequality (2.17) for we get

Taking into account (2.6), we obtain the inequality

in

There-

115

from which Theorem 1 immediately follows. Proof of Theorem 2. We make use of the inequality (2.2) in Lemma 1, taking into account the fact that may tend to infinity as slowly as we please. We obtain, under the conditions in Theorem 2 that the the inequality holds for all points with where is a constant depending on and R. Therefore,

Consequently, taking into account (2.16) and (2.17), we obtain

Theorem 2(1) now immediately follows from (2.20), see also [4]. Theorem 2(2) can be proved similarly, if we use (2.8) in Lemma 1 instead of (2.2). 3. Functions in the unit disk If is meromorphic in the unit disk and satisfies (2.12), then Theorem 1 and Theorem 2 both admit corresponding counterparts. To see this, it is enough to note that by Lemma 2, the inequalities (2.18) and (2.20) remain true, if we replace by 4. Relation to the comparability property Both Theorem 1 and Theorem 2 make the use of in sets of of The comparability property, mentioned in Lemma 1, connects this quantity with the distances between and Thus, one can develop versions of these theorems by defining new notions of new exceptional values in terms of the distances.

Acknowledgements The second author’s research was partially supported by an RGC grant (project no. HKUST 6134/00P).

116

References l. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Barsegian G., Estimates of derivatives of meromorphic functions on sets of J. London Math. Soc. (2) 34 (1986), 534–540. Barsegian G., On meromorphic solutions of algebraic differential equations, Complex Analysis and Math. Physics, Proceedings, Krasnoyarsk, June, 1987. Barsegian G., On a method of study of algebraic differential equations, Bull. Hong Kong Math. Soc. 2 (1998), 159–164. Barsegian G.A, Estimates of higher derivatives of meromorphic functions and the multiple points in the second main theorem of R. Nevanlinna, Bull. Hong Kong Math. Soc. 2 (1999), 341–346. Barsegian, G., Laine, I. and Yang C. C., Stability phenomenon and problems for complex differential equations with relations to shared values, Mat. Stud. 13 (2000), 224–228. Barsegian G., Laine I. and Yang C. C., On a method of estimating derivatives in complex differential equations, J. Math. Soc. Japan 54 (2002), 923–935. Barsegian G.A. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications, I, On the magnitudes of functions on the sets of of derivatives, Complex Variables Theory Appl. 41 (2000), 293–313. Barsegian G.A. and Yang C.C., Some new generalizations in the theory of meromorphic functions and their applications, II, On the derivatives of meromorphic and inverse functions, Complex Variables Theory Appl. 44 (2001), 13–27. A remark on the Ahlfors theory of covering surfaces, Teor. Funkcional. Anal. i No. 20, (1974), 70–72, 174. (Russian) Miles J., A note on Ahlfors’ theory of covering surfaces, Proc. Amer. Math. Soc. 21 (1969), 30–32. Nevanlinna R., Eindeutige analytische Funktionen, Springer, 1936.

MAXIMUM MODULUS POINTS, DEVIATIONS AND SPREADS OF MEROMORPHIC FUNCTIONS

E. CIECHANOWICZ Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Poland, [email protected]

AND I.I. MARCHENKO Department of Mathematics, Kharkov State University, 310000 Kharkov, Ukraine Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Poland, [email protected]

Abstract. We consider the influence of the number of maximum modulus points over the spread and the magnitude of deviation of meromorphic functions. Mathematics Subject Classification 2000: 30D35; 30D30. Key words and phrases: meromorpic function, subharmonic function, spread, deviation

1. Introduction We shall use standard notations of value distribution theory such as see [5]. For a meromorphic function define quantity

we The

is called Petrenko’s magnitude of deviation of the meromorphic function at Petrenko obtained a sharp upper estimate of the magnitude of deviation of a meromorphic function of finite lower order, see [8].

© 2004 Kluwer Academic Publishers

118

Theorem A then for each

If

is a meromorphic function of finite lower order

Let be a meromorphic function and let be a positive nondecreasing convex function of for such that We shall denote by the number of component intervals of the set

possessing at least one maximum modulus point of the function over, let us denote We set:

Let now and

More-

be a positive nondecreasing continuous function such that We denote The quantity

is called the spread of the meromorphic function first introduced by Edrei [2]. Baernstein received a sharp lower estimate of spread in 1973, see [1]: Theorem B then

If

is a meromorphic function of finite lower order

In this paper, we obtain an upper estimate of in terms of Apart from that, we show upper estimates of in terms of and for meromorphic functions of finite lower order. Our main results are as follows: Theorem 1 have

For a meromorphic function

of finite lower order

we

119

Theorem 2 Then

Let

be a meromorphic function of finite lower order

Theorem 3 For every meromorphic function of finite lower order have the inequality

The above estimates in case when duced by one of us in 1995, see [7].

and

we

were intro-

2. Auxiliary results

Let again be a meromorphic function and decreasing convex function of such that consider the function

Lemma 1 Proof. Let that

The function

is a

function in

and be entire functions without common zeros such . Then it is easy to see that

The function is a convex function of is a subharmonic function in see [9]. Also

is a subharmonic function in

is a

be a positive nonWe first

function in

for

Thus

As in [1], define

Therefore

120 where E is a measurable set and is the Lebesgue measure of E. Now for each consider the set:

and let where is the symmetric rearrangement of the set see [6]. The function is non-negative and non-increasing in the interval even in and for each fixed equimeasurable with Moreover, it satisfies the relations:

where From Baernstein’s theorem in [1], the function monic in continuous in for each fixed

Let

When

is subhar-

and logarithmically convex in Moreover,

be a real-valued function of a real variable

and define

is twice differentiable in r, then

Lemma 2 For almost all and for all has neither zeros nor poles in

such that the function we have

121

Proof. Let us assume that is a number satisfying the hypothesis. Since is a non-increasing function of the derivative exists for almost all Let us choose such that exists. If then for all and so As is a convex function of we see that Therefore the lemma is proved in the case when or when Let us assume now that and By [1], there exists a set such that

Moreover

Let us now consider the function Then the set is finite. If it were not there would exist a convergent sequence such that As is chosen so that there are neither zeros nor poles of on the circle the function is an analytic function of for Applying the uniqueness theorem, we can state that if then for all This would mean that for all As a result which is a contradiction. Therefore the set is indeed finite. This, together with our assumption that leads us to the conclusion that also the set is finite. As a result,

where Let us now consider for

We have

Since the set that

the function, see [4]:

and

for all

is an open subset of the circle As

Hence

it implies it follows again

122

from the uniqueness theorem that the family of intervals is finite. Let denote the number of those intervals. The function is harmonic on a certain neighborhood of the circle as has neither zeros nor poles on this circle. Therefore

Finally, it follows from our previous considerations that

Following the same lines as in the proof of Lemma 1 in [7], we arrive at the following conclusion

By the definition, is the number of component intervals of the set possessing at least one maximum modulus point of On the other hand, is the number of component intervals of the set and Therefore, we have Also and so we finally receive

In order to proceed we need one more lemma: Lemma 3 [8] Let be a meromorphic function of lower order Then for each there exist sequences tending to infinity such that and that for all we have

123

3. The upper estimate of the magnitude of deviation

Proof of Theorem 1. If So let us assume that

then the theorem is obviously fulfilled. Then for every we have

Let us consider the case of

numbers

and

and

Now we choose

satisfying the inequalities

Moreover, as in [3], we put

Applying the Fatou lemma, we receive

It follows from this inequality that

is an increasing function in

where

is the left derivative of

is a convex function of

and so

Therefore, for almost all

at the point

From the inequal-

ity (3.1) and Lemma 1 it follows that for almost all

By definition,

takes integral values only. Thus for

there is

From this and from (3.2) it follows that for almost all

124

If there are neither zeros nor poles of on the circle for the function fulfills a Lipschitz condition in Therefore also fulfills a Lipschitz condition on see [6]. This implies that the function is absolutely continuous on Integrating twice by parts we receive

This way we obtain the inequality

Dividing this inequality by integrating it by parts over the intervals defined in Lemma 3 and then applying suitable estimates, we obtain that for all

Therefore, there exists such that the definition of we receive that there is a sequence for

Let us take

We have

From such that

125

As

and

we receive

Therefore

As and were chosen arbitrarily and the following results: If then for every there is and and therefore

As this is true for any

we obtain This means that

we infer that

If next

and

then there exists such a

that

and thus

for every

Therefore in this case,

If finally and then There exists such a that Therefore, for there is This means that for we have as Therefore, for every we have

as receive

Let us set

and

in (3.5). Hence for all

we

126

Thus

From this we can easily obtain the assertion in the case of The proof in the case of can be conducted similarly, see [7] for details. 4. The upper estimates of spread

Proof of Theorem 2. First assume that If the theorem is obviously true, so we also assume that Then for any we have Moreover, we assume that and are such that Choose then such that

Since

we have for we obtain by (3.5) that

as let

Let us now choose Then

as

As

such that for

there is

Moreover, as

there is

and

Then

127

Since

As

we obtain

by passing to the limit with

and

Hence

This way we receive

Next we let Hence for every

and and

fulfill the inequality

Therefore we have

This completes the proof of Theorem 2 as in all other cases, the proof is similar to the proof of Theorem 1. Proof of Theorem 3. As the proof is straightforward for the cases when or we may assume that and We may also assume that Let be a number such that

128 Then, as since

for

Let now

by (3.5) as

This means that, for

Moreover,

again,

This way we

there is

Now, as have

as

we have we again have

Thus we obtain

As and

This way we have that for every

This means that

therefore, passing to the limit with we receive

and

129

which completes the proof of this theorem. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Baernstein A., Integral means, univalent functions and circular symmetrization, Acta Math. 133 (1974), 139–169. Edrei A., Sums of deficiencies of meromorphic functions, J. Analyse Math. 14 (1965), 79–104. Essen M. and Shea D.F., Applications of Denjoy integral inequalities and differential inequalities to growth problems for subharmonic and meromorphic functions, Proc. Roy. Irish Acad. Sect. A 82 (1982), 201–216. Gariepy R. and Lewis J.L., Space analogues of some theorems for subharmonic and meromorphic functions, Ark. Mat. 13 (1975), 91–105. Gol’dberg A. A. and Ostrowskii I.V., Distribution of values of meromorphic functions, Izdat. “Nauka”, Moscow 1970. (Russian) Hayman W.K., Multivalent Functions, Cambridge University Press, Cambridge 1958. Marchenko I. I., On the magnitudes of deviations and spreads of meromorphic functions of finite lower order, Mat. Sb. 186 (1995), 391–408. Petrenko V.P., Growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 414–454. (Russian) Ronkin L.I., Introduction into the theory of entire functions of many variables, Izdat. “Nauka”, Moscow 1971. (Russian)

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COMPOSITION THEOREMS, MULTIPLIER SEQUENCES AND COMPLEX ZERO DECREASING SEQUENCES

THOMAS CRAVEN Department of Mathematics, University of Hawaii Honolulu, HI 96822, [email protected]. edu AND GEORGE CSORDAS Department of Mathematics, University of Hawaii Honolulu, HI 96822, [email protected]. edu

Abstract. An important chapter in the theory of distribution of zeros of polynomials and transcendental entire functions pertains to the study of linear operators acting on entire functions. This article surveys some recent developments (as well as some classical results) involving some specific classes of linear operators called multiplier sequences and complex zero decreasing sequences. This expository article consists of four parts: Open problems and background information, Composition theorems (Section 2), Multiplier sequences and the Laguerre-Pólya class (Section 3) and Complex zero decreasing sequences (Section 4). A number of open problems and questions are also included. Mathematics Subject Classification 2000: Primary 26C10, 30C15, 30C10, Secondary 26D15, 26D10. Key words and phrases: Laguerre-Pólya class, multiplier sequences, zeros of entire functions, composition theorems

1. Introduction: Open problems and background information In order to motivate and adumbrate the results to be considered in the sequel, we begin here with a brief discussion of some basic (albeit fundamental) questions and open problems. Let denote the vector space (over

© 2004 Kluwer Academic Publishers

132 or ) of all polynomials of degree at most For (where S is an appropriate set of interest), let (S) denote the class of all polynomials of degree at most all of whose zeros lie in S. (The problems cited in the sequel are all open problems.) Problem 1.1 Characterize all linear transformations (operators)

where, for the sake of simplicity, we will assume that deg Remarks. We hasten to remark that Problem 1.1 is open for all but trivial choices of S (and perhaps, for this reason, it has never been stated in the literature, as far as the authors know). In fact, this problem is open in such important special cases when (i) (ii) S is a half-plane, (iii) S is a sector centered at the origin, (iv) S is a strip, say, or to cite a non-convex, but important, example (v) S is a double sector centered at the origin and symmetric about the real axis. New results about classes of polynomials are almost always of interest; but when such new results also extend, say, to transcendental entire functions, they tend to be significant. For example, when S is the open upper half-plane, the Hermite-Biehler theorem [64, p. 13] characterizes the polynomials all whose zeros lie in S. Moreover, this theorem extends to certain transcendental entire functions Levin [60, Chapter VII]. If S is the left half-plane, then results relating to Problem 1.1 would be important in several areas of applied mathematics (see for example, Marden’s discussion of dynamic stability [62, Chapter IX]). In this case, the known characterization of the Hurwitz polynomials (that is, real polynomials whose zeros all lie in the left half-plane [62, p. 167]) is undoubtedly relevant. (See also the work of Garloff and Wagner [37] concerning the Hadamard products of stable polynomials.) It is interesting to note from an historical perspective, that finding just one new T satisfying (1.1) can be significant. For example, if S is a convex region in and T = D, where then by the classical Gauss-Lucas theorem T satisfies (1.1) (cf. [62, p. 22]). In the sequel, as we consider some special cases of Problem 1.1, we will encounter some other notable linear transformations which satisfy (1.1). Problem 1.2 Characterize all linear transformations (operators) such that where and are real polynomials (that is, the Taylor coefficients of are all real) and denotes the number of nonreal zeros of counting multiplicities.

133

If then (1.2) is a consequence of Rolle’s theorem. If is a real polynomial with only real zeros and then (1.2) follows from the classical Hermite-Poulain Theorem [64, p. 4]. There are many other linear transformations T which satisfy inequality (1.2). Indeed, set and for an arbitrary real polynomial define

If

is a real polynomial with only real negative zeros and if then by a theorem of Laguerre (cf. Theorem 4.1 below)

where is an arbitrary real polynomial. Of course, differentiation is a linear transformation satisfying (1.2) and more: The polynomial has zeros between the real zeros of In [21] the following problem is raised. Problem 1.2a Characterize all linear transformations such that has at least one real zero between any two real zeros of This problem is solved in [21, Corollary 2.4] for linear transformations defined as in (1.3). They are precisely those for which is a nonconstant arithmetic sequence all of whose terms have the same sign. Problem 1.3 Characterize all linear transformations (operators) such that Recently, a number of significant investigations related to the above problems have been carried out by Iserles and Saff [49], Iserles and Nørsett [47] and Iserles, Nørsett and Saff [48]. In particular, in [47] and [48] the authors study transformations that map polynomials with zeros in a certain interval into polynomials with zeros in another interval. In [18], Carnicer, Peña and Pinkus characterize a class of linear operators T (which correspond to unit lower triangular matrices) for which the degree of the polynomials and are the same and A noteworthy special case of Problem 1.3 arises when the action of the linear transformation T on the monomials is given by for some The transformations which satisfy (1.4) are called multiplier sequences (cf. [73] or [72, pp. 100–124]). The precise definition is as follows.

134

Definition 1.4 A sequence of real numbers is called a multiplier sequence if, whenever the real polynomial has only real zeros, the polynomial also has only real zeros. In 1914 Pólya and Schur [73] completely characterized multiplier sequences. Their seminal work was a fountainhead of numerous later investigations. Applications to fields other than can be found in [19]. Among the subsequent developments, we single out the notion of a totally positive matrix and its variation diminishing property, which in conjunction with the work of Pólya and Schur, led to the study of the analytical and variation diminishing properties of the convolution transform by Schoenberg [77] and Karlin [50]. A by-product of this research led to conditions for interpolation by spline functions due to Schoenberg and Whitney [80]. (In regard to generating functions of totally positive sequences see, for example, [1], [2], [50]. Concerning the generating functions of Pólya frequency sequences of finite order, see the recent paper of Alzugaray [3]). In light of Problem 1.2, it is natural to consider those multiplier sequences which satisfy inequality (1.2). These sequences are called complex zero decreasing sequences and are defined as follows. Definition 1.5 ([24]) A sequence is said to be a complex zero decreasing sequence, or CZDS for brevity, if

for any real polynomial in the plural.)

(The acronym CZDS will also be used

As a special case of Problem 1.2 we mention the following open problem. Problem 1.6 Characterize all complex zero decreasing sequences. The aim of this brief survey is to provide a bird’s-eye view of some of the classical results as well as recent developments related to the aforementioned open problems. Since the so-called composition theorems ([62, Chapter IV], [64, Kapitel II]) play a pivotal role in the algebraic characterization of multiplier sequences, in Section 2 we examine some sample results which lead to the composition theorems. While a detailed discussion of the composition theorems is beyond the scope of this article, in Section 2 we include a proof of de Bruijn’s generalization of the Malo-Schur-Szegö Composition Theorem. In Section 3 we state the Pólya and Schur algebraic and transcendental characterization of multiplier sequences [73]. The latter characterization involves a special class of entire functions known as the Laguerre-Pólya class. We exploit this connection and use it as a conduit in

135

our formulation of a number of recently established properties of multiplier sequences. In Section 4 we highlight some selected results pertaining to the ongoing investigations of properties of CZDS and we list several open problems. Finally, we caution the reader that the selected bibliography is not intended to be comprehensive. 2. Composition theorems

A key step in the characterization of multiplier sequences rests on the composition theorems. In this section our aim is to succinctly outline some of the precursory ideas which lead to the Malo-Schur-Szegö Composition Theorem. Before stating this theorem, we briefly describe Laguerre’s Separation Theorem and Grace’s Apolarity Theorem, two results which are frequently invoked in the proofs of composition theorems for polynomials. (We remark parenthetically that there are other approaches to some of these theorems. Indeed, Schur’s original proof ([81] or [60, p. 336]) was based on properties of Sturm sequences. However, Sturm sequences are inapplicable for the determination of the nonreal zeros of a polynomial and thus this approach does not seem to lend itself to generalizations.) Given the extensive literature dealing with composition theorems for polynomials (also called Hadamard products of polynomials), our treatment is of necessity perfunctory and is limited to our goal of providing a modicum of insight into the foundation of the theory of multiplier sequences. (For additional citations we refer to Borwein and Erdélyi [14], Marden [62] and Obreschkoff [64] and the references contained therein.) In order to motivate Laguerre’s Separation Theorem, we associate with each polynomial a “generalized” derivative called the polar derivative (with respect to defined by

Note that if deg then is a polynomial of degree When then we define to be the ordinary derivative. Now by the classical Gauss–Lucas Theorem [62, §6], any circle which contains in its interior all the zeros of a polynomial also contains all the zeros of What is the corresponding result for polar derivatives? By considering circular regions (i.e., closed disks, or the closure of the exterior of such disks or closed half-planes), which are “invariant” under Möbius transformations, Laguerre obtained the following invariant form the Gauss-Lucas Theorem ([14, p. 20], [62, §13], [64, §4]). Theorem 2.1 (Laguerre’s Separation Theorem) Let be a polynomial of degree

136

1. Suppose that all the zeros of lie in a circular region D. For all of the zeros of the polar derivative lie in D. 2. Let be any complex number such that Then any circle, C, passing through the points

and

either passes through

all the zeros of or separates the zeros of (in the sense that there is at least one zero of in the interior of C and at least one zero in the exterior of C). of

Suppose that (for fixed we obtain (assuming that

Then, solving (2.1) for

in terms

which appears as the “mysterious” point in Laguerre’s Separation Theorem. Marden [62, p. 50] gives two proofs using spherical force fields and properties of the centroid of a system of masses. For a simple, purely analytical proof we refer to A. Aziz [5]. A masterly presentation of Laguerre’s theorem, its invariance under Möbius transformations, (and some of its more recent applications) in terms of the notion of a generalized center of mass is given by E. Grosswald [39]. (See also Pólya and Szegö [74, Vol. II, Problems 101-120].) In order to state Grace’s Apolarity Theorem ([62, p. 61], [64, p. 23], [14, p. 23], [38]) it will be convenient to adopt the following definition. Definition 2.2 Two polynomials

where

are said to be apolar if their coefficients satisfy the relation

Theorem 2.3 (Grace’s Apolarity Theorem) Let and be apolar polynomials. If has all its zeros in a circular region D, then has at least one zero in D. Grace’s Apolarity Theorem can be derived by repeated applications of Laguerre’s Separation Theorem [62, p. 61]. This fundamental result relating the relative location of the zeros of two apolar polynomials, while remarkable for its lack of intuitive content, has far-reaching consequences. One such consequence is the following composition theorem.

137 Theorem 2.4 (The Malo-Schur-Szegö Theorem [62, §16], [64, §7]) Let

and set

1. (Szegö, [85]) If all the zeros of lie in a circular region K, and if are the zeros of then every zero of is of the form for some and some 2. (Schur, [81]) If all the zeros of lie in a convex region K containing the origin and if the zeros of lie in the interval (–1,0), then the zeros of also lie in K. 3. If the zeros of lie in the interval and if the zeros of lie in the interval (or in ), where then the zeros of lie in 4. (Malo [64, p. 29], Schur [81]) If the zeros of are all real and if the zeros of are all real and of the same sign, then the zeros of the polynomials and are also all real, where As a particularly interesting example of the last of these results, take to see that for any positive integer the polynomial transforms to with only real zeros. In [62], [64] and the references cited in these monographs the reader will find a number variations and generalizations of Theorem 2.4 (see also the more recent work of A. Aziz [6], [7] and Z. Rubinstein [76]). Among the many related results, we wish to single out here Weisner’s sectorial version of Theorem 2.3 [86]; that is, composition theorems for polynomials whose zeros lie in certain sectors. Weisner’s proofs are based on the Gauss-Lucas Theorem and Laguerre’s Separation Theorem. In [16], N. G. de Bruijn further extended Weisner’s results and obtained an independent, geometric proof of a generalized Malo-Schur-Szegö Composition Theorem. We conclude this section with de Bruijn’s result which deserves to be better known. The details of the proof given below are sufficiently different from de Bruijn’s original proof to merit their inclusion here. Let denote an open sector with vertex at the origin and aperture Similarly, set If we denote the “product” sector by where where

138

The sector is defined as will denote the open left half-plane by

In the sequel we

Theorem 2.5 (Generalized Malo-Schur-Szegö Composition Theorem [16]) Let and and let

If has all its zeros in the sector zeros in the sector then

and if has all its has all its zeros in the sector

Remark 2.6 (Rotational independence) We claim that it suffices to prove the theorem in the special case when each sector has as its initial ray. Indeed, suppose that the zeros of lie in and the zeros of lie in where and are defined above. Then the zeros of the polynomials and lie in and respectively. In this case, by assumption, the zeros of the composite polynomial (which is now) lie in the sector But then has its zeros in as desired. A similar argument shows that if the theorem holds for any particular and then it holds for any rotations of those sectors.

Lemma 2.7 Theorem 2.5 holds when

and

are half-planes.

Proof. We consider the case when By Remark 2.6, it suffices to prove that if all the zeros of and lie then cannot vanish on the positive real axis. In order to prove this assertion, set

where Re Re for all Fix Then, logarithmic differentiation yields

Thus, then

and fix with

(Indeed, if But then this would contradict (2.6), since

139

Re

Therefore, all the zeros of lie in the open left half-plane By the same argument we see that all the zeros of

lie in

Continuing in this manner, we find that all the zeros of

lie in the open left half-plane

and, since axis.

was arbitrary,

Thus, (cf. (2.3))

does not vanish on the positive real

Proof of Theorem 2.5. By Lemma 2.7 and Remark 2.6, the theorem is true for half-planes. Let and be two half-planes, with initial rays and respectively, and terminal rays and respectively. Then all the zeros of lie in that is, they lie off the ray By Remark 2.6, it suffices to prove the result for sectors and whose initial rays lie on the positive Thus, we have to show that all the zeros of lie in the open sector bounded by the rays and We apply Lemma 2.7 for each Thus Therefore, the zeros of cannot lie on the rays in the closed sector from But this leaves all the zeros in We observe that continuity considerations show that Theorem 2.5 remains valid when the open sectors are replaced by closed sectors, provided that we append the condition that the polynomial is not identically zero. From Theorem 2.5 we can deduce several corollaries (cf. [16]). For example, if the zeros of the polynomial all lie in the sector and if the zeros of are all real, then the zeros of lie in This follows from two applications of Theorem 2.5: First let represent the closed upper half-plane, and then let represent the closed lower halfplane. In particular, the Malo–Schur result (see part (4) of Theorem 2.4) is a special case of this, where has only real zeros and the zeros of are all real and of the same sign.

140

3. Multiplier sequences and the Laguerre-Pólya class

It follows from part (4) of Theorem 2.4 that if the polynomial has only real negative zeros, then the sequence is a multiplier sequence, where if (see Definition 1.4). In this section we state several necessary and sufficient conditions for a sequence to be a multiplier sequence. The transcendental characterization of these sequences is given in terms of functions in the Laguerre-Pólya class (see Definition 3.1), while the algebraic characterization rests on properties of a class of polynomials called Jensen polynomials (Definition 3.4). In addition, we discuss a number of topics related to multiplier sequences and functions in the Laguerre-Pólya class: the closure properties of functions in the LaguerrePólya class, the Turán and Laguerre inequalities, the complex analog of the Laguerre inequalities, iterated Turán and Laguerre inequalities, the connection between totally positive sequences and multiplier sequences, the Gauss-Lucas property and convexity properties of increasing multiplier sequences, the Pólya–Wiman Theorem and the Fourier–Pólya Theorem, the Pólya–Wiman Theorem and certain differential operators and several open problems (including a problem due to Gauss). Definition 3.1 A real entire function the Laguerre-Pólya class, if

where zeros in then will write for all

is said to be in can be expressed in the form

is a nonnegative integer and the sum and if has all its (or ), then we will use the notation (or If (or for all is said to be of type I in the Laguerre-Pólya class, and we We will also write if and

In order to clarify the above terminology, we remark that if then or but that an entire function in need not belong to Indeed, if where denotes the gamma function, then but This can be seen, for example, by looking at the Taylor coefficients of Remark 3.2 (a) The significance of the Laguerre-Pólya class in the theory of entire functions stems from the fact that functions in this class, and only these, are the uniform limits, on compact subsets of of polynomials with

141

only real zeros (Levin [60, Chapter VIII]). Thus it follows that the LaguerrePólya class is closed under differentiation; that is, if then for In fact a more general closure property is valid. Indeed, let

denote differentiation with respect to

suppose that the entire functions the action of the differential operator

and

are in

and If

is defined by

and if the right-hand side of (3.2) represents an entire function, then the function An analysis of various types of infinite order differential operators acting on functions in is carried out in [23]. (b) To further underscore the importance of the Laguerre-Pólya class, we cite here a few selected items from the extensive literature dealing with the differential operator where In connection with the study of the distribution of zeros of certain Fourier transforms, Pólya characterized the universal factors ([68] or [72, pp. 265–277]) in terms of where Subsequently, this work of Pólya was extended by de Bruijn [17] who studied, in particular, the operators and Benz [11] applied the operator to investigate the distribution of zeros of certain exponential polynomials. The operators play a central role in Schoenberg’s celebrated work [79] on Pólya frequency functions and totally positive functions. Hirschman and Widder [44] used to develop the inversion and representation theories of certain convolution transforms. More recently, Boas and Prather [13] considered the final set problem for certain trigonometric polynomials when differentiation D is replaced by Theorem 3.3 ([73], [60, Chapter VIII], [64, Kapitel II]) Let where for 1. (Transcendental Characterization.) T is a multiplier sequence if and only if

2. (Algebraic Characterization.) T is a multiplier sequence if and only if

142

We remark that the Taylor coefficients of functions in the LaguerrePólya class have analogous characterizations. It is the sign regularity property of the Taylor coefficients of a function in (that is, the terms all have the same sign or they alternate in sign) that allows us to invoke the Malo-Schur Composition Theorem (part (4) of Theorem 2.4) and thus deduce the remarkable algebraic characterization (3.4) of multiplier sequences. Definition 3.4 Let

be an arbitrary entire function. Then

the Jensen polynomial associated with the entire function defined by

is

If

we will write The Jensen polynomials associated with arbitrary entire functions enjoy a number of important properties (cf. [22], [34]). For example, the sequence is generated by that is,

Moreover, it is not difficult to show that for

holds uniformly on compact subsets of then where the polynomials

[22, Lemma 2.2]. Observe that, if for each

called Appell polynomials (Rainville [75, p.

145]), are defined by

If

of Jensen polynomials associated with a function from the generating relation (3.6) that the sequence multiplier sequence for each fixed

is a sequence then it follows is itself a

We next consider several necessary and sufficient conditions for a real entire function

to belong to the Laguerre-Pólya class.

143 Theorem 3.5 ([22, Corollary 2.6]) Let by (3.7). Let

where

is the for

be an entire function defined

Jensen polynomial associated with Then if and only if

Suppose that

In particular, if for then the sequence is multiplier sequence if and only if (3.9) holds. In [22, Theorem 2.5, Corollary 2.6, Theorem 2.7] the reader will find other formulations of Theorem 3.5 expressed in terms of Jensen polynomials. In order to state a different type of characterization of functions in we consider, for each fixed the Taylor series expansion of where is a real entire function. Then an elementary calculation shows (cf. [28, Remark 2.4]) that, for each fixed

where

is given by the formula

Theorem 3.6 ([65], [34, Theorem 2.9], [28, Theorem 2.2]) Let be a real entire function whose Taylor series expansion is given by (3.7). Suppose that where and the genus of is 0 or 1. Then if and only if for all In particular, if for then the sequence is a multiplier sequence if and only if for all Since the Laguerre-Pólya class is closed under differentiation (cf. Remarks 3.2 (a)), it follows from Theorem 3.6 that for all and for all By specializing to the case when we obtain the following necessary conditions for to belong to Corollary 3.7 Let be an entire function defined by (3.7). If then the following inequalities hold.

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1. (The Turán Inequalities [22].)

2. (The Laguerre Inequalities [22].)

While the Turán and Laguerre inequalities are some of the simplest conditions that a function in must satisfy, the verification of the Laguerre inequalities, in general, is a nontrivial matter. For higher order inequalities of the type (3.12) see S. Karlin and G. Szegö [51]. Other extensions and applications may be found in M. Patrick [65] and H. Skovgaard [84]. We next proceed to describe various ramifications, extensions, generalizations and open problems related to these fundamental, albeit basic, inequalities. First, we note that there is a complex analog of the Laguerre inequalities which, in conjunction with appropriate growth conditions, characterizes functions in Theorem 3.8 (Complex Laguerre Inequalities [34, Theorem 2.10]) If a real entire function has the form where and the genus of is 0 or 1, then if and only if

Is there a real variable analog of Theorem 3.8? That is, can the Laguerre inequalities (3.12) be strengthened with some supplementary hypotheses to yield a sufficient condition? To shed light on this question, for each we associate with a real entire function the real entire function

Now it is not difficult to show that where Also, if then it follows from an extension of the Hermite-Poulain Theorem ([67, §3] or [72, p. 142]) that for all and so by Corollary 3.7, for all If and if is not of the form C exp then it is known that for all [32, Theorem I]. The main results in [32] are converses of this implication under some additional assumptions on the distribution of zeros of The proofs involve the study of the level sets of that is, the sets

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The analysis of the connections between the Laguerre expression of the level set Re and the zero set of is the dominant theme of this paper. Also in this paper the authors state that they “do not know if the converse of Theorem I (cited above) is valid in the absence of additional assumptions” [32, p. 379]. Here we note that the strict inequality for all is necessary as the following example shows. Let Then an elementary, but tedious, calculation shows that

and equals 0 only if but where Problem 3.9 Let that

and Thus If we replace the differential operator then we are led to the following problem.

for all by

be a real entire function of order less than 2. Suppose

If (3.15) holds, is tion on the growth of

(See [23, p. 806] for the reasons for this restric)

We next explore some other avenues that might provide stronger necessary conditions than those stated in Corollary 3.7. To this end, we consider iterating the Laguerre and Turán inequalities. Definition 3.10 For any real entire function

and for

set

set

Note that with the above notation, we have

for

and and that for The authors’ earlier investigations of functions in the Laguerre-Pólya class [22], [25], [28] have led to the following open problem. Problem 3.11 ([28, §3]) If are the iterated Laguerre inequalities valid for all That is, is it true that

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In the formulation of Problem 3.11, the restriction to the class is necessary, since simple examples show that (3.16) need not hold for functions in For example, but a calculation shows that is negative for all sufficiently small positive values of In [22, Theorem 2.13] the authors have shown that (3.16) is true when that is, the double Laguerre inequalities are valid. The proof there is based on certain polynomial invariants and Theorem 3.6. A somewhat shorter proof, which also depends on Theorem 3.6 is given in [28, Theorem 3.5]. Theorem 3.12 ([22, Theorem 2.13], [28, Theorem 3.5]) If then for

A particularly intriguing open problem arises in the special case when in (3.16). Problem 3.13 ([28, §3]) Is it true that

We next turn to the iterated Turán inequalities. Definition 3.14 Let be a sequence of real numbers. We define the iterated Turán sequence of via and

Thus, if we write

then

is just

evaluated

at In [28, §4] the authors have shown that for multiplier sequences which decay sufficiently rapidly all the higher iterated Turán inequalities hold. The main result of [28] is that the third iterated Turán inequalities are valid for all functions of the form where Theorem 3.15 ([28, Theorem 5.5]) Let set

so that

and

and

for

Then

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An examination of the proof of Theorem 3.15 shows that the restriction that has a double zero at the origin is merely a ploy to render the, otherwise very lengthy and involved, computations tractable. We next touch upon the characterization of entire functions in purely in terms of their Taylor coefficients. To this end, we consider the entire function

and recall the following definition. Definition 3.16 A real sequence is said to be a totally positive sequence, if the infinite lower triangular matrix

is totally positive; that is, all the minors of A of all orders are nonnegative. In [1, p. 306], M. Aissen, A. Edrei, I. J. Schoenberg and A. Whitney characterized the generating functions of totally positive sequences. A special case of their result is the following theorem. Theorem 3.17 ([1, p. 306]) Let be the entire function defined by (3.20). Then is a totally positive sequence if and only if An immediate consequence of Theorem 3.17 is the following corollary. Corollary 3.18 ([1, p. 306]) Let

Then if and only if the sequence totally positive sequence.

is a

Suppose that the generating function (3.20) is an entire function. Then, in light of Theorem 3.3(1), the sequence is a multiplier sequence if and only if the sequence is a totally positive sequence. Remarks. Totally positive sequences were first introduced in 1912 by M. Fekete and G. Pólya [36]. For a concise survey of totally positive matrices

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we refer to T. Ando [4]. The connection between totally positive sequences and combinatorics is treated in F. Brenti’s monograph [15]. S. Karlin’s monumental tome [50] on total positivity, while mostly concerned with totally positive kernels, also treats totally positive matrices and I. J. Schoenberg’s theory of variation diminishing transformations [78]. From the extensive literature treating total positivity and related topics, here we merely mention the recent work of M. Alzugaray [3] and O. M. Katkova and [52] investigating the zero sets of generating functions of multiply positive sequences. (These are sequences which have the property that the minors of the Toeplitz matrix (3.21), less than or equal to some fixed order, are all nonnegative.) Increasing multiplier sequences enjoy a number interesting geometric properties some of which we now proceed to sketch here. To facilitate our description, we introduce the following terminology. Definition 3.19 A sequence of real numbers is said to possess the Gauss-Lucas property, if whenever a convex region K contains the origin and all the zeros of a complex polynomial then all the zeros of the polynomial also lie in K. The proof of the complete characterization of sequences which enjoy the Gauss-Lucas property hinges on the Malo-Schur-Szegö Composition Theorem (cf. Theorem 2.4(1)) and on the fact that the zeros of the Jensen polynomials associated with an increasing multiplier sequence must all lie in the interval [0,1] (see [20, Theorem 2.3]). Theorem 3.20 ([20, Theorem 2.8]) Let be a nonzero sequence of real numbers. Then T possess the Gauss-Lucas property if and only if T is a multiplier sequence and The classical example of this theorem is its application to the sequence T = {0,1,2,... }, since for any polynomial This, and other examples, suggest that the operators T may be viewed as generalized forms of differential operators. The problem of extending the foregoing results to transcendental entire functions whose zeros lie in an unbounded convex region appears to be very difficult. However, for transcendental entire functions of genus zero, we have the following consequence of Theorem 3.20. Corollary 3.21 ([20, Corollary 3.1]) Let be an increasing multiplier sequence. Let K be an unbounded convex region which contains the origin and all the zeros of an entire function of genus zero. Then the zeros of the entire function also lie in K.

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We remark that the extension of these results to real entire functions of less restricted growth, but all whose zeros are real, is still open (cf. S. Hellerstein and J. Korevaar [42]). Turning to the convexity properties of multiplier sequences, we first note that all multiplier sequences are eventually monotone; that is, from a certain point onward the multiplier sequence T is either increasing or decreasing (cf. [20, Proposition 4.4]). In the sequel it will be convenient for us to adopt the following standard notation (the for forward differences. (A caveat is in order. The symbol used in (3.8) has a different meaning.) Definition 3.22 For any real sequence and

we define

Proposition 3.23 ([20, Proposition 4.2]) Let If then

Moreover,

Since the Laguerre-Pólya class is closed under differentiation, it follows that if then

If we assume that then is also an increasing sequence by Proposition 3.23 and thus we conclude that for each fixed nonnegative integer Now for inequality (3.23) says that is convex. We conclude this section with a few remarks concerning two famous conjectures, related to functions in the Laguerre-Pólya class, which have been recently solved. These long-standing open problems, known in the literature as the Pólya–Wiman conjecture and the Fourier–Pólya conjecture, have been investigated by many eminent mathematicians. The history associated with these problems is particularly interesting. (See, for example, G. Pólya [71] or [72, pp. 394–407] for a general discussion of the theme, and

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[70] or [72, pp. 322–335] for a comprehensive survey which covers almost everything in this area up to 1942.) The Pólya–Wiman conjecture has been established by T. Craven, G. Csordas, W. Smith [29], [30], Y.-O. Kim [54], [55]. We shall refer to their result as the Pólya–Wiman Theorem. Recently, H. Ki and Y.-O. Kim [53] provided a truly elegant proof of this theorem. Theorem 3.24 (The Pólya–Wiman Theorem [29], [30], [55], [54], [53]) Let be a real entire function, where and suppose that the genus of is at most 1. If has only a finite number of nonreal zeros, then its successive derivatives, from a certain one onward, have only real zeros, that is for all sufficiently large positive integers Theorem 3.24 confirms the heuristic principle according to which the nonreal zeros of the derivatives of a real entire function move toward the real axis when the order of is less than 2. The dual principle asserts that the nonreal zeros of the derivatives move away from the real axis when the order of is greater than 2. A long-standing open problem related to this dual principle may be stated as follows. If the order of a real entire function is greater than 2, and if has only a finite number of nonreal zeros, then the number of the nonreal zeros of tends to infinity as (G. Pólya [71]). Significant contributions to this problem were made by B. Ja. Levin and [61] and extended by S. Hellerstein and C. C. Yang [43]. In particular, S. Hellerstein and C. C. Yang showed that the conjecture is true for real entire functions of sufficiently large order (see also T. Sheil-Small [82]). The Fourier–Pólya conjecture (established in [53]) asserts that one can determine the number of nonreal zeros of a real entire function of genus 0 by counting the number of critical points of has just as many critical points as couples of nonreal zeros. When and all its derivatives possess only simple zeros, then the critical points of are the abscissae of points where has positive minima or negative maxima. The definition of critical points is more elaborate if there are multiple zeros ([53], see also Y.-O. Kim [56], [57]). We next consider a few sample results which pertain to investigations related to Theorem 3.24. In [23] the authors analyze the more general situation when the operator D in the Pólya–Wiman Theorem is replaced by the differential operator where need not belong to Indeed, if is a real power series with zero linear term and if is any real polynomial, then for all sufficiently large positive integers More precisely, the authors proved the following result.

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Theorem 3.25 ([23, Theorem 2.4]) Let

be a real power series with and Let be any real polynomial of degree at least one. Then there is a positive integer such that for all In fact, can be chosen so that all the zeros are simple. If the linear term in (3.25) is nonzero, then simple examples show that the conclusion of Theorem 3.25 does not hold without much stronger restrictions on [23, §3]. To rectify this, the authors consider and a real entire function having only a finite number of nonreal zeros (with some restriction of the growth of or as in Theorem 3.26 below). If has at least one real zero, then for all sufficiently large positive integers The proof of the following theorem is based on several technical results ([23, Lemma 3.1, Lemma 3.2] and [83, p. 41 and p. 106]) involving differential operators. Theorem 3.26 ([23, Theorem 3.3]) Let and be real entire functions of genus 0 or 1 and set and where If has only a finite number of nonreal zeros and has at least one real zero, then there is a positive integer such that for all A separate analysis of the operator shows that, not only does Theorem 3.26 hold, but that the zeros become simple. In fact, if where the order of is less than two, then has only real simple zeros. Theorem 3.27 ([23, Theorem 3.10]) Let and suppose that the order of is strictly less than 2. Let for all Then, for each fixed and the zeros of are all simple. Corollary 3.28 ([23, Theorem 3.11]) Let be a real entire function of order strictly less than 2, having only a finite number of nonreal zeros. If then with only simple zeros for all sufficiently large The question of simplicity of zeros is pursued further in [23, §4]. The authors proved that if and are functions in the Laguerre-Pólya class of order less than two, has an infinite number of zeros, and there is a bound on the multiplicities of the zeros of then has only

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simple real zeros [23, Theorem 4.6]. In [23, p. 819] the question was raised whether or not the assumption (in [23, Theorem 4.6]) that there is a bound on the multiplicities of the zeros of is necessary. That is, if and if has order less than two, then is it true that has only simple real zeros? The study of the “movement” of the zeros under the action of the infinite order differential operators was initiated by G. Pólya ([67] or [72, pp. 128– 153]) and N. G. de Bruijn [17] in their study of the distribution of zeros of entire functions related to the Riemann (For recent results in this direction see [33] and [31].) In [17], de Bruijn proved, in particular, that if is a real entire function of order less than two and if all the zeros of lie in the strip then the zeros of satisfy if and Im if This result may be viewed as an analog of Jensen’s theorem on the location of the nonreal zeros of the derivative of a polynomial [62, §7]. Problem 3.29 Is there also an analog of Jensen’s theorem for when is an arbitrary function (not of the form ) in the Laguerre-Pólya class? Finally, there is also an interesting connection between the ideas used to prove the Pólya–Wiman Theorem (for entire functions of order less than 2) [29, Theorem 1] and a question that was raised by Gauss in 1836 [29, p. 429]. Let be a real polynomial of degree and suppose that has exactly nonreal zeros, Then Gauss’ query is to find a relationship between the number and the number of real zeros of the rational function

If has only real zeros, then for all and consequently in this special case the answer is clear. Now it follows from [29, Theorem 1] that if for some the polynomial has only real zeros, then has precisely real zeros. On the basis of their analysis, the authors in [29, p. 429] stated the following conjecture. Problem 3.30 Let be a real polynomial of degree suppose that has exactly nonreal zeros,

and Prove that

where denotes the number of real zeros, counting multiplicities, of the rational function defined by (3.26).

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Gauss’ question has been studied by several authors (see the references in [29]). For recent contributions dealing with Problem 3.30 we refer to K. Dilcher and K. B. Stolarsky [35]. 4. Complex zero decreasing sequences (CZDS) It follows from Definition 1.5 that every complex zero decreasing sequence is also a multiplier sequence. If is a sequence of nonzero real numbers, then inequality (1.5) is equivalent to the statement that for any polynomial has at least as many real zeros as has. There are, however, CZDS which have zero terms and consequently it may happen that deg When counting the real zeros of the number generally increases with the application of T, but may in fact decrease due to a decrease in the degree of the polynomial. For this reason, we count nonreal zeros rather than real ones. The existence of a nontrivial CZDS is a consequence of the following theorem proved by Laguerre and extended by Pólya ([69] or [72, pp. 314-321]). We remark that in the next theorem, part (2) follows from (1) by a limiting argument. Theorem 4.1 (Laguerre [64, Satz 3.2])

1. Let be an arbitrary real polynomial of degree and let be a polynomial with only real zeros, none of which lie in the interval Then 2. Let be an arbitrary real polynomial of degree let and suppose that none of the zeros of lie in the interval Then the inequality holds. 3. Let then the sequence is a complex zero decreasing sequence. As a particular example of a CZDS, we can apply Theorem 4.1(2) to the function to obtain One of the main results of [24] is the converse of Theorem 4.1 in the case that is a polynomial. The converse fails, in general, for transcendental entire functions. Indeed, if is a polynomial in then and are transcendental entire functions which generate the same sequence but they are not in For several analogues and extensions of Theorem 4.1, we refer the reader to S. Karlin [50, pp. 379–383], M. Marden [62, pp. 60–74], N. Obreschkoff [64, pp. 6–8, 42–47]. A sequence which can be interpolated by a function that is, for will be called a Laguerre multiplier sequence or a Laguerre sequence. It follows from Theorem 4.1 that Laguerre sequences are multiplier sequences.

154 With the terminology adopted here, the Karlin-Laguerre problem [8], [24] can be formulated as follows. Problem 4.2 (The Karlin-Laguerre problem.) Characterize all the multiplier sequences which are complex zero decreasing sequences (CZDS). This fundamental problem in the theory of multiplier sequences has eluded the attempts of researchers for over four decades. In order to elucidate some of the subtleties involved, we need to introduce yet another family of sequences related to CZDS. The reciprocals of Laguerre sequences are examples of sequences which are termed in the literature as and are defined as follows (cf. L. Iliev [46, Ch. 4] or M. D. Kostova [58]). Definition 4.3 A sequence of nonzero real numbers, is called a if

whenever for all We remark that if is a sequence of nonzero real numbers and if is an entire function, then a necessary condition for to be a is that for all real (Indeed, if for then continuity considerations show that there is a positive integer such that for ) In [46, Ch. 4] (see also [58]) it was pointed out by Iliev that are the positive semidefinite sequences. There are several known characterizations of positive definite sequences (see, for example, [63, Ch. 8] and [87, Ch. 3]) which we include here for the reader’s convenience. See also [24, Theorem 1.7], where the first item should refer only to positive definite s. Theorem 4.4 Let be a sequence of nonzero real numbers. Then the following are equivalent. 1. (Positive Definite Sequences [87, p. 132]) For any polynomial not identically zero, the relation for all implies that

2. (Determinant Criterion [87, p. 134])

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3. (The Hamburger Moment Problem [87, p. 134]) There exists a nondecreasing function with infinitely many points of increase such that

The importance of in our investigation stems from the fact that a necessary condition for a sequence to be a CZDS is that the sequence of reciprocals be a Thus, for example, the reciprocal of a Laguerre multiplier sequence is a As our next example shows, there are multiplier sequences whose reciprocals are not Example 4.5 ([24, p. 423]) Let 3.3, T is a multiplier sequence since

Next, let determinant det

Then by Theorem

Then a calculation shows that the is

Therefore, by (4.2) we conclude that is not a and a fortiori the multiplier sequence T is not a CZDS. It is also instructive to exhibit a concrete example for which inequality (1.5) fails. To this end, we set Then a calculation shows that

Now it can be verified that and hence again it follows that the multiplier sequence T is not a CZDS. In light of Example 4.5, the following natural problem arises. Problem 4.6 (Reciprocals of multiplier sequences.) Characterize the multiplier sequences with for which the sequences of reciprocals, are

156 One of the principal results of [24, Theorem 2.13] characterizes the class of all polynomials which interpolate CZDS. The proof of the next theorem requires several preparatory results involving properties of both CZDS and Theorem 4.7 ([24, Theorem 2.13]) Let be a real polynomial. The sequence is a complex zero decreasing sequence (CZDS) if and only if either 1. and all the zeros of are real and negative, or 2. and the polynomial has the form

where

for each

and

is a fixed positive integer.

We remark that in part (2) of Theorem 4.7, the assumption that for each is necessary. Indeed, set and in (4.4), so that If then the sequence has the form and thus the terms of the sequence eventually become positive even though It follows that T cannot even be a multiplier sequence. A similar claim can be made for sequences arising from polynomials of the form with In general, if a sequence, of positive real numbers grows sufficiently rapidly, then it is a For example, recently the authors proved that if and if then is a positive definite sequence [27]. (The question whether or not the constant is best possible remains open.) Thus, applying this criterion to sequences of the form where is a positive integer, we see that such sequences are positive definite sequences. Furthermore, it is known that the sequence of reciprocals (where is a positive integer, is a multiplier sequence [24, p. 438]. However, it is not known whether or not these multiplier sequences are CZDS. For ease of reference, and to tantalize the interested reader, we pose here the following concrete question. Problem 4.8 (a) Is the sequence a CZDS? (b) More generally, if is a positive multiplier sequence with the property that is a is it true that is CZDS? In order to establish the existence of additional classes of CZDS in [24, §4] the authors first generalized a classical theorem of Hutchinson [45] (see also Hardy [40] or [41, pp. 95-99], Petrovitch [66] and the recent paper by Kurtz [59, p. 259]) and obtained the following results.

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Theorem 4.9 ([24, Theorem 4.3]) Let with for and suppose that the Turán inequalities, hold for N – 1, where

Then the polynomial has only real, simple negative zeros. Corollary 4.10 ([24, Corollary 4.9]) Let with for and suppose that where

Then

and

are entire functions of order zero and

In order to expedite our exposition, we shall also introduce the following definition. Definition 4.11 A sequence of nonnegative real numbers will be called a rapidly decreasing sequence if satisfies inequality (4.6). The sequence is rapidly decreasing if and this sequence is a Laguerre sequence for any Sequences of the form where and is a positive integer, are multiplier sequences, but these sequences cannot be interpolated by functions For indeed, if then

where and Then from the standard estimates of the canonical product (see, for example, [12, p. 21]), we deduce that for any there is a positive integer such that We infer from (4.7) and (4.8) that complex zero decreasing sequences which decay at least as fast as cannot be interpolated by functions in By way of applications of Corollary 4.10, we proceed to state two results which show how rapidly decreasing sequences can be used to generate complex zero decreasing sequences.

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Corollary 4.12 ([24, Corollary 4.7]) Let rapidly decreasing sequence. Then for each fixed

Moreover, if where Jensen polynomial associated with the sequence for that is, for any polynomial for where Corollary 4.13 ([24, Corollary 4.8]) Let sequence and let

be a

then

is the is a CZDS we have

be a rapidly decreasing

Then the sequence is a CZDS. We remark that if is a CZDS with for then the sequence where may not be a CZDS for some To verify this claim, consider the sequence Then it follows that T is a CZDS [24, Proposition 3.5]. A calculation shows that Let so that But has real zeros (both of which are positive) if and only if Hence by Theorem 4.7, is not a CZDS for any In contrast to the previous examples, it is possible to exhibit a CZDS for which the sequence is a CZDS for all where Let Then is a CZDS and for each fixed is a CZDS (cf. [24, Lemma 5.3]). The principal source of the difficulty in characterizing CZDS is that, today, the only known, essentially nontrivial CZDS are the multiplier sequences that can be interpolated by functions in We use the terms “essentially nontrivial” advisedly to circumvent trivial examples of the following sort. Let Then, the sequence {2,2,2,...} is clearly a CZDS, but More sophisticated examples fostered a renewed scrutiny of the Karlin-Laguerre problem, and the investigation of when a CZDS can be interpolated by functions in has led to the following two theorems ([8], [9], [10]). Theorem 4.14 ([9, Theorem 2]) Let

be a CZDS. If

159

then there is a function

where interpolates the sequence

of the form

and that is,

such that for

Theorem 4.15 ([8, Theorem 3.6]) Let be an entire function of exponential type. Suppose that is a CZDS, where Let denote the (Phragmén–Lindelöf) indicator function of that is,

where If expressed in the form

where

then

is in

and

can be

and

These theorems are complementary results in the following sense. Theorem 4.14 asserts that if a CZDS (of positive terms) does not decay too fast (cf. (4.10)), then the sequence can be interpolated by function in having only real negative zeros. In contrast, Theorem 4.15 says that if for some entire function, of exponential type, the sequence is a CZDS and if does not grow too fast along the imaginary axis (cf. (4.11)), then has only real negative zeros. If a multiplier sequence does decay rapidly (cf. (4.12)), then the question whether or not such a sequence can be a CZDS remains an open problem.

Problem 4.16 If

then is

(so that

and if

is a CZDS?

The proof of Theorem 4.14 is rather involved and technical and therefore, due to restrictions of space, it would be difficult to convey here the flavor of the arguments used in [9]. By confining our attention to some special cases of Theorem 4.14, we propose to sketch here some of the techniques and results that can be used to establish converses of Laguerre’s

160

theorem (Theorem 4.1). In the case of polynomials, the converse of Laguerre’s theorem is an immediate consequence of Theorem 4.7 since this theorem completely characterizes the class of all polynomials which interpolate CZDS. On the other hand, the converse of Laguerre’s theorem fails, in general, for transcendental entire functions, as the following example shows. Example 4.17 Let be a polynomial in quence is a CZDS). Then, as noted earlier,

(so that the se-

and

are transcendental entire functions which both interpolate the same sequence but these entire functions are not in Thus, in the transcendental case additional hypotheses are required in order that the converse of Laguerre’s theorem hold. The main result in [26, Theorem 3.9] shows that the converse of Laguerre’s theorem is valid for (transcendental) entire functions of the form where and is a real polynomial which has no nonreal zeros in the left half-plane. The proof hinges on a deep result of Schoenberg (see Theorem 4.19 below) on the representation of the reciprocals of functions in terms of Pólya frequency functions. These functions are defined as follows. Definition 4.18 A function is a frequency function if it is a nonnegative measurable function such that

A frequency function K is said to be a Pólya frequency function if it satisfies the following condition: For every two sets of increasing real numbers and the determinantal inequality

holds.

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Theorem 4.19 (Schoenberg [79, p. 354]) Suppose that where and is not of the form Then the reciprocal of if can be represented in the form

where

is a Pólya frequency function such that

the integral converges up to the first pole of

if

and

Conversely, suppose that

is a Pólya frequency function such that for and the integral converges for Re Then this integral represents, in the halfplane Re the reciprocal of a function where is not of the form Theorem 4.20 ([26, Theorem 3.5]) Let where is not of the form Let be a polynomial with only real zeros, and suppose that Then the sequence is a CZDS if and only if has only real negative zeros. If has only real negative zeros, then and T is a CZDS, by Laguerre’s theorem. Conversely, suppose that T is a CZDS. With reductio ad absurdum in mind, assume that has a positive zero. Since T is a CZDS, the sequence is a and so the application of this sequence to the positive function must give (see the remarks after Definition 4.3)

for all Since is not of the form we may invoke Schoenberg’s theorem (Theorem 4.19) and therefore we can express as

where is a Pólya frequency function such that for Now a somewhat complicated analysis of the behavior of shows that Consequently, is not a as and so we have obtained the desired contradiction. The next preparatory result, whose proof also depends on Schoenberg’s theorem, provides information about the oscillation properties of entire functions under the action of certain

162

Proposition 4.21 ([26, Proposition 3.7]) Let Suppose that with if Then the function

and

and is not of the form

changes sign infinitely often in the interval With the aid of the foregoing preliminary results, we proceed to prove the following theorem. Theorem 4.22 ([26, Theorem 3.8]) Suppose that where and is not of the form Let be a real polynomial all of whose zeros lie in the right half-plane Re Let If the sequence is a CZDS, then all the zeros of are real. Proof. Assume the contrary so that may be expressed in the form where and and Re Then the polynomial gives rise to the entire function where is a polynomial. We next approximate the entire function by means of the polynomials where and (see the remarks following Definition 4.3). We note, in particular, that has exactly the same real zeros as has. Moreover, as uniformly on compact subsets of If we set then by Proposition 4.21, the function

has infinitely many sign changes in the interval Also, as converges to uniformly on compact subsets of Thus, for all sufficiently large each of the approximating polynomials has more real zeros than has. Since T is a CZDS, and since consequently, for all sufficiently large, the polynomial has more real zeros than has. This is the desired contradiction. Combining Theorem 4.22 with Theorem 4.20 (for the details see [26, Theorem 3.9 and Proposition 3.1]) yields the following converse of Laguerre’s theorem.

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Theorem 4.23 ([26, Theorem 3.9]) Suppose that where Let be a real polynomial with no nonreal zeros in the left half-plane Re Suppose that and set Then is a CZDS if and only if has only real negative zeros.

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NEVANLINNA THEORY IN AN ANNULUS

RISTO KORHONEN University of Joensuu, Department of Mathematics, P.O. Box 111, FIN-80101 Joensuu, Finland, Present address: Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK, [email protected]

Abstract. A concrete presentation of Nevanlinna theory in a domain has been offered by Bieberbach. He applied Green’s formula to prove the first main theorem and the lemma of the logarithmic derivative for meromorphic functions outside a disc of radius R. Apart from this work, Nevanlinna theory outside a disc has been considered in the form of brief remarks only in various articles. The purpose of this paper is to collect these comments into a coherent presentation, and to generalize these results for functions meromorphic in an open annulus. We define annulus versions of the Nevanlinna functions allowing accumulation of poles also to the inner boundary, and prove analogues of Nevanlinna’s main theorems including the lemma of the logarithmic derivative. Instead of using Green’s formula, we base our reasoning on a theorem due to Valiron. Mathematics Subject Classification 2000: 30D35. Key words and phrases: Nevanlinna theory; meromorphic function; annulus

1. Introduction

The importance and elegance of Nevanlinna’s theory of value distribution has inspired many authors to find modifications and generalizations to the original theory since its birth in 1920’s. Two of the the most important variants was undoubtedly given by L. Ahlfors in 1935. He used an idea of specially constructed conformal metrics to give his own geometric version on Nevanlinna theory [1]. Not yet satisfied with that, he went on to show that essential parts of Nevanlinna theory can be extended to a large class

© 2004 Kluwer Academic Publishers

168

of functions, which Ahlfors called quasiconformal [2]. Following the same procedure a couple of years later, G. af Hällström established analogues of Nevanlinna’s theorems for some classes of multiply-connected regions [8]. Despite of these very powerful generalizations, it is sometimes more convenient to work in simple domains due to straightforward notations. For example, analyzing functions which are meromorphic in the unit disc is practical by using a specific version of Nevanlinna theory. This theory was composed in a series of articles appeared in 1922–1925, see [10] for instance. In addition, the study of value distribution of meromorphic functions in a half-plane has been given essentially two different approaches by R. Nevanlinna [11] and M. Tsuji [12], respectively. In [11], Nevanlinna considered meromorphic functions in closed sectorial regions, while the Tsuji approach has been modified for sectors by S. Wang in his thesis [14]. Nevanlinna theory outside of a disc of radius R has been briefly considered by L. Bieberbach in [5, pp. 97–107] in connection with the Malmquist theorem. Here Bieberbach proved the first main theorem and the lemma of the logarithmic derivative by using Green’s formula. S. Bank and I. Laine noted in [3], without giving any specific details, that all usual results of Nevanlinna theory, including results of Tumura–Clunie type, remain valid in corresponding form outside the disc of radius R, in other words, in the annulus Since the method Bieberbach used to prove Nevanlinna’s theorems outside of the disc of radius R relies on the use of Green’s formula, it was necessary for him to assume the function to be meromorphic on the boundary It is immediately seen that then, in fact, the function is meromorphic in some annulus where On one hand such assumption is convenient, since it helps to avoid any problems caused by possible accumulation of poles to the boundary. On the other hand, essential information about the behavior of the function close to the boundary is lost in the process. By using Valiron’s factorization theorem and by defining Nevanlinna functions conveniently, we are able to omit this rather restrictive condition. In fact, we present detailed proofs of main theorems of Nevanlinna theory in the domain thus generalizing the results in [5, pp. 97–107]. This enables more delicate analysis near the inner boundary of the considered annulus. 2. The Nevanlinna functions

Suppose that

is meromorhic in Then the counting function of

where is defined by

169

where and annulus The proximity function of is

where usual manner by

counts the number of poles in the each pole according to its multiplicity.

The characteristic function of

is defined in a

where Note that if is fixed, the characteristic function is essentially the same as the characteristic function defined in [5, p. 98]. We shall denote by any quantity satisfying

as and for which

possibly outside of an exceptional set

if

and

if

Furthermore,

describes the ramification of a meromorphic function in the outer growth order of is defined by

if

and by

Moreover,

170

if

Finally, the inner growth order of

is defined by

The essential difference to Bieberbach’s notations can be seen when we keep fixed and allow to move freely on the interval This procedure gives us more flexibility while studying functions with essential singularities inside the disc as the following example demonstrates. Example 2.1 Let Then has an isolated essential singularity at but it is analytic outside of origin. Furthermore,

for all Hence the outer growth order of is but the inner growth order is It is also worth to point out that is decreasing function of if we keep fixed. Obviously this could never happen for the usual characteristic function in the complex plane. Remarks. 1. If is analytic in where and does not have an essential singularity at infinity, then is analytic in the disc In this case, the inner growth order of is equal to

which is the growth order of in the disc 2. If is analytic in and does not have an essential singularity at infinity, then is meromorphic in the complex plane. The inner growth order of is then equal to

which is the growth order of

171 3. Valiron’s theorem

We begin with the original theorem by Valiron [13, p. 15] (see also [9, pp. 101–102]), which is the starting point of our reasoning. Theorem A (Valiron) Let be analytic in with an essential singularity at Then G may be represented as

where is analytic in and is entire and transcendental. Since its first appearance, Theorem A has been subject to many remarks in the literature motivated by applications in Nevanlinna theory. First, it was noted without proof in [3] that this theorem can be modified in such a way that it applies for functions which are analytic at a neighborhood of in other words outside of a disc of some radius R. Second, the assumption that G has an essential singularity at is redundant by [4, Section 2]. Finally, in [6] it was mentioned that Theorem A remains true also for meromorphic functions outside of a disc of radius R. Previous observations and comments, with some additional minor modifications, are gathered up in the following version of the Valiron’s original theorem. Although the proof is a simple modification of [13, pp. 14–15], see also [9, pp. 101–102], some caution is needed near the boundary of the considered annulus. Hence we include the details of the proof here. Theorem B (Valiron) Let be meromorphic in and let Then G may be represented as

where

(a)

is meromorphic in closed annulus (b) is meromorphic in closed annulus (c) Proof. Choose any poles or zeros in

and analytic and non-zero in a and analytic and non-zero in a

such that Write G in the form

where H is analytic and non-zero in

and

does not have any

172

where and are the canonical products formed with the zeros and poles, respectively, of G in Similarly,

where and are the canonical products formed with the zeros and poles, respectively, of G in Denoting we have that is analytic on and so we may write

Let have

where

By integrating (3.2) in A, we

is a convergent Laurent series in

Moreover,

where and are analytic in and By combining equations (3.3) and (3.4), and denoting

and we have

for all

Since the right hand side of

respectively.

173

is single-valued in must be an integer. Therefore (3.7) holds for all To complete the proof, we need to show that functions and have all properties listed in (a) and (b), respectively. Since and are meromorphic in also is meromorphic on the same domain by (3.5). In particular, and are analytic, and is non-zero in Therefore, since is analytic and non-zero in By a similar reasoning, is meromorphic in and analytic and non-zero in

4. Nevanlinna’s main theorems

As mentioned earlier, Bieberbach proved a version of the first main theorem for meromorphic functions outside a disc of radius R by using Green’s formula. Chiang and Gao proved essentially the same result in [6, pp. 279–280], but their reasoning relied on the use of Valiron’s theorem. These results are contained in the following theorem, which is the first main theorem for meromorphic functions in an annulus. Theorem 4.1 Let Then

be a meromorphic function in

Proof. Take

and let

By Theorem B, we may write

where is meromorphic in bounded in Keep first fixed, and assume that

and analytic, non-zero and Moreover, is meromorphic in In this case,

where over,

is the usual proximity function in a disc of radius

where

is the usual counting function in a disc of radius

More-

Similarly,

174 and

Combining (4.1), (4.2), (4.3), (4.4) and the first main theorem, we have

for all Fix now

and suppose that

Then

Moreover, by performing a change of variables

we have

175

Since

and

is meromorphic in

we have

by combining (4.5), (4.6) and (4.7). Finally, since

for all

we obtain by (4.5) and (4.8)

for all

The following lemma of the logarithmic derivative is also essential part of Nevanlinna theory. The proof is again based on Theorem B. Lemma 4.2 Let Then

be a transcendental meromorphic function in

and

as

Proof. By Theorem B, we have

where

is meromorphic in Clearly,

Keep first

fixed, and let

and

Then by (3.5)

is meromorphic in

176 is analytic, and hence bounded, in Thus

as

since Keep now fixed, and let

for some

By (3.6)

is analytic, and hence bounded, in Now, since is meromorphic in (4.9) and (4.10)

for some we have by

as The second main theorem in follows by Theorem 4.1 and Lemma 4.2 applied with the standard reasoning, see [7], for instance. Theorem 4.3 Let be a meromorphic function in let be distinct points. Then

as

and

let

and

177

Similar reasoning yields most of the standard results in Nevanlinna theory. For instance, it is easy to prove by using Lemma 4.2 and elementary induction, that for any transcendental meromorphic function in

for all as and Equation (4.11) in turn yields, again with the standard reasoning, the following Clunie lemma for meromorphic functions in an annulus. Lemma 4.4 Let satifying

be a transcendental meromorphic function in

where and are polynomials in meromorphic coefficients in say

for all If the total degree of derivatives is then

as

and its derivatives with such that

as a polynomial in

and its

and

5. Concluding remarks

Most of the known results about Nevanlinna theory in the complex plane, or in the unit disc, may now be immediately stated and proved in an analogous form for an annulus. Since the proofs of these results would be almost identical to the proofs of the original theorems, we are satisfied to give only the Clunie lemma as an example. Lemma 4.4, along with the rest of the machinery of Nevanlinna theory, gives a large set of tools to our disposal, which enables the study of both the growth, and the value distribution of solutions of complex differential equations on an annulus. Furthermore, considering the situation in an annulus allows us to deal with a class of functions, which is somewhat broader than the field of meromorphic functions. It is defined as follows. Definition 5.1 Let a function be meromorphic, except for essential singularities These singularities may or may not be isolated, but, however, they must not accumulate in the complex plane. The class of all such functions is denoted by

178

In fact, the class forms a field with respect to usual addition and multiplication, and it has the field of meromorphic functions as its subfield. Now, by using the concepts of Nevanlinna theory in an annulus, the growth and value distribution of any may be analyzed in each annuli

separately. For instance,

and in

belongs to

but

does not, since essential singularities of accumulate to the origin. In addition, choosing the origin as a center point of the annulus does not play a special role in our considerations. Define

where and the annulus multiplicity. Moreover, define

counts the number of poles in each pole according to its

where We may prove all main results of the present paper for (5.1) and (5.2) by using identical reasoning as to above. This modification enables, for instance, local growth and value distribution considerations around any isolated essential singularity in the complex plane. Acknowledgements

Partially supported by the Academy of Finland (project 50981) and INTAS (project 99-00089). The work was completed in the Department of Mathematical Sciences in Loughborough University. References 1.

Ahlfors L.: Über eine Methode in der Theorie der meromorphen Funktionen, Soc. Sci. Fennica, Comment. Phys.-Math. 8 (1935), No. 10, 1–14.

179 2. Ahlfors L.: Zur Theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157–194. 3. Bank S. B. and I. Laine: Representations of solutions of periodic second order linear differential equations, J. Reine Angew. Math. 344 (1983), 1–21. 4. Bank S. B. and J. Langley: Oscillation theorems for higher order linear differential equations with entire periodic coefficients, Comment. Math. Univ. St. Paul. 41 (1992), 65–85. 5. Bieberbach L.: Theorie der gewöhnlichen Differentialgleichungen, Springer–Verlag, Berlin–Heidelberg–New York, 1965. 6. Chiang Y. M. and S. A. Gao: On a problem in complex oscillation theory of periodic second order linear differential equations and some related perturbation results, Ann. Acad. Sci. Fenn. Math. 27 (2002), 273–290. 7. Hayman W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964. 8. af Hällström G.: Über meromorphe Funktionen mit mehrfach zusammengängenden Existenzgebieten, Acta Acad. Aboensis 12 (1940), no. 8, 100 pp. 9. Laine I.: Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993. 10. Nevanlinna R.: Zur Theorie der meromorphen Funktionen, Acta Math. 46 (1925), 1–99. 11. Nevanlinna R.: Über die Eigenschaften meromorpher Funktiones in einem Winkelraum, Acta Soc. Sci. Fenn. 50 (1925), 1–45. 12. Tsuji M.: On Borel’s directions of meromorphic functions of finite order, Tohoku Math. J. 2 (1950), 97–112. 13. Valiron G.: Lectures on the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949. 14. Wang S.: On the sectorial oscillation theory of Ann. Acad. Sci. Fenn. Math. Diss. 92 (1994).

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ON STRONG ASYMPTOTIC TRACTS OF FUNCTIONS HOLOMORPHIC IN A DISK

I.I. MARCHENKO Department of Mathematics, Kharkov State University, 310000 Kharkov, Ukraine Institute of Mathematics, University of Szczecin, 70-451 Szczecin, Poland, [email protected]

AND I.G. NIKOLENKO Department of Mathematics, Kharkov State University, 310000 Kharkov, Ukraine [email protected]

Abstract. For entire functions of finite order the Ahlfors classical theorem about finiteness of the set of asymptotic values is well known. In 1999, the first author introduced the concept of the strong asymptotic value of entire functions and obtained an analogue of the Ahlfors theorem for distinct strong asymptotic spots of entire functions of infinite order [14]. In 2001 we introduced the concept of a strong asymptotic spot in a point for functions holomorphic in the disk. For such asymptotic spots we obtained an analogue of the Ahlfors theorem for functions holomorphic in the disk. In the case of holomorphic functions of order MacLane proved that the set of distinct asymptotic tracts corresponding to the point is finite [12]. He also obtained an estimate for their quantity. We introduce the concept of a strong asymptotic tracts for functions holomorphic in a disk and obtain an analogue of the Ahlfors theorem for their quantity for functions holomorphic in a disk. Mathematics Subject Classification 2000: 30D30, 30D35. Key words and phrases: asymptotic values; strong asymptotic tracts

© 2004 Kluwer Academic Publishers

182 1. Notations and results We first recall the definition of an asymptotic tract of a function holomorphic in the disk, see [3], [10], [20] and [12]. To this end, suppose that for a finite value a non-empty component of the open set

can be determined uniquely for each implies that and (2) an asymptotic tract corresponding to follows by replacing the condition It is now easy to see that the set

Whenever (1) then is called If a similar definition above by

is a nonempty, connected and closed subset of the circle The set K is called the end of the tract. If K is a point then such tract is called pointwise. Two tracts and are distinct if for small enough. If they are automatically distinct. Let now be a holomorphic function in the unit disk of order here is the Nevanlinna characteristic function. MacLane has shown that the set of distinct asymptotic tracts corresponding to a point is finite, see [12]. He also obtained an estimate for their quantity. We next introduce the concept of a strong asymptotic tract for a function of an arbitrary order, holomorphic in the disk and we obtain an analogue of the classical Ahlfors theorem about finiteness of the set of strong asymptotic tracts. Definition 1. An asymptotic tract of a function holomorphic in the disk exists a continuous curve for and if the function on this curve in the sense that

where

is a strong asymptotic tract if in this disk there such that tends to a value as fast as

stand for the maximum modulus of

on the circle

183

Theorem 1. Let be a non-constant function, holomorphic in the unit disk, having finite lower order and distinct strong asymptotic tracts Then

where integer part of a real number.

and where [ ] denotes the

Corollary 1. Let be a function of finite lower order holomorphic in the unit disk such that Then the set of distinct strong asymptotic tracts is finite. We remark that the magnitudes have been investigated in [19], [18], [11]. The estimate in Theorem 1 is attained by the function

where

is the Mittag-Leffler function of finite order

see [7, p. 111].

Theorem 2. If a function of infinite lower order, holomorphic in the unit disk, has distinct strong asymptotic tracts then

where and where is the spherical area, counting multiplicities of the covering, of the image on the Riemann sphere of under Corollary 2. Let be a function holomorphic in the unit disk such that Then the set of distinct strong asymptotic tracts is finite. It is easy to see that the set of strong asymptotic tracts

is

infinite for the function Therefore, the condition is essential. Again, we remark that the magnitudes have been investigated in [2], [4], [13], [15] and [17]. The estimate in Theorem 2 is attained by the function

184 where is the entire function of infinite order defined in [9, p. 81–82]. For poinwise tracts corresponding to a point of modulus Theorem 1 and Theorem 2 are consequences of the results by the authors in [16].

2.

Proof of Theorem 1

Let function Let us choose consider the functions

be the distinct strong asymptotic tracts of the such that We

It is easy to see that the functions functions in Following [1], define

are subharmonic

where E is a Lebesgue measurable set and is its linear measure. By [1], it follows that are subharmonic functions in continuous in and logarithmically convex in for any and any fixed Moreover,

where is the symmetrically decreasing rearrangement of see [8]. As in [18], we now define

From the properties of the functions the following statement:

we conclude

Lemma 1.

The function is subharmonic in continuous in mically convex in for any and any fixed

logarithMoreover,

185

From the definitions of strong asymptotic tracts and of the functions and from the equality (4), we infer that

To prove Theorem 1 we require an additional result from [18]: Lemma 2. For each fixed number positive numbers, and exists

there exist two sequences of such that and for each there

such that for

It is sufficient to prove Theorem 1 in the special case only when the function is twice continuously differentiable in Indeed, if this is not case, we may approximate the function by a monotone family of indefinitely differentiable subharmonic functions uniformly converging to in compact subsets of see [18]. For each number consider

as in [5] and [6]. Applying the differential operator function and using the subharmonicity of the function we obtain

to the

186

We now multiply the inequality (6) by and integrate it over the intervals defined as in Lemma 2). Then

By Lemma 2, we have

Using the definition of the function from [18], we obtain

Therefore, there exists

the relations in (5) and Lemma 5

such that

Hence

Since the number

is arbitrary, we get

Taking the minimum of the right hand side with respect to we conclude that

proving Theorem 1 for It remains to settle the case function

For each

we consider the

187

Similarly to the proof of the case

Passing to the limit as

we obtain

we have

completing the proof of Theorem 1. Remark. The proof of Theorem 2 can be carried out similarly, making use of the Pòlya peaks for meromorphic functions of infinite order, see [2], [15] and [16]. Acknowledgements This research has been partially supported by the grant INTAS-99-0089. References l. Baernstein A., Integral means, univalent functions and circular symmetrization, Acta. Math. 133 (1974), 139–169. 2. Bergweiler W. and Bock H., On the growth of meromorphic functions of infinite order, J. Anal. Math. 64 (1994), 327–336. 3. Boutroux P., Sur l’indétermination d’une fonction uniforme au voisinage d’une singularité transcendente, Ann. Sci. École Norm. Sup. (3) 25 (1908), 318–370. 4. Eremenko A., An analogue of the defect relation for the uniform metric, Complex Variables Theory Appl. 34 (1997), 83–97. 5. Essen M. and Shea D.F., Applications of Denjoy integral inequalities to growth problems for subharmonic and meromorphic functions, research announcement, Proceedings of the Symposium on Complex Analysis (Univ. Kent, Canterbury, 1973), pp. 59–68. London Math. Soc. Lecture Note Ser., No. 12, Cambridge Univ. Press, London, 1974. 6. Gariepy R. and Lewis J.L., Space analogues of some theorems for subharmonic and meromorphic functions, Ark. Math. 13 (1975), 91–105. 7. Gol’dberg A. A. and Ostrovskii I.V., The distribution of values of meromorphic functions, Izdat. “Nauka”, Moscow, 1970, 591 p. (Russian) 8. Hayman W.K., Multivalent functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 48, Cambridge University Press, Cambridge, 1958. 9. Hayman W.K., Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1964. 10. Iversen F., Recherches sur les fonctions inverses des fonctions méromorphes, Thèse, Helsingfors, 1914, p. 1–67. 11. Kritov A.V., On growth meromorphic functions and analytical curves in the unit disk, Manuscript in VINITI 1657–81. Dep. (Russian) 12. MacLane G.R., Asymptotic values of holomorphic functions, Rice Univ. Studies 49 (1963), 83 p. 13. Marchenko I.I., On the growth of entire and meromorphic functions, Mat. Sb. 189 (1998), 59–84. (Russian)

188 14. Marchenko I.I., On the asymptotic values of entire functions, Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), no. 3, 133–146. (Russian) 15. Marchenko I.I. and Nikolenko I.G., Growth of meromorphic in the disk functions of infinite order, Vestnik Khar’kov. Univ., Ser. Math. Mech. 475 (2000), 113–125. (Russian) 16. Marchenko I.I. and Nikolenko I.G., On strong asymptotic spots of functions holomorphic in the disk, Mat. Fiz. Anal. Geom. 9, no. 3 (2002), 369–384. (Russian) 17. Marchenko I.I. and Nikolenko I.G., On the values of deviations for functions meromorphic in the disk, Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 2002, no. 2, 25–28. (Russian) 18. Marchenko I.I. and Scherba A.I., On the magnitudes of deviations of meromorphic functions, Mat. Sb. 181, no. 1 (1990), 3–24. (Russian) 19. Petrenko V.P., The growth of meromorphic functions, Kharkov: “Vishcha shkola”, 1978. 136 p. (Russian) 20. Valiron G., Lectures on the general theory of integral functions, Toulouse Privat, 1923.

A NEW TREND IN COMPLEX DIFFERENTIAL EQUATIONS: QUASIMEROMORPHIC SOLUTIONS

G. A. BARSEGIAN

Institute of Mathematics of National Academy of Sciences of Armenia 24-b Marshal Bagramian ave., Yerevan 375019, Armenia, [email protected] A. A. SARKISIAN Institute of Mathematics of National Academy of Sciences of Armenia 24-b Marshal Bagramian ave., Yerevan 375019, Armenia, [email protected] AND C. C. YANG Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China, [email protected]

Abstract. In complex differential equations, one usually studies analytic or meromorphic solutions. We start a new trend by considering quasimeromorphic solutions for generalized algebraic differential equations of the first order. In particular, the classical Goldberg result that any meromorphic solution of a first order algebraic differential equation must be of finite order will been extended here to solutions of first order generalized algebraic differential equations. Mathematics Subject Classification 2000: 30D30; 34M99. Key words and phrases: Key-words: complex differential equations, quasimeromorphic functions, value distribution.

© 2004 Kluwer Academic Publishers

190 1. Introduction and results

We introduce the class of functions and briefly show that some of the main results of the value distribution theory (Ahlfors version) remain valid. In the present paper, as an application of the theory, we consider solutions of generalized algebraic differential equations of the first order. Complex (in particular algebraic) differential equations have been studied very intensively for their diverse applications in many other mathematical topics. Moreover, there is a strong tendency to consider different type generalizations of analytic functions in many topics earlier connected or devoted to analytic or meromorphic functions. In the present paper we start to consider quasimeromorphic solutions of some generalizations of classes of algebraic differential equations that relate to linear or nonlinear partial differential equations. The topic of algebraic differential equations is the most developed part in the theory of complex differential equations. It studies differential equations of the form

where P is a polynomial in each of its arguments. An important part of the theory of algebraic differential equations deals with analytic or meromorphic solutions of the following system of equations:

the second equation here means that

is analytic (meromorphic), and

where We extend the class of solutions of (1.1) by deleting the second condition and by introducing the so-called quasimeromorphic solutions. Namely, we consider complex valued functions which will be called quasimeromorphic if is twice differentiable in and and furthermore, can be expressed as where is meromorphic and is a quasiconformal homeomorphism; it is just a generalization of the quasiconformal functions (see Lehto [11], p. 239). Thus, if is quasimeromorphic, then in any finite disk there are finitely many points such that and is quasiconformal in some neighborhood of

191

each type

Moreover, if

is quasimeromorphic then satisfies an equation of (see [11], p. 239), where

is the so-called formal (or complex) derivative. If is analytic then becomes the usual derivative Note that, in general, a solution of equation may not be differentiable in We will consider quasimeromorphic solutions of partial differential equations or what is the same, solutions of

Here the solutions will be K-quasimeromorphic if a positive number K > 1 (see Ahlfors [1], p. 12–13).

with

A classical result due to Gol’dberg [10] says that any solution of the equation (1.1) meromorphic in the complex plane must be of finite order. In particular, this means that both the Nevanlinna characteristic function or the Ahlfors-Shimizu spherical characteristic function (see [12]) are of finite order. In the present paper we study the growth of quasimeromorphic solutions of (1.2) in terms of the Ahlfors-Shimizu spherical characteristic function We shall give upper bounds for such solutions of (1.2) with satisfying the following properties: 1.

Instead of the polynomial P in (1.1) we consider functions which are required to be polynomial in only, with no restrictions to other arguments. Thus

2. Furthermore, for the coefficient functions we assume that for any given there is a finite set of complex values lying in a disk and there are some positive constants such that

and that

192

whenever

and

Remark 1 Note that the functions substantially generalize the corresponding algebraic differential equations (1.1), where

and are polynomials with respect to Note, that in this case for any value lying on a positive distance from zeros of the polynomial with the first term of degree T, we have

for any point

such that

and

Remark 2 In particular, as such a function consider

where are arbitrary functions meromorphic in superposition.

we may

and

means

Thus it is pertinent to use the following Definition 1 We say that is a generalized algebraic differential equation if F satisfies (1.3), (1.4) and (1.5). Theorem 1 Any solution K-quasimeromorphic in algebraic differential equation is of finite order

of a generalized

Immediately, according to Remark 2 we have Corollary 1 Let be a polynomial with respect to variables and which has coefficients that are meromorphic functions in with respect to that is can be expressed as Then any meromorphic solution in of the equation is of finite order Note that the equation in Corollary 1 becomes an ordinary algebraic differential equation if are constants. Example The following is an example of such a generalized equation, which admits a solution

193

Corollary 2 (Gol’dberg, [10]) Any solution meromorphic in order algebraic differential equation must be of finite order. Definition 2

We say that a function in if is for all where

of a first

given in the complex plane is in the disks

It is easy to see that Theorem 1 immediately follows from the following more general result, which deals with solutions. Theorem 2 Any solution algebraic differential equation is of finite order.

in

of a generalized

We next consider equations involving not only but also Given a quasimeromorphic function in let be a polynomial with respect to variables Then there exists a polynomial and a number such that

for all As an example, we may take, for instance, where are polynomials with degrees and We may also consider equations with numerous other conditions, say, taking into account, in the previous formula not only degrees but also coefficients of polynomials. Substituting in the variable by we obtain another type of generalized algebraic differential equation

where it is assumed that the corresponding coefficients satisfy (1.3), (1.4) and (1.5). Theorem 3 Let be a polynomial satisfying (1.7) and let be a K-quasimeromorphic function that satisfies a generalized algebraic differential equation (1.8) and the conditions (1.3), (1.4) and (1.5). Then is of finite order. Theorem 2 is an immediate consequence of Theorem 3 as one can see by taking and observing (1.7). 2. Lemmas and proofs of the results

We shall use the theory of covering surfaces developed by L. Ahlfors and the so called method of estimating derivatives. Ahlfors theory was first applied

194

to algebraic differential equations in [3]. The method has been derived in [3] from the so-called proximity property of see [2]. The method easily results in some new types of problems and generalizations. For instance, a new stability phenomenon was studied in [6] and geometric locations of of solutions of equations in [7]. New generalizations and further applications of the method can be found in [5], [8] and [9], The classical Ahlfors theory of covering surfaces gives a metric-topological approach to the main conclusions of the Nevanlinna theory. A great advantage of this theory is that it enables one to extend geometric main conclusions of the Nevanlinna value distribution theory to the more general classes of K-quasimeromorphic functions This is what we shall prove next. We note here the connection between the main magnitudes and where is the spherical area of and is the spherical length of the boundary of This connection plays a crucial role in the Ahlfors theory of covering surfaces. Lemma 1 ([12, Ch. 13]) Let Then for any arbitrary

holds for

where

be a K-quasimeromorphic function in

is a set of finite logarithmic measure.

It follows from (2.1) that

The above condition is called a regular exhaustion. It appears that for function satisfying the condition above, geometric main conclusions of value distribution theory can be derived from the Ahlfors theory. By adopting Ahlfors arguments we easily conclude now that the condition of regular exhaustion is fulfilled for much larger classes of functions. Indeed, note that Ahlfors derives (2.1) from the following inequality

Now assume that the function is for all such that Denote by the set of values

in all the disks is a constant, for which

195

Then by following the idea due to Ahlfors, we obtain by integrating (2.3) over the set of values such that

This means that the set quently, Lemma 2 Let any arbitrary

for

be a

where

is of finite logarithmic measure. ConseThus we have function in

Then for

is a set of finite logarithmic measure.

Lemma 2 means that all functions are regularly exhausting. For further applications, we need the first main theorem of Ahlfors theory in a special case. Let be the image of under mapping by a K-quasimeromorphic function and let be the spherical image of on the Riemann sphere. Let D be a domain on the Riemann sphere, be the spherical area of D and be the sum of areas of all parts of lying over D. The magnitude

is called the “averaged number of sheets of the surface

lying over D”.

Lemma 3 (First Fundamental Theorem of Ahlfors [12]) Let quasimeromorphic function in Then

be a K-

where

From the three lemmas above we conclude the following key lemma: Lemma 4 Let be a K-quasimeromorphic function in arbitrary constant

Then for any

196 where and mic measure. Moreover, if is a then for any arbitrary

where logarithmic measure.

and

where

is a set of finite logarithfunction in

where

is a set of finite

We shall apply Lemma 4 to prove Theorem 2. To this end, recall first that all roots of an algebraic equation

lie in a disk equation (1.2) we get for any value

see [13]. Applying this to the

We choose now a disk mark 1, such that the distance

R defined as in Rebetween D and all the points are positive. For all values we consider their preimages so that By taking into account (1.4), (1.5), (2.6) and (2.7) we have that for any

Recalling the Jacobian of

we note that the element of the spherical area of mapping is here is the Euclidean area element. Therefore

and

where

197

Now, taking into account (2.8) and (2.9), we have

Starting with a value

we have

and by the inequality (2.6) in Lemma 4,

on

Finally, we conclude that starting with a new value and only, the following inequality holds:

depending

for Since the characteristic function is monotonically increasing and has a finite logarithmic measure, it follows that for all values greater than some value

completing the proof of Theorem 2.

198

To prove Theorem 3 it is enough to note that, by arguing similarly as in the proof of the inequality (2.7), we get an upper bound for Then, due to (1.7), we get an upper bound for as well. Then Theorem 3 follows by repeating similar arguments as in the proof of Theorem 2. Examples of equations with quasimeromorphic solutions. Let T be an affine non-degenerating transformation that can be expressed in a complex form as

Denote its inverse transformation by

and by

the transformation

Note that if morphic function we have

Now let

that is

is a meromorphic function, then is a quasimeroCalculating formally the derivative of

denote a meromorphic solution of the Riccati equation:

where are polynomials. By applying the transformation to both sides of (2.11) and by (2.10), we obtain

where the right-hand-side may be rewritten as a function In turn, applying transformation we can rewrite as Thus the function is a quasimeromorphic solution of the following equation

It is easy to check that for the function

199

the conditions (1.3), (1.4) and (1.5) in Theorem 1 are fulfilled. Acknowledgements The first author would like to thank the Abdus Salam International Centre for Theoretical Physics (ICTP) for kind hospitality and support. The third author’s research was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (project no. HKUST 1643/00P). References 1. Ahlfors L., Lectures on quasiconformal mappings, Van Nostrand C. Inc., TorontoNew York-London, 1966. of meromorphic functions, Math. 2. Barsegian G., The property of closeness of Sb. (N.S.) 120(162) (1983), 42–67. (Russian) 3. Barsegian G., Estimates of derivative of meromorphic functions oh sets of J. London Math. Soc. 34 (1986), 534–540. 4. Barsegian G., Applications of L. Ahlfors theory of covering surfaces to the study of algebraic differential equations of the second orders, Differentsial’nye Uravneniya 23 (1987), 341–344. 5. Barsegian G., Begehr H. and Laine I., Stability phenomenon for generalizations of algebraic differential equations, Z. Anal. Anwendungen 21 (2002), 495–503. 6. Barsegian G., Laine I. and Yang C.C., Stability phenomenon and problems for complex differential equations with relations to shared values, Mat. Stud. 13 (2000), 224–228. 7. Barsegian G., Laine I. and Yang C.C., Locations of values of meromorphic solutions of algebraic differential equations, Complex Variables Theory Appl. 46 (2001), 323– 336. 8. Barsegian G., Laine I. and Yang C.C., On a method of estimating derivatives in complex differential equations, J. Math. Soc. Japan 54 (2002), 923–935. free 9. Barsegian G.A. and Yang C.C., On the cross-road of value distribution, boundary theories and applied mathematics, preprint. 10. Goldberg A.A., On one-valued integrals of differential equations of the first order, Ukrain. Mat. Ž. 8 (1956), 254–261. 11. Lehto O. and Virtanen K.I., Quasiconformal mappings in the plane, Springer-Verlag, New York-Heidelberg, 1973. 12. Nevanlinna R., Eindeutige analytische Funktionen, Springer, 1936. 13. Pólya G. and Szegö G., Aufgaben und Lehrsätze aus der Analysis. Zweiter Band. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954.

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ON THE FUNCTIONAL EQUATION P(F) = Q(G)

HA HUY KHOAI Institute of Mathematics, Hanoi, Vietnam AND YANG C.C. Department of Mathematics, Hong Kong University of Science and Technology, Kowloon, Hong Kong, China, mayang@uxmail. ust.hk

Abstract. We prove that for a generic pair (P, Q) of polynomials P of degree and Q of degree where are satisfying some conditions, for meromorphic functions implies We also give another proof of the statement saying that a generic polynomial of degree at least 5 is a uniqueness polynomial for meromorphic functions. Mathematics Subject Classification 2000: 32H20, 30D35. Key words and phrases: functional equation, uniqueness polynomial, meromorphic function, unique range set

1. Introduction

As a simple application of his value distribution theory, Nevanlinna proved that a nonconstant meromorphic function (in the complex plane is uniquely determined by the inverse image of five distinct values (including possibly the infinity), ignoring multiplicities. It follows that if two nonconstant meromorphic functions satisfy for five distinct values and then Gross [3] extended the study by considering pre-images of a set and posed the question “Is there a finite set S so that an entire function is determined uniquely by the pre-image of the set S, counting multiplicities?” Concerning developments on this question and its related problems, we refer the reader to a survey article by Hua

© 2004 Kluwer Academic Publishers

202

and Yang [5]. In particular, as a connection to the study of the uniqueness problem, Li and Yang [6] introduced the following definition: Definition 1. A nonconstant polynomial is said to be a unique polynomial for entire functions (meromorphic functions) if whenever for two non-constant entire (meromorphic) functions it implies that Such a polynomial will be called UPE and UPM, respectively. To formulate some recent results on UPE and UPM, let us recall the following notations: (i) A statement S(P) holds for a generic polynomial of degree if there exists a proper algebraic subset such that S(P) holds for whenever (ii) A statement S(P,Q) holds for a generic pair of polynomials (P,Q), P of degree Q of degree if there exists a proper algebraic subset such that S (P, Q) holds for

whenever In [6] it is proved that a generic polynomial of degree 4 is a UPE, and any polynomial of degree four or less can’t be UPM. Later on, Yang and Hua ([9]) showed Theorem A. Let and

Then P is a UPM, if is not equal to 0.

The following result was obtained by Shiffman ([8]) : Theorem B. If is a UPM.

is a generic polynomial of degree at least 5, then P

Now we call a poynomial a strong uniqueness polynomial for meromorphic functions if implies for all nonconstant meromorphic functions and a nonzero constant Recently, H. Fujimoto obtained the following result: Theorem C. If P is a strong UPM.

is a generic polynomial of degree at least five, then

203

In this paper, we study a more general question. Namely, we study meromorphic solutions of the functional equation where P, Q are two given nonlinear polynomials. Without loss of generality for a pair (P, Q) of polynomials given by (1), (2), we assume that Then we shall prove the following Theorem 1. For a generic pair (P, Q) of polynomials, the functional equation has no pair of nonconstant meromorphic function solutions satisfy one of the following conditions: i) ii) iii)

and

if

The main tool in this paper will be the singularity theory. Moreover, we show that by our method one can obtain Theorems A, B and C as well. 2. Proof of Theorem 1

For simplicity, we denote Let (P, Q) be a pair of polynomials given by (1), (2). Consider the curve in defined by the following equation:

We call

the associated curve of the pair (P, Q).

Lemma 1. There exists an algebraic subset point then the curve Proof. Let

be a singular point of

Then the point following algebraic subset

Let

such that if the is non-singular.

We have

belongs to the

204

be the natural projection. We set is a finite set. Hence, the map

is proper. Therefore, proving Lemma 1.

Obviously, if

is an algebraic subset, and

Lemma 2. Assume that the curve of the curve is given by

then

is a proper subset,

is non-singular. Then the genus

Proof. The proof of Lemma 2 uses standard arguments of algebraic geometry only. However, for the convenience of the reader, we describe the reasoning in detail. First of all, we recall the notion of the Newton polygon for later use. Let C be an algebraic curve of degree in defined by the equation

The set

is called the support of We denote by the convex hull of the set {Supp We now denote by the Newton polygon of at infinity, i.e., the part of the boundary of which does not contain the segments [0,.] and [.,0] of the coordinate axes. Then the Newton main part at infinity of is defined by

Let be an edge of and be the main part of with respect to Then is said to be non-degenerate if the system of equations has no solutions in If for all is non-degenerate, then we say that is non-degenerate with respect to the Newton polygon at infinity. Finally, is convenient, if contains two points with the coordinates (0,.) and (.,0). A result of ([4]) says that if the curve C is non-singular, and its Newton polygon is non-degenerate and convenient, then the genus

205

of C is equal to the number of points with entire coordinates, which are contained inside of For our case, the curve has a very simple Newton polygon: this is the triangle with the vertices at (0,0), It is clear that the Newton polygon of is non-degenerate and convenient, and the genus of can be calculated by the above mentioned result of By an elementary argument we see that the number of entire points inside of the triangle is

where

is the greatest common divisor of

proving Lemma 2.

Proof of Theorem 1. Let (P, Q) be a generic pair of polynomials such that the associated curve X is non-singular. It suffices to take (P, Q) with as in Lemma 1. Assume that the equation (3) has a solution where are nonconstant meromorphic functions. Then the mapping

has the image contained in X. If X is hyperbolic, then From this it follows that a contradiction with the genericity of the pair (P, Q). Hence, it remains to prove that under the assumptions of Theorem 1, the associated curve X is hyperbolic. By Lemma 2, the genus of X is given by

An easy computation shows that if satisfy the hypothesis of Theorem 1, then and, therefore, X is hyperbolic. 3. Some remarks

Remark 1. We can give more concrete conditions for the meaning of “genericity”. For example, let

Corollary 1. Assume that for and satisfy the conditions of Theorem 1. Then for meromorphic functions implies that

206

Proof. From the condition for it follows that the associated curve of (P, Q) is non-singular. Then Corollary 1 is proved by using Lemma 2 and the proof of Theorem 1. Remark 2. By using our method we give another proof of Theorems B and C saying that a generic polynomial of degree at least 5 is a strong uniqueness polynomial. Let C be a nonzero constant, and let X be the curve defined by the following equation We are going to show that for a generic polynomial P of degree if then X is hyperbolic, and for C = 1, X has two irreducible components: the curve and a hyperbolic curve. (1) The case For a generic polynomial P, the equation has distinct roots and the curve X has a singularity at the points only such that

By the genericity of P and at most for one point The singularity of X at this point is given by

which is an ordinary double singularity. First we claim that X is irreducible. Otherwise, where are two curves of degree and of multiplicities at equal to By Bezout’s theorem ([1]) (because the tangent cone of X is the union of distinct lines). Since we must have a contradiction. Since X is irreducible, the genus of X is given by the following formula ([7]):

where

is defined by

where is the Milnor number the number of irreducible branches at that When we have hyperbolic.

and where is From (7), it follows and the curve Y is

207 (2) The case C = 1. In this case the curve X can be decomposed in two curves: the curve and the following curve of degree

By an argument similar to that one in Lemma 1, we can show that when P is generic, is non-singular. Moreover, the Newton polygon of is nondegenerate and convenient. Then the genus of is equal to Therefore, is hyperbolic, if Remark 3. Let now be a polynomial satisfying the conditions of Theorem A. By an argument similar to that one in Remark 2, we can prove that in this case the associated curve is decomposed in two irreducible curves, where and is a non-singular curve with a non-degenerate and convenient Newton polygon. Theorem A now immediately follows. Acknowledgements The work of the second author was partially supported by a UGC grant of Hong Kong. References 1. Beauville A., Surfaces algébriques complexes, Astérisque 54, 1978. 2. Fujimoto H., On uniqueness of meromorphic functions sharing finite sets, Preprint. 3. Gross F., Factorization of meromorphic functions and some open problems, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), pp. 51–67, Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977. 4. A.G., Newton polyhedra and the genus of complete intersections, Funktsional. Anal. i Prilozhen. 12 (1978), no. 1, 51–61. (Russian) 5. Hua X.H. and Yang C.C., Uniqueness problems of entire and meromorphic functions, Bull. Hong Kong Math. Soc. 1 (1997), no. 2, 289–300. 6. Li P. and Yang C.C., Some further results on the unique range sets of meromorphic functions, Kodai Math. J. 18, 1995, 437–450. 7. Milnor J., Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. 8. Shiffman B., Uniqueness of entire and meromorphic functions sharing finite sets, Preprint. 9. Yang C.C. and Hua X.H., Unique polynomials of entire and meromorphic functions, Mat. Fiz. Anal. Geom. 4 (1997), no. 3, 391–398.

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VALUE DISTRIBUTION OF THE HIGHER ORDER ANALOGUES OF THE FIRST PAINLEVÉ EQUATION

YUZAN HE Institute of Mathematics, Academia Sinica, Beijing 100080, China

Abstract. The higher order analogues of the first higher order Painlevé equation arise from an exact reduction of the higher order In this paper we examine value distribution properties of the meromorphic solutions of Mathematics Subject Classification 2000: 30D35; 34M55. Key words and phrases: Painlevé equation; meromorphic solution; value distribution

1. Introduction

Let us consider the Korteweg–de Vries equation, abbreviated hierarchy in the form

where

and

The transformation

reduces the equation [2]

to the following algebraic differential equation

where the inverse of D, i.e.

© 2004 Kluwer Academic Publishers

and

denotes

210

For

Therefore we get

we have

which is nothing but the first Painlevé equation, and we call higher order analogue of the first Painlevé equation of order For we have

the

In [2], V. I. Gromak investigated and obtained several basic analytic properties which are similar to that of the first Painlevé equation. Recently Y. Li and W. Yuan [4] got some results on value distribution properties for the special case the higher order analogue of the first Painlevé equation. In this paper, we discuss the higher-order analogues of the first Painlevé equation in general case. We first show that there is only one leading term in and get its precise expression. We then obtain some value distribution properties of the meromorphic solutions of Through this paper, we use the standard notations of the Nevanlinna theory such as etc [3]. 2. Main Results

Proposition 2.1 The differential polynomial of the left hand side of has only one leading term of the form where All coefficients in are constants except for the term Proposition 2.2 For

rational solutions cannot exist.

Proposition 2.3 All poles of the meromorphic solution of ble. Theorem 2.4 Let for all

be a meromorphic solution of

are douThen

Corollary 2.5 Let be a meromorphic solution of Then there exists a set E in of finite linear measure, such that for all we have

Theorem 2.6 Let

be a meromorphic solution of

The

211 where

3. Proofs of theorems

Proof of Proposition 2.1. Recall the definition of the degree of a differential monomial defined by The total degree of a differential polynomial with constant coefficients is It is easy to show that the total degree of a differential polynomial remains unchanged in differentiation and in the reversed operation as well. We are going to prove Proposition 2.1 by induction. For we have

Hence, has only one dominant term whose degree is greater than the degree of the remaining terms. So does Therefore the conclusion in Proposition 2.1 holds for For we have

Clearly, the degree of the leading term of is greater than the degree of all other terms. Proposition 2.1 also holds for Now we may assume that the conclusion holds for and we write in the form

where the total degree of

and for each term in

212

We wish to show that the assertion holds for

In fact,

and Since is and, the total degree of and of by the above argument, derivating or integrating a differential polynomial doesn’t change the total degree, the total degree of is Besides for each term in

where

Noting the expression of we have Now

for each term in

can be rewritten in the form

where is a differential polynomial with constant coefficients and its total degree for each term in Proof of Proposition 2.2. Let be a rational solution of where and are polynomials with degree and respectively. Then as Substituting into (3.1), we have

213

Set where tion in

of degree has one dominant term the leading term is just In both cases we have which contradicts (3.2). Proof of Proposition 2.3. Suppose Then is a pole of

is a pole of

is a rational funcIf only. If then as with multiplicity

with multiplicity Obviously, is a pole of with multiplicity and a pole of with multiplicity if is still a differential polynomial. Let be a pole of the solution of with multiplicity Substituting into the differential polynomial we observe that is a pole of with multiplicity By definition

According to the above argument, is a pole of each term on the right hand side of (3.3) with multiplicity and respectively. Since solves comparing both sides of we get To prove Theorem 2.4, we need the following lemmas. Lemma A (J. Clunie [1]) Let tion of the differential equation

be a transcendental meromorphic solu-

where are differential polynomials with total degree

214

and respectively and the coefficients phic functions satisfying then

and

are meromorfor all and If

Lemma B (A. Z. Mohon’ko and V. D.Mohon’ko [5]). Let

be an algebraic differential equation and a transcendental meromorphic solution of (3.4). If a constant doesn’t solve (3.4), then

Proof of Theorem 2.4. Suppose tion 2.1, can be rewritten

is a solution of

From Proposi-

By (3.1), the total degree of the right hand side of the above equation is By Lemma A, it follows

and so

Since

for each term in

any constant is a solution of solve By Lemma B, we have

and so

doesn’t

implying that Proof of Corollary 2.5. Let be a solution of Theorem 2.4 and the Nevanlinna’s first fundamental theorem

By

215

Then where

is a set in

with finite linear measure. Similarly,

where

is a set in

of finite linear measure. Set

Then

Proof of Theorem 2.6. Derivating (3.1), we get

where

Since we have as well. Hence for each term in the right hand side of the above equation, there is at least one for each From (3.5), we have

Noting we get

and

216

By the Nevanlinna’s first fundamental theorem

and so Since all poles of

are double, we have

Hence

Further, and noting (3.5), we have

and so

Acknowledgements

Partially supported by the National Nature Science Foundation of China (Grant No. 19971091) and the Väisälä Fund of the Finnish Academy of Sciences and Letters.

217

The author would like to express his thanks to the referee for reading the manuscript carefully and making valuable comments. References 1. Clunie, J., On integral and meromorphic functions, J.London Math. Soc. 37 (1962), 17–27. 2. Gromak, V. I., The first-higher order Painlevé equation, Differ. Equations 35 (1999), 37–41. 3. Laine, I., Nevanlinna theory and complex differential equations, W. de Gruyter, Berlin–New York, 1993. 4. Li, Y. Z. and Yuan, W. J., On analytic properties of order analogue of the first Painlevé equation, to appear. 5. Mohon’ko, A. Z. and Mohon’ko, V. D., Estimates of the Nevanlinna charateristics of certain classes of meromorphic functions and their applications to differential equations, Sibirsk Mat. Zh. 15 (1974), 1305–1322. (Russian) 6. Schubart, H., Zur Wertverteilung der Painlevéschen Transcendenten, Arch. Math. 7 (1956), 284–290. 7. Shimomura, S., Value distribution of Painlevé transcendents of the first and second kind, J. Anal. Math. 82 (2000), 333–346.

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SOME FURTHER RESULTS ON THE FUNCTIONAL EQUATION P(F) = Q(G)

CHUNG-CHUN YANG Department of Mathematics The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, [email protected] AND PING LI Department of Mathematics University of Science and Technology of China Hefei, Anhui 230026, China, [email protected]

Abstract. In this paper we study further the solvability, i.e. the existence of nonconstant meromorphic solutions of the functional equation where P, Q are two polynomials in Some criteria for such questions in terms of the degrees of P and Q as well as their coefficients are established. Mathematics Subject Classification 2000: 39B32, 30D30. Key words and phrases: functional equations; Hilbert problems; meromorphic functions

1. Introduction The famous unsolved Tenth Problem of Hilbert asks: To find an algorithm for finding all integral solutions of where is a polynomial in with integer coefficients. Later, in this connection, Thue proved that (a non-zero integer) has only a finite number of integral solutions, whenever is a homogeneous polynomial of degree in with integer coefficients, irreducible in the field of rational functions. For non-homogeneous the corresponding question would be more

© 2004 Kluwer Academic Publishers

220

difficult to handle. Let denote a plane cubic: where are integers. The Mordell-Weil Theorem [4] says that there exists a finite set of rational points on the such that every other rational point (i.e. points with both coordinates rational) can be “derived” from these, by the so-called tangent-chord process. It is natural to study the analogue the of Hilbert’s Tenth Problem in the field of meromorphic functions. More specifically, we ask what forms of the equation F a polynomial in and with complex coefficients) may or may not have nonconstant meromorphic solutions i.e. Earlier in 1920s, as a simple application of Nevanlinna’s value distribution theory, Nevanlinna himself proved that a nonconstant meromorphic function (in the complex plane) is uniquely determined by the inverse images of five distinct values (including infinity), ignoring multiplicities: If two meromorphic functions satisfy for five distinct values then In 1982, Gross-Yang [1] extended the study of the so-called uniqueness or unicity of meromorphic functions by considering the pre-image of a finite set of a function, i.e., what kind of finite set S such that whenever of (ignoring multiplicities) for two nonconstant meromorphic functions and implies We refer the reader to [3] and to the survey article [7] for more developments of unicity of meromorphic functions or mappings and related topics. In particular, as a connection to the study of the unicity problem, Li and Yang [3] introduced the following: Definition 1. A nonconstant polynomial is said to be a unique polynomial for entire (meromorphic) functions, if whenever for two nonconstant entire (meromorphic) functions and For the characterizations of such polynomials, we refer the reader to [6] or [8]. Recently, Ha and Yang [2] generalized the above studies by considering pairs of non-linear polynomials such that the only meromorphic solutions that satisfy are constants. By using, mainly, the singularity theory and the concept of genus for algebraic curves (in algebraic geometry), the following two results were obtained in [2]: Theorem A. For a generic pair (P, Q) of polynomials, the functional equation has no solution in the family of nonconstant meromorphic functions, if and satisfy one of the following conditions: i) ii) iii)

and

221

Remark. Here a statement S(P, Q) holds for a pair of generic polynomials of degrees and respectively, if there exists a proper algebraic subset such that S(P, Q) holds for

whenever The following is an application of Theorem A, which also gives a more concrete condition for the meaning of “generity”. Theorem B. Let

and

be defined as in Theorem A. Let

with and Assume further that for and and satisfy one of the conditions of Theorem A. Then for meromorphic functions the equation implies that both and are constants. Remark. A well-known result of Picard, see [5], p. 272, asserts that if the genus of a polynomial equation is greater than one, then there do not exist nonconstant meromorphic solutions (in the plane) such that Thus Theorem A essentially follows by computing the genus of the curve Generally, however, it is not easy to calculate the genus of such a curve. In this paper, we shall utilize value distribution theory to obtain some criteria, without calculating of the genus. Furthermore, our new results enable us to resolve the case when and not covered by Theorem A. When some necessary and sufficient conditions are derived to ensure the existence of nonconstant meromorphic functions that satisfy At the end, we propose a conjecture on meromorphic functions that are analogues to Hilbert’s Tenth Problem. 2. Lemmas

Lemma 1. Let and be two polynomials of degrees and respectively, where If there exists a zero of such that the equation has no multiple roots, and if there exist two nonconstant meromorphic functions and such that then is the greatest common factor of and and is a simple

222

zero of

Furthermore, the following inequalities hold:

where

is not a zero of

Proof. Suppose that is a zero of such that the equation has no multiple roots. If there exist two nonconstant meromorphic functions and such that

then we have This leads to

where is a quantity satisfying except possibly for a set of finite linear measure of Let be the greatest common factor of and Then there exist two relatively prime integers and such that for

Suppose that is a pole of of multiplicity Then from (6), is also a pole of of multiplicity of and i.e., Since and are relatively prime to each other, we see that divides Hence the multiplicity of any pole of is at least This implies

Since

is a root of such that

This and (6) yield

there exists a polynomial and

of degree

223

Let

denote a complex number that is not a zero of Since has no multiple roots, by Nevanlinna’s second fundamental theorem, we have

By (11),

By (9), we now have

The above inequalities and (7) lead to

which implies that

Since the above inequality is equivalent to On the other hand, by the hypothesis of Lemma 1, we have and thus Note that We then deduce that and Therefore, all the above inequalities become equalities. Hence, we get (1), (2), (3) and (4). Furthermore, which is the greatest common factor of and Since we see from (10) that is a simple zero of Let now be the meromorphic function defined by

From (1) and (3), we get

Therefore, we get from (6), and thus Hence On the other hand, from (4) we see that “almost all” poles

224

of have multiplicity and these poles are also the poles of with multiplicity Therefore, “almost all” the poles of are not poles of From (11), we have

where is a polynomial of degree that is a zero of Then by (11), we have has no multiple roots, we see that from (15) that is not a pole of Therefore, and so (5) follows. Lemma 2. Let

and

respectively, where

Suppose Since It follows

be two polynomials of degrees If there exists a multiple zero

and

of

such

that the equation has no multiple roots,, and if there exist two nonconstant meromorphic functions and such that then and is a zero of with multiplicity 2. Furthermore, the following inequalities hold:

where Proof. Since of degree

is not a zero of is a multiple zero of such that

there exists a polynomial and

Similar to the proof of Lemma 1, we can prove the conclusions of Lemma 2.

and

to get

Lemma 3. Let and be two polynomials of degrees and respectively, where If there exist two different zeros and of such that each of the equations has no multiple roots, and if there exist two nonconstant meromorphic functions

225 and of

such that then and Furthermore, the following inequalities hold:

where

are simple zeros

is not a zero of

Proof. Since of degree

If such that

is a zero of such that

there exists a polynomial and

then there exists a polynomial and

of degree

From this and similar arguments as used in the proof of Lemma 1, we can prove that to get Lemma 3. If then and have no common zeros, and both have no multiple zeros. By Nevanlinna’s second fundamental theorem, we have

From this we can prove that The conclusions of Lemma 3 follow, again by similar arguments as in the proof of Lemma 1.

3. Statements and proofs of main results Theorem 1. Suppose that and are two polynomials of degrees and respectively. Then there exist no nonconstant meromorphic functions and satisfying if

and

satisfy one of the following conditions:

226

and is not the greatest common factor of and and there exists a zero of such that has no multiple roots. (2) and there exist two different zeros and of such that have no multiple roots. (3) and there exists a multiple zero of such that has no multiple roots. (4) and there exists a zero of and a multiple zero of such that both have no multiple roots. (5) and there exist three zeros of such that the equations have no multiple roots. and there exists a zero of with multiplicity such (6) that has no multiple roots. (1)

Proof. From Lemma 1 we see that there exist no nonconstant meromorphic functions and satisfying (16) when and satisfy one of the conditions (1) or (3). If and satisfy the condition (2), and if there exist two nonconstant meromorphic functions and satisfying (16), then by Lemma 1 we would have and Hence which is impossible. Similarly, from Lemmas 2 and 3 we see that there exist no nonconstant meromorphic functions and satisfying (16) when and satisfy one of the conditions (4), (5) or (6). Corollary 1. Suppose that and are two polynomials of degrees and respectively. If or then there exist some nonconstant meromorphic functions and satisfying Proof. The conclusion is obvious for rewrite the functional equation

If

and in the form:

then we can

where

are complex numbers and If two of are equal, say, then (17) is equivalent to where Clearly, (17) can have nonconstant meromorphic solutions. If are distinct, then there exists an elliptic function satisfying Hence, in this case, (17) has nonconstant meromorphic solutions.

227

If

and in the form

then we can rewrite the functional equation

where of

are complex numbers and If two are equal, say, then (18) is equivalent to where Therefore, by the conclusion for the case we see that (18) has nonconstant meromorphic solutions. Suppose that are distinct. Then (18) is equivalent to

where

By the conclusion for the case we see that (18) and thus has nonconstant meromorphic solutions. If then and can be expressed as and respectively. Let

Then the equation

is equivalent to

By the conclusion for the case and the above equation has nonconstant meromorphic solutions. Hence the equation has nonconstant meromorphic solutions as well. For the special case

and

we have the following

228

Theorem 2. Suppose that and are two polynomials of degrees 3 and 4, respectively. Then the functional equation has nonconstant meromorphic solutions and if and only if there exists a zero of such that = 0 has at least one multiple root, and

where

are complex numbers such that the polynomial

has at least one multiple zero. Proof. Let

and

be the zeros of If both and have no multiple roots, then by Theorem 1 we see that the functional equation has no nonconstant meromorphic solutions and If for one of and say for has at least one multiple root, then there exist complex numbers and such that (19) and (20) hold. Therefore, the functional equation is equivalent to

Let

Then (22) can be rewritten as

If the polynomial defined in (21) has no multiple zeros, then there exist no nonconstant meromorphic functions and satisfying (23). Therefore, has no nonconstant meromorphic solutions. If the polynomial defined in (21) has at least one multiple zero then (23) is equivalent to

where are some complex numbers and By Corollary 1, the above equation has nonconstant meromorphic solutions. This completes the proof of Theorem 2.

229

For the case by Theorem 1 and by an argument similar to that in the proof of Theorem 2, we get the following Theorem 3. Suppose that both are polynomials of degree 4, and Then the functional equation has nonconstant meromorphic solutions and if and only if there exists a zero of such that has at least one multiple root, and

where

are complex numbers such that the polynomial

has at least one multiple zero. 4. Applications and a conjecture

Example 1. There exist no nonconstant meromorphic functions such that

and

For polynomials and we see that has roots and It is easy to see that both have no multiple roots. Hence, by Theorem 2, there exist no nonconstant meromorphic functions and satisfying Example 2. There exist no nonconstant meromorphic functions such that

and

where a, b are nonzero complex numbers. Since the polynomial b) has no multiple zeros. Hence, by Theorem 2, the functional equation has no nonconstant meromorphic solutions. Example 3. Let

where and are complex numbers. Then there exist two nonconstant meromorphic functions and such that

230

Example 4. Let

Then there exist two nonconstant meromorphic functions and such that Indeed, this is true for and By Lemmas 1, 2 and 3, we can get the following two results concerning the existence of meromorphic solutions of certain non-linear Briot-Bouquet type differential equations. Theorem 4. Suppose that and are two polynomials of degrees and respectively. If and if there exists a zero of such that the equation has no multiple roots, then there exists no nonconstant meromorphic function satisfying or where is positive integer. Remark. The fact that the equation has transcendental meromorphic solutions for suitable complex numbers and shows that the condition is necessary in Theorem 4. Theorem 5. Suppose that both and are polynomials of degree If there exists a multiple zero of such that the equation has no multiple roots, or if there exist two different simple zeros and of such that each of the equations has no multiple roots, then there exists no nonconstant meromorphic function satisfying or where is positive integer. Finally, we would like to pose the following conjecture analogous to the famous Mordell conjecture in number theory: Conjecture. If a diophantine equation, where F is a polynomial in and of degree greater than 3 with rational coefficients, has no or at most finitely many rational solutions, then the corresponding equation F has none or at most finitely many nonconstant meromorphic solutions and i.e. Note that here any two pairs of solutions and of the equation are considered to be the same, whenever is an arbitrary nonconstant entire function. References 1.

Gross F. and Yang C.C., On preimages and range sets of meromorphic functions, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 1, 17–20. 2. Ha H.K. and Yang C.C., On the functional equation this volume. 3. Li P. and Yang C.C., On the unique range set of meromorphic functions, Proc. Amer. Math. Soc. 124 (1996), 177–185. 4. Mordell L.J., Diophantine Equations, Academic Press, New York, 1969. 5. Nevanlinna R., Uniformiserung, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953.

231 6. Shiffman B., Uniqueness of entire and meromorphic functions sharing finite sets, Complex Variables Theory Appl. 43 (2001), no. 3-4, 433–449. 7. Yang C.C., Unicity and factorization of meromorphic functions, Proceedings of the Second Asian Mathematical Conference 1995 (Nakhon Ratchasima), 64–84, World Sci. Publishing, River Edge, NJ, 1998. 8. Yang C.C. and Hua X.H., Unique polynomials of entire and meromorphic functions, Mat. Fiz. Anal. Geom. 4 (1997), 391–398.

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RECENT TOPICS IN UNIQUENESS PROBLEM FOR MEROMORPHIC MAPPINGS

YOSHIHIRO AIHARA Numazu College of Technology, 3600 Ooka, Numazu, Shizuoka 410–8501, Japan, [email protected]

Abstract. In this article, we give a survey of recent development of uniqueness problem for meromorphic mappings. In particular, we give an overview of its applications to constructing problem of hyperbolic hypersurfaces in complex projective spaces. Furthermore, we give a review on some recent researches on unique range set for meromorphic functions of one complex variable. Mathematics Subject Classification 2000: 32H30; 32H04 Key words and phrases: Algebraic dependence, uniqueness problem, finiteness theorem, meromorphic mapping

Introduction The aim of the present article is to give a survey of recent development of the uniqueness problem for meromorphic mappings and related subjects. The study of the uniqueness problem of meromorphic mappings under condition on the preimages of divisors was first studied by G. Pólya and R. Nevanlinna ([29] and [38]). They proved the following famous five points theorem: Let and be nonconstant meromorphic functions on If for five distinct points in then and are identical (see also [31]). Such a theorem may be called an absolute unicity theorem in as much as the condition concerns set equality. On the other hand, G. Pólya-R. Nevanlinna also have a relative unicity theorem. These theorems add the requirement that, for each inverse image in question, and take their value there with the same multiplicity. For example, the following four points theorem is well-known: Let and be as above. If as

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234

divisors for distinct four points in then either or for an automorphism T in Aut determined by Until now, many researchers have studied unicity theorems for meromorphic functions on as well there have been many contributions in the multidimensional case. Some of relevant papers will be listed in references (cf. [1], [2], [3], [4], [9], [10], [12], [13], [14], [17], [23], [40], [43], [48] and [52]). In this article, we mainly focus our attention on the uniqueness problem for meromorphic mappings from an analytic covering space over into projective algebraic manifolds. We consider both of the absolute case and the relative case. We also give results on unique range sets for meromorphic function of one complex variable which are studied with activity. Furthermore, we give applications to construct hyperbolic hypersurfaces in complex projective spaces. In this article, for a convenience to the reader, we first give some known facts in the Nevanlinna theory of meromorphic mappings and in the Kobayashi theory of hyperbolic manifolds. In §2, we mainly discuss finiteness theorems for meromorphic mappings which are due to H. Fujimoto and Y. Aihara. In §3, we deal with the propagation of algebraic dependence and its applications. In particular, we deal with a unicity theorem in the absolute case from this point of view. We note that some interesting dependence theorems of different type were obtained by H. Fujimoto [17]. In §4, we introduce some theorems on the unique range set, due to H. Fujimoto and M. Shirosaki in particular. In §5 we finally give a method for constructing hypersurfaces in complex projective spaces with some interesting properties. Those hypersurfaces were constructed by M. Shirosaki ([43] and [44]), showing that the uniqueness problem of meromorphic mappings is closely related with the problem of constructing of Kobayashi hyperbolic projective hypersurfaces. The main parts in this article are §2, §3 and §5. We present a detailed introduction for each section below. 1. Preliminaries

In this section, we summarize known results in the Nevanlinna theory of meromorphic mappings and in the theory of hyperbolic manifolds. For details, see [26], [32] and [37]. In particular, we refer to Chapter 3 in [37] for properties on currents. Let be a finite analytic (ramified) covering space over and let be its sheet number. That is, X is a reduced irreducible normal complex space and is a proper surjective holomorphic mapping with discrete fibers. We denote by B the ramification divisor. Let

235

be the natural coordinate system in

For a (1,1)-current

where set

and set

of order zero on X, we

where denotes the characteristic function of Let M be a projective algebraic manifold and the canonical bundle of M. For a line bundle L over M, we denote by the space of all holomorphic sections of We denote by Pic(M) the Picard group over M. Let and We simply write for Then F is said to be ample provided that a line bundle is ample for some positive integer We fix an ample line bundle Now we set

It is easy to see that [F/L] < 0 if and only if is ample. Denote by a hermitian fiber metric in L and by its Chern form. Let be a meromorphic mapping. We set

as the characteristic function of with respect to L. We also define for in the following way. If is a positive integer with then we set

It is easy to see that is well-defined. Let be the complete linear system determined by L. We have the following Nevanlinna inequality for meromorphic mappings (cf. [32, p. 269]): Theorem 1.1 Let be a line bundle over M and let be a nonconstant meromorphic mapping. Then

for as

with

where O(1) stands for a bounded term

Let be a meromorphic mapping, and let Let Z be an effective divisor on X such that for distinct irreducible

236

hypersurfaces we set

in X and for nonnegative integers

For a positive integer

A meromorphic mapping is said to be dominant provided that rank The following second main theorem for dominant meromorphic mappings gives an essential computational way in the next section (cf. [32, Theorem 1]): Theorem 1.2 Let M be a projective algebraic manifold with and let be an ample line bundle. Suppose that are divisors in such that has simple normal crossings only. Let be a dominant meromorphic mapping. Then

where

except on a Borel subset with finite measure.

To apply Theorem 1.2, it is essential to give an estimate for by the characteristic function of In the case where and the ramification theorem due to H. Selberg is well-known (cf. [41]). In the case of meromorphic mappings we have the following ramification estimate proved by J. Noguchi. Definition 1.3 Let Y be a compact complex manifold. We say that a meromorphic mapping separates the fibers of if there exists a point in such that for any pair of distinct points In this case, X is said to be the proper existence domain of Assume that separates the fibers of and L is ample. Then there exist the least positive integer and a pair of sections such that the meromorphic function separates the fibers of Then we have the following ramification estimate due to J. Noguchi ([32, p. 277]): Theorem 1.4 (Noguchi) separates the fibers of

Suppose that is ample and Let be as above. Then

237 In the case where does not separate the fibers of we cannot estimate the growth of the ramification divisor in general. However, we have the following reduction theorem proved by J. Noguchi ([32, p. 272]): Theorem 1.5 (Noguchi) Let be a meromorphic mapping. Then there exist a finite analytic covering space a surjective proper holomorphic mapping and a meromorphic mapping separating the fibers of such that the following diagram

is commutative. Furthermore, if

is dominant, so is

From the above theorem, we can determine the proper domain of existence for an arbitrary meromorphic mapping For the theory of algebroid reduction, see also [51]. Remark 1.6 We note that is also a normal complex space. By making use of Theorem 1.4, we can easily obtain the following equalities (cf. [32, p. 273]):

Thus we also have for each positive integer Hence, by Theorems 1.2 and 1.4, we have the following: For an arbitrary dominant meromorphic mapping the inequality

holds. Here is the ramification divisor of the inequality

We also see that

holds. Therefore we can apply Theorems 1.2 and 1.4 for an arbitrary meromorphic mapping This observation is very useful and will be essentially used in the next section.

238 We finally give some results on Kobayashi hyperbolic space needed later. Let X be a reduced complex space and Y a Cartier divisor on X. Definition 1.7 A complex space is said to be Brody hyperbolic if it carries no nonconstant holomorphic curve from The following theorem is essentially due to R. Brody: Theorem 1.8 (1) Assume that X is compact. Then X is Kobayashi hyperbolic if and only if X is Brody hyperbolic. (2) If X and X \ Y are Brody hyperbolic, then X \ Y is complete hyperbolic and hyperbolically embedded in X. For the proof, see [26, p. 75]. The definition of Kobayashi hyperbolicity is well-known, while the notion of hyperbolically embedding is more complicated. Hence, for these definitions, see [26, p. 19 and p. 32]. 2. Unicity theorems and finiteness theorems

As mentioned in the Introduction, there have been many studies on the unicity of meromorphic mappings. The first remarkable results on the unicity of meromorphic mappings were obtained by H. Fujimoto. Indeed, he proved a number of excellent unicity theorems. For example, he proved the following brilliant theorem ([12, p. 1] and [13, p. 117]): Theorem 2.1 (Fujimoto) Let be nonconstant meromorphic mappings with the same inverse images of hyperplanes in general position. then there exists an automorphism T of such (1) If that and either or is linearly nondegenerate, then (2) If and either or is algebraically nondegenerate, then (3) If His proofs are based on the Borel identity. This result is in the relative case, that is, the multiplicities of the pull back divisors are considered. On the other hand, we have the following unicity theorem due to S. J. Drouilhet [9], which is the first several variable result in the absolute case (cf. [49]): Theorem 2.2 (Drouilhet) Let M be a projective algebraic manifold and A a smooth affine variety. Let be an ample line bundle over M and the hyperplane bundle over Let be a nonconstant holomorphic mapping. Let be hypersurfaces with normal crossings. Let be transcendental meromorphic mappings. Suppose that as a point set. If on Z and is ample, then

239

In the proof of this kind of unicity theorems, we use a second main theorem type inequality in an essentially computational way (cf. [1], [3], [4]). We discuss theorems of this type from the point of view of propagation of algebraic dependence in the next section. The finiteness theorem for meromorphic mappings was also studied by H. Cartan and R. Nevanlinna in 1920’s; they obtained a finiteness theorem for meromorphic functions on the complex plane ([7] and [30]). The finiteness theorem of Cartan-Nevanlinna states that there exist at most two meromorphic functions on that have the same inverse images with multiplicities for distinct three values in The unicity theorems for meromorphic mappings may be considered as a special case where the finite set of a family of meromorphic mappings reduces to a one point set. However, compared with those for unicity theorems, the study of finiteness theorems are not well studied. In 1981, H. Fujimoto generalized the theorem of Cartan-Nevanlinna to the case of meromorphic mappings of into complex projective spaces by making use of Borel’s identity ([14], IV and [15]). He proved the finiteness of families of linearly nondegenerate meromorphic mappings of into with the same inverse images for some hyperplanes. In his results, the number of hyperplanes in general position is essential and must be larger than a certain number depending on the dimension of the projective spaces. The theorem of Fujimoto has been extended to the case of meromorphic mappings into a projective algebraic manifold ([15] and [16]). Note that there has been an essential problem in the multidimensional case in this point. Namely, in the case where a given divisor is irreducible, what kind of condition yields the finiteness of families of meromorphic mappings? In this section, we mainly deal with the finiteness problem for meromorphic mappings of into a compact complex manifold M and for a divisor D on M. To state our results, we first give some definitions. For details, see [2]. Let be a fixed line bundle over M, and let be linearly independent holomorphic sections of with Throughout this section, we assume that for some positive integer where are effective divisors on M. Set

where Let D be a divisor defined by phic mapping by

Definition 2.3 Let we write

We define a meromor-

be a nonnegative integer. For divisors

and

on

240

if there exists a divisor on such that case of (mod 0) if and only if Let E be a nonzero effective divisor on

the set of all meromorphic mappings

in the special We denote by

such that

Definition 2.4 We say that a meromorphic mapping has the Zariski dense image if is not included in any proper analytic subset of M. Let

denote the subset of all with the Zariski dense image. The main result of the present article is as follows ([2, Theorem 4.1]): Theorem 2.5 If rank the number of mappings in depending on D only.

and

Let and set Then we have the following corollary:

then is bounded by a constant for a positive integer

Corollary 2.6 Let be an automorphism of and a meromorphic mapping with the Zariski dense image. Suppose that rank and (mod If then there exists a positive integer depending on D only such that In the proof of Theorem 2.5, we essentially use a generalized Borel identity. Namely, the following lemma, due to H. Fujimoto and M. Green, plays a central role in the proof (see [11, Corollary 6.4] and [21, p. 70]): Lemma 2.7 Let be nonzero meromorphic functions on satisfying the functional equation

Suppose that for all there exists a decomposition of indices, (1) Every contains at least two indices,

and

Then such that

241

(2) the ratio of (3)

and is nonzero constant if and only if for every

For the proof of the above theorem, we also use a generalization of Fujimoto’s finiteness theorem. Namely, we generalize Fujimoto’s finiteness theorem as follows. Definition 2.8 A meromorphic mapping is said to be linearly nondegenerate if is not included in any proper linear subspace of Let be effective divisors on hyperplanes in general position in Let

and let

be

be the set of all linearly nondegenerate meromorphic mappings into such that

of

for Then we have the following theorem that is a generalization of the finiteness theorem of Fujimoto ([2, Theorem 4.1]): Theorem 2.9 Suppose that either the number of mappings in constant depending on only.

or

Then is bounded by a

In [15], H. Fujimoto proved this theorem in the case of The proof of the above theorem is complicated. Hence we refer [2] and [15] for the proof. We note that there exists an algorithm to give the upper bound for the number of mappings in the family For instance, we have

We do not know whether the upper bound is sharp or not in the case of It is an interesting problem to determine the least upper bound for the numbers of mappings in For the family we have the following result ([2, Theorem 4.5]): Theorem 2.10 Suppose that rank the number of mappings in depending on D only.

If then is bounded by a constant

242

We now give the proofs of Theorem 2.5. For a positive integer we denote by the set of all meromorphic functions on such that for some nonzero meromorphic functions on For let denote the set of all nonvanishing holomorphic functions on Let Then we have a relation

where and for Applying Lemma 2.7 to the above relation, we have a decomposition of indices. Since and have the Zariski dense images, we have that for all Thus we obtain

for of Aut T of

where is a permutation of Let be the subgroup generated by all diagonal matrices. So, an automorphism belongs to if and only if

for some system in

where We set

is a homogeneous coordinate

Then we have that for some note that

and for

with

We

where The above argument shows that there exist finitely many linearly nondegenerate meromorphic mappings with that satisfy the following property: For an arbitrary there exist and such that For we define

To prove that is finite for each

Let

is finite, it suffices to show that be fixed. We define effective divisors

243 on

as follows. Let be the zero divisor of for and where is a reduced representation of Take the following hyperplanes in general position in

Let be the set of all linearly nondegenerate meromorphic mappings such that (mod for Thanks to Theorem 2.9, is bounded by a constant depending on s only. For by the definition of we have Let be an arbitrary meromorphic mapping in Since rank it is easy to see that

is finite. Therefore we have the desired conclusion. We give here examples of pairs (M, D) satisfying the assumptions of the above theorems. We first consider the case where In the following two examples, let and let be a homogeneous coordinate system in As usual, we denote by [H] the hyperplane bundle over Example 2.11 Let D be a Fermat hypersurface defined by

Put

and and

for It is easy to see that rank

Then

be a set of monomials with Example 2.12 Let (for the definition, nonnegative rational exponents that is see [28]). Let be the smallest positive integer such that all exponents of are integers. Put and for Then and Since is we may assume that for Hence rank Next we give an example of (M, D) such that example is due to J. Noguchi:

The following

Example 2.13 Let and be smooth elliptic curves. We denote by (resp. ) the identity of an abelian group (resp. ). Let (resp. ) be a point in (resp. ). Let be the line bundle

244 over determined by a divisor for By Abel’s theorem, and are linearly equivalent. Hence there exist holomorphic sections and such that and Set We define a line bundle by where are the natural projections. Put and Then for all By the construction of it is easy to see that for some effective divisors on M and rank We finally give the following unicity theorem ([2, Theorem 6.2]): Theorem 2.14 Assume that there exist big line bundles such that and for some holomorphic sections of Let be meromorphic mappings with the Zariski dense images whose ranks are not less than Suppose that the following conditions are satisfied: (1) rank (2) (3) as point sets (say Z), (4) on Then there exists a positive integer depending on only such that if on In the proof of the above theorem, we use the following ([2, Lemma 2.4]) Lemma 2.15 Let be as in Theorem 2.14. Let be a meromorphic mapping with the Zariski dense image and assume that rank Put Suppose that rank and Then

where except on a Borel subset

with finite measure.

Remark 2.16 M. Shirosaki ([43] and [44]) constructed hypersurfaces and in which have interesting properties. Indeed, let and be algebraically nondegenerate meromorphic from into If (or then His construction is elementary. Furthermore, is Kobayashi hyperbolic and has a remarkable arithmetic property which concerns with Lang’s conjecture. We note that has many irreducible components. We give a sketch of the construction and properties of those hypersurfaces in §5.

245 3. Algebraic dependence of meromorphic mappings In this section, we give a summary of the author’s recent research on the algebraic dependence of meromorphic mappings from the point of view of Nevanlinna theory. In [29] and thereafter methods used in proving relative case theorems have been essentially different from those used in the absolute case. In the proof of absolute unicity theorems, we use a second main theorem for meromorphic mappings in an essentially computational way (cf. [1], [3], [4] and [9]). Note that the second main theorem for meromorphic mappings is established in a few cases. Hence, we deal with the case of dominant mappings. In what follows, we consider the following setting. Let be a finite analytic covering space and M a projective algebraic manifold. Let be dominant meromorphic mappings from X into M. Suppose that they have the same inverse images of given divisors on M. We first give conditions under which are algebraically related. We consider the propagation of algebraic dependence of meromorphic mappings and their applications to the uniqueness problem. Roughly speaking, our results say that if these mappings satisfy the same algebraic relation at all points of the set of the inverse images of divisors and if the given divisors are sufficiently ample, then they must satisfy this relationship identically. These results are considered as the propagation theorems of algebraic dependence. The propagation of dependence from a proper analytic subset to the whole space was first studied by L. Smiley [48] (cf. [50, p. 176]). There have been several studies on the propagation of dependence (cf. [10], [23] and [52]). So far, this problem has been studied under conditions on the growth of meromorphic mappings. For example, W. Stoll [52] proved some interesting theorems on the propagation of dependence of meromorphic mappings under a condition on the growth of mappings in different settings. In his results, at least one of the mappings must grow quicker than the ramification divisor B of That is, it is assumed that

However, there can be a few restricted cases only where meromorphic mappings satisfy these conditions even if dim M = 1 (cf. [33] and [39]). In this section, we first give criteria for the propagation of algebraic dependence of meromorphic mappings from X into M under the condition on the existence of meromorphic mappings separating the fibers of Thanks to the theory of algebroid reduction of meromorphic mappings, we can always find such a mapping. Thus it seems that our condition is more natural and essential than the above mentioned conditions. The theorem

246 on algebroid reduction of meromorphic mappings and the ramification estimate due to J. Noguchi [32] are essentially important in the proofs of our results. In some of our criteria, we assume more complicated conditions, but they have wider ranges of applicability. These criteria are actually corollaries of Lemmas 3.2 and 3.3, which are fundamental for our study. We also consider the case where given divisors may determine distinct line bundles and give their applications. We note that certain kind of unicity theorems such as results in [3] and [9] may be considered as a special case of theorems on the propagation of dependence. In these theorems we can see that, for two meromorphic mappings with the same inverse images of divisors as point sets (say Z) satisfying on Z, the algebraic relation on Z propagates to the whole space X. We give some unicity theorems from this point of view in §3. In §4, we study the uniqueness problem of holomorphic mappings into smooth elliptic curves. In particular, we give some conditions under which two holomorphic mappings are related by endomorphism of elliptic curves. For more details, see [5] and [6]. 3A. Propagation of algebraic dependence We first give a definition of the algebraic dependence of meromorphic mappings. Let M be a projective algebraic manifold and an ample line bundle over M. Set For meromorphic mappings we define a meromorphic mapping by

where are the indeterminacy loci of A proper algebraic subset of is said to be decomposable if there exist algebraic subsets and such that Definition 3.1 Let S be an analytic subset of X. Nonconstant meromorphic mappings are said to be algebraically dependent on S if there exists a proper algebraic subset of such that and is not decomposable. In this case, we also say that and are on S . Let be divisors in such that has simple normal crossings only. Let be hypersurfaces in X such that dim for any We define a hypersurface S in X by Let Z be an effective divisor on X, and let be a positive integer. If for distinct irreducible hypersurfaces in X and for nonnegative integers then we define the support of Z with order at

247 most

by

Assume that coincides with for all with where is a fixed positive integer. Let be the set of all dominant meromorphic mappings such that is equal to for each with Let and be big line bundles over M. We define a line bundle over by where are the natural projections on the factor. Let be a big line bundle over In the case of we assume that there exists a positive rational number such that is big. If then we take Let be the set of all hypersurfaces in X such that for some and that is not decomposable. Assume that separates the fibers of Since L is ample, there exist a positive integer and a pair of sections such that a meromorphic function separates the fibers of for all such mappings We denote by the least positive integer among those We assume that there exists a line bundle, say in such that is either big or trivial for Set We define by

Then, by making use of Theorems 1.2 and 1.4, we have our basic result, from which we see that, if is sufficiently big, then the algebraic dependence on S propagates to the whole space X. Lemma 3.2 Let and that and are on X. Note that

be arbitrary mappings in and on S. If is big, then

Suppose and are

is negative in the above theorem. In the case we get same conclusion if there exists at least one positive (see [6, Theorem 2.7]). Now, let us consider a more general case. Let and be ample line bundles over M. Let be positive integers and assume that has only normal crossings, where Let Z be a hypersurface in X. Let be a family of dominant meromorphic mappings such that

248 for some by

In the case where

for

we define

Then we have one more fundamental result for our study. Lemma 3.3 Let and Suppose that are on X.

be arbitrary mappings in and on Z. If is big, then

are

Now, we give criteria for the propagation of algebraic dependence of dominant meromorphic mappings. We first give a corollary of Lemma 3.2. For we define [F/L] by

Set

We also set

Then we have the following criterion for the propagation of algebraic dependence: Corollary 3.4 Let are positive and if

then

are

Suppose that they are

on S. If

on X.

By making use of Lemma 3.3, we also have the following two criteria. Set

We also set

for

Then we have the following criterion:

249 Corollary 3.5 that are

then

Let

are

and

Suppose

on X.

Set

Then we also have the following:

Corollary 3.6

then

be arbitrary mappings in on Z. If all and if

are

Let

be as in Corollary 2.5. If all

and if

on X.

Remark 3.7 (1) The case, where either all or all are especially important from the viewpoint of the Nevanlinna theory. We now consider the case where for some We first note that Supp Set and for Then it is easy to see that the proofs of Lemmas 3.2 and 3.3 also work in the case where for some Hence the conclusions of the above propositions are still valid for the case where some of the We also note that the proof of Lemma 3.2 also works in the case where some of the are empty sets. (2) If is positive, we cannot conclude the propagation of dependence under the condition on the existence of deficient divisors. We now give an example. Let and Suppose that is the hyperplane bundle. Let L = H and We now take

Let

Set be a meromorphic function defined by and Then it is clear that and are for all and put Picard’s deficient values of Let Now, we see that and If is the diagonal of then It is clear that and Note that the proofs of the above theorems also work in the case where some of are empty sets. 3B. Unicity theorems for meromorphic mappings In this section we give some unicity theorems as an application of criteria for dependence by taking line bundles of special type. For the details

250 of results in this direction, see [1], [3], [4], [9] and [49]. We keep the same notation as in §2. Let be a meromorphic mapping with rank We denote by H the hyperplane bundle over Take We also take Then we see that

We fix

Set

where is the locus of indeterminacy of A set said to be generic with respect to and provided that

of divisors is

for at least one where denotes the locus of indeterminacy of We assume that is generic with respect to and in what follows. Let be the set of all mappings such that on S. Then we have the following unicity theorems by Lemma 3.2 and by the uniqueness of analytic continuation (cf. [3, Theorem 2.1]): Theorem 3.8 Suppose that just one mapping

is big. Then the family

contains

In the case where is not ample, we cannot prove in general. However we can show the unicity theorem for under an additional condition on the existence of Nevanlinna’s deficient divisors. Indeed, we have the following unicity theorem ([3, Theorem 2.15]), which shows that the existence of Nevanlinna’s deficiency imposes a strong restriction on the behavior on meromorphic mappings: Theorem 3.9 and

Suppose that

If for at least one one mapping

is generic with respect to

then the family

and

contains just

We next consider the case dim M = 1. Assume that M is a compact Riemann surface with genus In the case we have the following unicity theorem for meromorphic functions on X by Theorem 3.8, which is closely related to the uniqueness problem of algebroid functions (cf. [1, Theorem 3.3]). Theorem 3.10 mappings. Let

Let be distinct points in

be nonconstant holomorphic The following hold.

251 (1) Suppose that and are identical on X. (2) Suppose that and are identical on X.

for all

If

for all

then If

then

Note that the above theorem is sharp in the case Example 3.11 We consider the integral

on the unit disc in Set and Then maps the unit disc onto the square By Schwarz’s reflection principle, the inverse function of can be analytically continued over the complex plane and the resulting function is doubly periodic. Let and Set and Then for all but The uniqueness problem of holomorphic mappings into a compact Riemann surface with positive genus is not well studied (cf. [1], [10] and [40]). In the case of we will discuss the uniqueness for holomorphic mappings into smooth elliptic curves in the subsection 3C below. We now consider the case where Note that Riemann-Roch’s theorem shows In this case, we have the following unicity theorem (cf. [1, Theorem 3.6)]): Theorem 3.12 Let be nonconstant holomorphic mappings. Let be distinct points in M. Then the following claims hold: for all If (1) Suppose that then and are identical on X. (2) Suppose that for all If then and are identical on X. Note that under the conditions of Theorems 3.10 and 3.12, at least one is not empty. 3C. Holomorphic mappings into smooth elliptic curves We consider the case where M is a smooth elliptic curve E. The uniqueness problem of holomorphic mappings into elliptic curves was first studied by E. M. Schmid [40]. He obtained the following unicity theorem: Let be nonconstant holomorphic mappings, where R is an open Riemann surface of a certain type. Then there exists a nonnegative integer depending on

252 R only such that, if for distinct points in E, then and are identical. In the special case we have So far, there have been few studies only on the uniqueness problem of holomorphic mappings (cf. [10] and [40]). In this section, we consider the problem of determining a condition which yields for an endomorphism of the abelian group E. We first note the following fact: If separates the fibers of then we can take (cf. [33, p. 286]). Let Since we identify the Chern class of L with an integer. We now consider the infimum [F/L] of the set of rational numbers such that is ample. We note that if Hence the conclusions of Lemma 3.3, Corollaries 3.5 and 3.6 are still valid provided that and all are identical. We also note that is not necessarily a rational number in this section. It is well-known that

We denote by the point bundle determined by Let Let be nonconstant holomorphic mappings. We denote by End (E) the ring of endomorphisms of E. If E has no complex multiplication, it is well-known that Hence for some integer We now seek conditions which yield for some Let and consider a curve

in E × E. Let be the line bundle determined by In this subsection, denotes the infimum of rational numbers such that is ample. Then we essentially use the following theorem proved by T. Katsura (see [6, §6]): Theorem 3.13 (Katsura)

Let

be as above. Then

By the above theorem, we have the following corollary (cf. [47, p. 89]): Corollary 3.14 defined by

Let

be an integer. If then

is an endomorphism

By making use of Lemma 3.3, we have the following Theorem 3.15 Let and be as above. Let of points and an endomorphism of E. Set number of points in is also Suppose that for some If

be a set Assume that the then

253 In the above theorem, we assume that the cardinality of the point equals However, it may happen that For example, if and there exists at least one pair such that is point, then In this case, by making use of Corollary 3.6, we have the following

set

Theorem 3.16 Let be nonconstant holomorphic mappings. Let be a set of points and Set Assume that the number of points in is Suppose that If then Corollary 3.17 Let and X be as in Theorem 3.16. Let be a set of points and set for some integer Assume that the number of points in is Suppose that If then We do not know whether Theorem 3.16 is sharp or not. However, if the condition is not satisfied, then it is not necessarily true that Example 3.18 Let be an endomorphism defined by Define by and where is the universal covering mapping. Let Then It is clear that In this case, and Thus we have

and The following unicity theorem is a direct consequence of Theorem 3.15: Theorem 3.19 Let be distinct points in E. Let nonconstant holomorphic mappings. Suppose that for all where If and are identical. In the case of

be then

, we have the following

Corollary 3.20 Let be distinct points in E. Let be nonconstant holomorphic mappings. Suppose that for all If then and are identical. Remark 3.21 If we choose special points of E, we obtain an example which yields that Corollary 3.20 is sharp. Indeed, let be twotorsion points in E and let be the Weierstrass function. If for it is easy to see that by Nevanlinna’s four

254 points theorem. Hence or Since is an automorphism of E, it is acceptable that and are essentially identical. In this example, it seems that the structure of the function field of E affects strongly on the uniqueness problem for holomorphic mappings. 4. Unique range set problem In this section, a discrete subset S of is called a unique range set for meromorphic (or entire) functions provided that there exists no pair of distinct nonconstant meromorphic (or entire) functions such that they have the same inverse images of S counted with multiplicities. F. Gross and C. C. Yang first proved that the set

is a unique range set for entire functions ([22]). After their study, H.-X. Yi ([54]) proved the existence of unique range set with finitely many points as follows: Theorem 4.1 (Yi) Let and be two positive integers having no common factors and such that Let and be two nonzero constants such that the algebraic equation has no multiple roots. Set Then S is a unique range set for entire functions. In [43], M. Shirosaki generalized the above theorem for meromorphic functions. Let be a homogeneous coordinate system in Then he obtained the following Theorem 4.2 Let and be two positive integers with and which have no common factors. Then the zero set of the polynomial

is a unique range set for meromorphic functions. He also proved a more precise result as follows: Theorem 4.3 Let be a homogeneous polynomial as in Theorem 4.2. Let and be algebraically nondegenerate holomorphic mappings of into with reduced representations and respectively. If holds for some entire function

without zeros, then

255 where

is an entire function such that

There have been many efforts to find unique range sets which are as small as possible ([27], [55] and [56]). We next discuss uniqueness polynomials. In relation to the unique range set problem, B. Shiffman, C. C. Yang and X. Hua studied polynomials satisfying the condition that there exists no pair of distinct nonconstant meromorphic (or entire) functions and with see their papers [42] and [53]. For a finite set it is necessary for S to be a unique range set for meromorphic (or entire) functions that the associated polynomial

satisfies this condition. Now, we give the following definition: Definition 4.4 Let be a nonconstant monic polynomial. Then is called a uniqueness polynomial if implies for any nonconstant meromorphic functions and any nonzero constant We also call uniqueness polynomial in the broad sense if implies for any nonconstant meromorphic functions and In the [19], H. Fujimoto gave some sufficient conditions for uniqueness polynomials as well as for unique range sets. Let be a monic polynomial without multiple zeros whose derivative has mutually distinct zeros with multiplicities respectively. Then we say that satisfies the condition (H) if Under this assumption, he proved the following result: Theorem 4.5 If then satisfying the hypothesis (H) is a uniqueness polynomial in the broad sense. Moreover, he also proved the following theorem for uniqueness polynomials: Theorem 4.6 For a polynomial with (H) is a uniqueness polynomial provided that

satisfying the hypothesis

Corollary 4.7 Generic polynomials of degree at least five are uniqueness polynomials. Let be as above, that is ia a monic polynomial without multiple zero whose derivative has mutually distinct zeros

256 with multiplicities satisfies the condition (H).

respectively. Suppose that

Then the following is the main theorem in [19]: Theorem 4.8 (Fujimoto) Assume that ness polynomial, then there is a permutation such that

If

is not a uniqueof

We note that Theorem 4.6 is an immediate consequence of Theorem 4.8. In [20], H. Fujimoto give a geometric proof for Theorem 4.8. He also give an improvement of the above result: Theorem 4.9 (Fujimoto) A polynomial satisfying the hypothesis (H) is a uniqueness polynomial in the broad sense if and only if

Remark 4.10 Note that the assumption in Theorem 4.9 is always satisfied if Moreover, this holds if for and when and for 5. Concrete projective hypersurfaces In this section, we mainly deal with the results obtained by M. Sirosaki in [43] and [44]. In [43], he constructed a homogeneous polynomial for each and hypersurfaces in defined by with the uniqueness property as mentioned in Remark 2.16. Furthermore, in [44], he constructed another hypersurface which has the uniqueness property. This hypersurface is Kobayashi hyperbolic and hyperbolically embedded in

5A. Uniqueness property We first consider the case Let and be two positive integers with and which have no common factors. Define again For in

we also define, inductively, a homogeneous polynomials with degree by

257 We denote as the hypersurface in defined by have the following unicity theorem ([43, Theorem 2]): Theorem 5.1 Let mappings of into

and If

Then we

be algebraically nondegenerate holomorphic then and are identical.

Proof. Let and be reduced representations of and respectively. To prove Theorem 5.1, we will show that, if

holds for some entire function

without zeros, then

where is an entire function such that We give a proof by induction on In the case of we can show the desired property by making use of the second main theorem for meromorphic functions of one complex variable and the estimate for ramifications of meromorphic functions. The method for the proof is standard in the unique range set problem of one complex variable. For the proof of this case, see [43, pp. 292– 295]. Next we consider the case To show that with has the above property, we need the following lemma ([43, Lemma 1]: Lemma 5.2 Let be the homogeneous polynomial as above. Let and be nonconstant holomorphic mappings of into with reduced representations and respectively. If

holds for some meromorphic function zeros.

then

is an entire function without

By the assumption in Theorem 5.1, there exists a nonvanishing holomorphic function such that

where and are reduced representations of and respectively. For let A and B be entire functions such that and are reduced. Then we see that

258 by Lemma 5.2; here A / B is an entire function without zeros. Hence, by the reasoning above, we get

where is an entire function with Put and we get the conclusion for Assume that the result is true for and consider the case for In this case, by the assumption we have the identity

It follows from the result for

that

and where is an entire function such that induction, we have where is an entire function such that obtain that

By the hypothesis of

By

we

Thus we have the assertion. We give another homogeneous polynomial different from and define tively:

above. Let induc-

with degree for Let be a hypersurface defined by Then we have the following unicity theorem ([44, Theorem 3.1]): Theorem 5.3 Let and be algebraically non-degenerate holomorphic mappings of into with representations and respectively. If

for some meromorphic function

then

259 where

is a

root of unity.

For another example of a projective hypersurface which yields unicity of holomorphic mappings, see [45]. 5B. Hyperbolicity of In this subsection we prove theorems which show constantness of holomorphic mappings. To this end, we need some lemmas. Lemma 5.4 Let and be entire functions at least one of which is not identically equal to zero. If

for some entire function

then

is constant.

The proof is based on the ramification theorem for meromorphic functions, see [44, p. 32]. The next lemma is easily obtained by Lemma 5.4 (see also [44, p. 32]). Lemma 5.5 Let and be entire functions at least two of which are not identically equal to zero, and let C be a nonzero constant. If

then

is constant.

We now have the following lemma ([44, Theorem 4.3]): Lemma 5.6 Let be an integer and Suppose that at least two of identically equal to zero and that is constant. Then is constant.

entire functions. are not

The proof, which is somewhat complicated, follows by induction on For details, see [44, p. 33]. Theorem 5.7 Let be a positive integer and a holomorphic mapping of into with a reduced representation If then is constant. In particular, is Kobayashi-hyperbolic. Proof. We give a proof by induction on The case of is trivial and the case of is proved easily by Lemma 5.5. Assume that the result for holds. Let If we put then If then the conclusion is obvious. Hence assume that there exists such that Note that there exist at least two such By the hypothesis of

260 induction, is constant. Hence by Lemma 5.6, we conclude that is constant. Now, Brody’s theorem yields that is Kobayashi hyperbolic. Next, we consider the hyperbolicity of Lemma 5.8 Let be a holomorphic mapping of with a reduced representation If for some entire function then is constant. The proof is done by making use of Lemmas 5.4 and 5.5. See [44, p. 33]. Theorem 5.9 Let be a holomorphic mapping of reduced representation If function then is constant. In particular, hyperbolic and hyperbolically embedded in

into with a for some entire is complete

Proof. We give a proof by induction on The case of is proved by Lemma 5.4 and the case of by Lemma 5.8. Assume that the result holds for If we put then Since the case does not occur, there exists such that By the hypothesis of induction, is constant. If there exist at least two such that then Lemma 5.6 yields the conclusion. Hence we consider the case where there exists the only one such If then where K is a nonzero constant. Hence where is a root of K. Therefore is constant by Lemma 4.1. In this case, if then and where are nonzero constants. If then and where are nonzero constants. In the case where and we have the conclusion by using is hyperbolically embedded in and is complete hyperbolic by Theorem 1.8. The idea of this construction is using uniqueness polynomial, being different of other constructions (e.g., [28]). For a construction of this type, see also [46]. 5C. An arithmetic property of We finally give an arithmetic property of due to J. Noguchi [36] concerning with Lang’s conjecture. In 1974, S. Lang [24] conjectured that an algebraic variety defined over an arbitrary algebraic number field K carries only finitely many K-rational points if the complex space with some

261 embedding is Kobayashi hyperbolic (see also [25]). This conjecture was established for curves and a subvariety of Abelian varieties by Faltings. Note that an analogue over function field has been proved by J. Noguchi ([34] and [35]). So far, there had been no example, nor an existence theorem of a projective hypersurface of dimention greater than one that carries the above arithmetic property. Let be as above. Set

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263 42. Shiffman B., Uniqueness of entire and meromorphic functions sharing finite sets, Complex Variables Theory Appl. 43 (2001), 433–449. 43. Shirosaki M., On polynomials which determines holomorphic mappings, J. Math. Soc. Japan 49 (1997), 289–298. 44. Shirosaki M. On some hypersurfaces and holomorphic mappings, Kodai Math. J. 21 (1998), 29–34. 45. Shirosaki M., A hypersurface which determines linearly nondegenerate holomorphic mappings, Kodai Math. J. 23 (2000), 105–107. 46. Shirosaki M., Hyperbolic hypersurface in the complex projective spaces of low dimensions, Kodai Math. J. 23 (2000), 243–233. 47. Silverman J., The Arithmetic of Elliptic Curves, Springer-Verlag, Berlin– Heidelberg–New York, 1991. 48. Smiley L., Dependence theorems for meromorphic maps, Ph.D. Thesis, Notre Dame Univ., 1979. 49. Smiley L., Geometric conditions for the unicity of holomorphic curves, Contemporary Math. 25 (1983), 149–154. 50. Stoll W., The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Proc. Value Distribution Theory, Joensuu 1981 (eds. I. Laine et al.), pp. 101–219, Lect. Notes in Math. 981, Springer-Verlag, Berlin–Heidelberg–New York, 1983. 51. Stoll W., Algebroid reduction of Nevanlinna theory, Complex Analysis III, Proc. 1985–1986, (ed. C. A. Berenstein), pp. 131–241, Lect. Notes in Math. 1277, Springer-Verlag, Berlin–Heidelberg–New York, 1987. 52. Stoll W., On the propagation of dependences, Pacific J. Math. 139 (1989), 311–337. 53. Yang C. C. and Hua X., Unique polynomials of entire and meromorphic functions, Mat. Fiz. Anal. Geom. 4 (1997), 391–398. 54. Yi H.-X., A question of Gross and the uniqueness of entire functions, Nagoya Math. J. 138 (1995), 169–177. 55. Yi H.-X., The unique range sets of entire or meromorphic functions, Complex Variables Theory Appl. 28 (1995), 13–21. 56. Yi H.-X., Some further results on uniqueness of meromorphic functions, Complex Variables Theory Appl. 38 (1999), 375–385.

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ON INTERPOLATION PROBLEMS IN

CARLOS A. BERENSTEIN Department of Mathematics University of Maryland College Park, MD 20742, USA, [email protected] AND BAO QIN LI Department of Mathematics Florida International University Miami, FL 33199, USA, [email protected]

Abstract. We will consider under what conditions an analytic variety in is an interpolating variety for weighted spaces of entire functions, which is one of fundamental problems in several complex variables. Interest in this area arises from connections and applications of such questions to other problems such as representation of solutions of partial differential equations, deconvolution, and the Nullstellensatz. We shall discuss some of recent results on this subject, with a special attention given to those by the authors and their collaborators. Mathematics Subject Classification 2000: 32E30, 32A15, 32A22, 46E25. Key words and phrases: entire function; holomorphic mapping; interpolation; interpolating variety; weight; Jacobian; counting function.

1. Introduction The interpolation problems we consider can be phrased in a broad fashion. Let be the Hörmander algebra of entire functions in satisfying

© 2004 Kluwer Academic Publishers

266 for some A > 0, where is a plurisubharmonic weight function in (cf. [9], [26]). We will study under what (necessary and sufficient) conditions an analytic variety V in is an interpolating variety for in the sense that every analytic function on V satisfying an estimate of the form for some A, B > 0 always has an entire extension to a function in We will then say that V is an interpolating variety for We also consider the problem for the space of entire functions of minimal type satisfying that for every In this article, our attentions will be mainly given to discrete varieties, which is the original setting for the classical interpolation theorem in one complex variable, going back to Weierstrass and Mittag-Leffler. In this situation, the problem is to find necessary and sufficient interpolation conditions so that there always exists an entire function in the weighted space that has prescribed values, or more generally, prescribed finite collection of Taylor coefficients with the growth constrains. The interpolation problem is one of the central problems in several complex variables. Interest in this area arises from connections and applications of such questions to other subjects such as harmonic analysis, number theory, and system theory (see e.g. [2], [3], [4], [8], [9], [12], [13], [14], [15], [35], etc.). Note that necessary interpolation conditions and sufficient interpolation conditions require different tools and techniques. For example, the “standard” tool, estimates for the operators, has been very useful in obtaining sufficient interpolation conditions, but it does not provide methods for necessary conditions. Diverse methods from different areas are needed. In this survey paper, we shall discuss and present some of recent results in this subject, with a special attention given to those by the authors and their collaborators. We shall state the problems in more details and include some preliminaries in the second section. In the third section, we shall consider necessary and sufficient interpolation conditions for the Hörmander algebra. In the fourth section, we shall consider the growth of the “size” of interpolating varieties, and its connections with the transcendental Bézout problem. The fifth section is concerned with the geometry of interpolating varieties. In the last section, we consider interpolation for entire functions of minimal type and some of its applications. 2. Preliminaries Let us fix the notations and notions we will use in the paper. First of all, we consider a weight to be a plurisubharmonic function satisfying the following conditions:

267 and there exist positive constants that Let

and

be the space of entire functions in

such that

implies

We define

and

We mention here two basic but important examples of weights: and They correspond to the space of all entire functions of order and of finite type and the space of Fourier transforms of distributions with compact support in respectively (cf. [19]). When the space of entire functions of infraexponential type. Note that it is not the specific conditions on which are important, but rather their consequences for the ring The condition (2.1) implies that contains the polynomials, and (2.2) implies that is closed under differentiation. One can replace the condition (2.2) by the following Hörmander condition ([26], [27]): There exist four positive constants such that implies that The space carries a natural topology as an inductive limit of Banach space, while the space has a natural projective limit structure. Let V be an analytic variety in If for every analytic function on V satisfying that for some A, B > 0 on V there exists an entire function such that on V, then V is called an interpolating variety for When is discrete, one of the most interesting cases, one describes sequences as “analytic functions” on V , and then that on V becomes that for The notion of interpolation stated above can be phrased using the restriction mapping from the space of analytic functions on V. It is clear that the space of analytic function on V satisfying for some A, B > 0. In general, the space is too large. The interpolation problem is to determine when the mapping is surjective from to In the discrete case, associated to a given discrete variety there is a unique closed ideal in

268 Two entire functions

in can be identified modulo I if and only if The quotient space can be identified with the space A(V) of all sequences of complex numbers, which can be described as “analytic functions” on V. The map is the natural restriction map from into A ( V ) . The interpolation problem for is then to determine when is surjective from to the space of sequences satisfying that for some A, B > 0. The interpolation problem for is to determine when is onto from to the space of sequences satisfying that for every for some A > 0. Due to applications to other areas such as harmonic analysis, one also needs to consider the problem with multiplicities, which can be formulated in a natural way. Note that the problem with multiplicity is not simply a direct extension of the problem without multiplicity, as it is well-known that multiplicities often cause considerable difficulties in the problems. However, to simplify the exposition, we will not include multiplicity problems here. We refer the reader to [2], [4], [9], [10], [37], [38], [34], etc. for results and references in this direction. Interpolation for other weighted spaces of entire functions have been also studied. We refer the reader to [16], [25], [40], [41], [45], [46], etc. for related results. We will also need the following counting functions, which were originally introduced in Nevanlinna theory (see e.g. [48], [23]). Let V be an analytic variety of pure dimension We define the counting function

where

is the ball in

centered at the origin and with radius and is the usual Kähler form. We also define the integrated counting function

In the same way, one can define the counting function with respect to the ball centered at a point and with radius

and

3. Interpolation for the Hörmander algebra Interpolation problems have been studied extensively for a long time. Many interesting results were obtained by imposing conditions upon the weight,

269

the variety, or by changing the growth conditions. In particular, the problems for the Hörmander algebra are studied in a systematic way by Berenstein and Taylor in [9] with the purpose of generalizing Ehrenpreis’s Fundamental Principle for systems of convolution equations. Their relations and the background of the technique are extensively documented in the article [8]. Various interpolation results were obtained in [9], especially for complete intersections defined by so-called slowly decreasing entire functions, where a vector function is called slowly decreasing if and only if there exist such that (i) the connected components of the set

are bounded and (ii) if is a component of

then

for all

The notion of slowly decreasing ideas and the condition that the ideas are a complete intersection play key roles in the results. We refer the reader to [9] for various examples of slowly decreasing functions. We start with the following theorem of Berenstein and Taylor ([9]): Theorem 3.1. Let be slowly decreasing. Let be a discrete set and assume that each zero is simple; that is, det Then is an interpolating variety for if and only if there exist constants such that

where df is the Jacobian matrix of F. The theorem was due to Leont’ev [29] when and to Ehrenpreis and Malliavin [20] when and and to Berenstein and Taylor [10] for general weights when Generalizations to complete intersections defined by more than slowly decreasing functions can be found in a more recent work of Berenstein, Kawai and Struppa [4]. However, in practice, a discrete variety is generally not a complete intersection of some slowly decreasing functions. It is a natural goal to study necessary and sufficient interpolation conditions that apply to arbitrarily given varieties in When various results in this direction have been known (cf. [2], [6], [10], [43], etc.). When a necessary and sufficient interpolation condition in terms of the “directional derivatives” of

270

defining functions was found by Berenstein and Li ([5]), from which the following theorems follows: Theorem 3.2. Let be a discrete variety in and an integer. Then V is an interpolating variety for if and only if there exists an entire holomorphic mapping with such that and for each there exists a minor J of the Jacobian of f such that for some

Theorem 3.3. Let an interpolating variety for holomorphic mapping and for some

be a discrete variety in Then V is if and only if there exists an entire with such that

This theorem gives a necessary and sufficient interpolation condition, which applies to arbitrary discrete varieties. The method has been used for interpolation in the unit ball of (see [40], [38]), and for disjoint union of affine subspaces ([42]) in Theorem 3.3 also plays an important role in our work on estimating the “size” of interpolating varieties, which we are about to discuss in the next section. 4. Size of interpolating varieties Let

be a discrete variety in

Then the counting function

measures the “size” (or “volume”) of the variety V within the ball We shall consider how the counting function grows as In the one-dimensional case, the size of an interpolating variety is well bounded by using the weight. When the upper bound on the size of interpolating varieties is closely tied to the wellknown transcendental Bézout problem in for entire holomorphic maps The classic theorem of Bézout in algebraic geometry says that the degree of the intersection of algebraic sets does not exceed the product of the degree of the sets:

271

If

is entire, the growth of the maximum modulus plays the role of the degree of a polynomial. Assuming that it follows from the Jensen formula that for any there exists a C > 0 depending on such that

where This generalizes the bound on the numbers of zeros of a polynomial. Suitably interpreted, the above estimate carries over to bound the size of the analytic hypersurface for entire functions in For an entire holomorphic map from to noting that the formulas (4.1) and (4.2) would suggest an bound on in terms of i.e., This so-called transcendental Bézout problem was originally due to Griffiths (c.f. [21], [22]) and has been studied by many authors. It is well-known that the fields of algebraic geometry and complex analysis have a parallel development and frequently share the same underlying general principles. However, the analytic version of the Bézout theorem in algebraic geometry is not true for entire holomorphic maps when by a counter-example of Cornalba and Shiffman [18]. It is generally impossible to estimate in terms of the maximum modulus alone. On the other hand, such an estimate does hold “on the average” in certain sense, as shown by Gruman and Stoll ([24], [47]). It also holds for some special types of mappings such as exponential mappings, as shown by Berenstein and Yger ([12]). By a result of Carlson ([17]), the set of for which no such an estimate is possible for the zeros of is pluripolar. In particular, it was shown by Ji in [28] that

if

on where C is a positive constant depending on only. In [36], Li and Taylor showed that

if the Jacobian has a lower bound of the form (3.1). A similar bound on the size of interpolating varieties then follows from the lower bound on the Jacobian, thanks to Theorem 3.3 ([36]). The problem was further studied in [31], where Li applied the estimate in [36] to give a precise estimate on in terms of the maximum modulus and the Jacobian of

272 Theorem 4.1. Let be an entire holomorphic map and a discrete set. Then for any

where C is a positive constant depending on

only.

The actual value of C was also given in [31]. Above, denotes the supremum of on Note that if then the above gives the estimate on the counting function for an entire holomorphic map The estimate was shown to be precise in certain sense. In particular, if on V, then Theorem 4.1 implies that for any

(compare with (4.3)). A refined version of Theorem 4.1 involving a comparison function which can be well-fitted to specific classes of mappings can be found in [32]. Together with Theorem 3.3, Theorem 4.1 immediately yields the following estimate on the growth of the size of interpolating varieties: Theorem 4.2. Let

be an interpolating variety for

Then

The estimate (4.4) is sharp up to a constant term O(1) and the number in the estimate is best possible. To see this, let and Then it is easy to verify that

where One can check that V satisfies the conditions in Theorem 3.2 with the weight Thus, V is an interpolating variety for It is easy to verify that and and thus that We see that the upper bound in (4.4) is sharp up to a constant term and is best possible. It also shows that the estimate in Theorem 4.1 is sharp up to a constant multiplier (cf. [36]). 5. Geometry of interpolating varieties We have seen in the previous section that the counting function, which is a geometric quantity describing the size of the varieties, can be estimated

273

for interpolating varieties. It furnishes a general geometric characteristic to discrete interpolating varieties. It is a natural goal to study both necessary and sufficient geometric characterizations of interpolating varieties. However, this seems to be a very hard problem (see [11] and references therein). Even simple looking variations of weights often lead to considerable difficulties. We present two theorems in this direction. The following theorem of Berenstein and Li gives a complete solution to the above problem in the one-dimensional case, using counting functions, for slowly decreasing radial weight Theorem 5.1. [6] A discrete variety is an interpolating variety for if and only if for some constants A, B > 0,

and

Although Theorem 5.1 can be generalized to other radial weights, the necessary and sufficient conditions do not coincide, which is a phenomenon characteristic of functions of infinite order ([6]). Earlier results in this direction, especially for the weight can be found in [43], and related results for entire functions with prescribed indicator of growth can be found in [25]. In the higher dimensional case, a similar sufficient condition was found by Berenstein, Chang and Li in [1] for manifolds of codimension one. Theorem 5.2. Let If

for some

with

on

for some A,B > 0 , then V is an interpolating variety for Unlike the one-dimensional case, the condition (5.1) in Theorem 5.2 is no longer necessary for interpolation (cf. [1]). For example, consider and where

and is a Liouville number. Then V is an interpolating variety (see [11] or [10]). However, one can show that the condition (5.1) in Theorem 5.2 does not hold.

274 6. Interpolation and gap theorems Due to applications to the theory of Dirichlet series and some other problems such as representation of solutions of infinite-order linear partial differential equations (see e.g. [2, Ch. 6] and [4]), one needs to study interpolation for the space of entire functions of minimum type, instead of the space Recall that is the space of entire functions satisfying that Although the interpolation problems for and are related, the latter one is evidently more delicate due to extra difficulties caused by the zero type. The space has a natural projective limit structure. Under its natural locally convex topology, it becomes a nuclear Fréchet algebra. Algebras of this type naturally appear in complex and functional analysis. In particular, when the spaces of entire functions of infraexponential type are of considerable interest. The applications discussed in this section mainly concern this weight. It will be assumed that weights in this section are slowly decreasing radial weights, which contain the weight as a special example. The following necessary and sufficient conditions were recently found by Li in [33]: Theorem 6.1. Let interpolating variety for morphic mapping and

be a discrete variety. Then V is an if and only if there exists an entire holowith such that

In the special case geometric conditions in terms of counting functions of the variety V were found by Berenstein, Li and Vidras ([7]), partially motivated by the following well-known gap theorem of Pólya and Levinson [30], [2]. It has interesting applications to the theory of Dirichlet series: Theorem 6.2. Let be a sequence of complex numbers converging to infinity and satisfying that

as

and for some c > 0 and any

275 Then there exists an entire function that for all as

vanishing at

and satisfying

and

as

Clearly, the conclusion of the above theorem is equivalent to the statement that V is an interpolating variety for the space of entire functions of infraexponential type. The sequence V given in a geometrical manner is thus a special interpolating variety for A much more general theorem was given in [7]: Theorem 6.3. Let V be a discrete variety in C . Then V is an interpolating variety for if and only if

and

As a consequence of Theorem 6.3, we obtain the following Theorem 6.4. Let be a sequence of non-zero complex numbers converging to infinity and satisfying that for some

as

and for some c > 0 and any

Then there exists an entire function that for all as

vanishing at

and satisfying

and

as

Theorem 6.4 (for the weight Levinson theorem (for the weight

is more general than the Pólyaand also removed the conditions

276

(6.1) and (6.2) in the Pólya-Levinson theorem. A different proof of Theorem 6.4 for the weight can be found in [49] or [2], where a function F was constructed, whose zero set containing is an interpolating variety for the space of functions of infraexponential type. We conclude this paper by the following gap theorems due to Berenstein, Kawai and Struppa [4]. Let be sequences of non-zero real numbers such that

and

for a constant C > 0. Let variety defined by

is an interpolating variety for numbers satisfying that for every Theorem 6.5.

Let

be polynomials in

such that the

Let there is a

be complex such that

denote the infinite series given by

where is defined above. Then and for any real vector operator of infinite order

is a well-defined hyperfunction on there is a linear differential

such that

The important point of this theorem is that G is of local character and so the behavior of the function at distinct points is related by a local

277

operator. The function satisfies a system of infinite-order partial differential equations whose characteristic variety is contained in an interpolating variety V. An immediate consequence is the following regularity theorem: Theorem 6.6. Suppose that the function lytic in a neighborhood of the origin of

in Theorem 6.4 is real anaThen is real analytic in

Acknowledgements

Both authors are supported in part by NSF grants. References 1. Berenstein C.A., Chang D.C., and Li B.Q., Interpolating varieties and counting 2. 3. 4. 5. 6. 7. 8. 9.

10. 11.

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JET BUNDLES AND ITS APPLICATIONS IN VALUE DISTRIBUTION OF HOLOMORPHIC MAPPINGS

PEI-CHU HU Department of Mathematics, Shandong University, Jinan 250100, Shandong, P. R. China, [email protected] AND CHUNG-CHUN YANG Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, [email protected]

Abstract. In this paper, we have established the technique of the higher dimensional jets and applied the results to study value distribution of holomorphic mappings. As applications, we have also generalized the results of holomorphic curves obtained by Ochiai, Noguchi and Green-Griffiths to the higher dimensional cases. Mathematics Subject Classification 2000: 32A22, 32H30. Key words and phrases: jet bundle; holomorphic mapping; Nevanlinna theory

1. Introduction Interest in classical questions about solutions of a system of equations raised by Bloch [2] was revived by Ochiai [15]. Both introduced jets into the picture. Bogomolov [3] then used symmetric differentials. This was followed by the work of Green-Griffiths [6] from the point of view of jet differentials and curvature, and a conjecture was given by Lang [10]. Using the main theorem A of Ochiai [15] obtained by using jets, Noguchi [12], [13] proved a theorem of Second Main Theorem type for holomorphic curves into smooth complex projective algebraic variety. The purpose of this paper is to generalize the

© 2004 Kluwer Academic Publishers

282 theory to the higher dimensional cases by developing the corresponding jet theory. The main technique of the theory above is the jet theory based on germs of holomorphic curves. That is the foundation of value distribution theory of holomorphic curves of Ochiai and Noguchi. To generalize the theory to the higher dimensional cases, the first step is to establish the technique of jets based on germs of holomorphic mappings from polydiscs into complex manifolds (see Section 2). As an application, we extend the main theorem of Ochiai as follows (see Section 4): Theorem 1.1. Let X be a quasi-projective algebraic manifold of dimension and be a system of closed regular rational 1-forms on X. Assume that integers and satisfy the condition:

Let respect to

be a holomorphic mapping which is not degenerate with Put

Then for a rational function on X such that can be defined, the meromorphic function is algebraic over the field generated by

Let N be a smooth complex projective algebraic variety of dimension and D a hypersurface in N. We denote by the sheaf of germs of logarithmic 1-forms along D over N (see Section 5). Let be a holomorphic mapping, denote the characteristic function of relative to a Kähler form on N and the counting function for D (see Section 3). By using Theorem 1.1, we extend the main theorem of Noguchi [12] (for the precise statement, see Theorem 5.4) as follows: Theorem 1.2. Assume that there exists a system of logarithmic 1-forms such that the n-forms

are linearly independent over Let ping which is not degenerate with respect to condition (1). Then there is a positive constant

be a holomorphic mapwhere satisfies the such that

283

where

is a small term satisfying

The condition (1) implies that cases, that is, If

and has extremal

in fact, Noguchi [12] obtained

where is the counting function for D without counting multiplicities. Noguchi [13] also proved (3) for holomorphic curves where is the punctured disc with center at the infinity of the sphere Riemann. For the higher dimensional cases, we also prove (2), but we can’t obtain (3) for (see Section 6). 2. Jet bundles

Firstly we state some terminologies. Let V be a complex vector space of dimension Let be the symmetric tensor product of V which is a linear subspace of the tensor product of V with

For write

let if define

be the symmetric tensor product and For

If is a base of V, then then (5) is the usual product. Let

is a base of

be the standard coordinates on

If and write

284

Now let V be the complex vector space of dimension over Thus for a smooth mapping differential

spanned by the (total)

is given by

For any positive number we let be the open disc of radius Further, if is a sequence of positive numbers, the polydisc is given by the product of the discs

Let X be a complex manifold of dimension Given we consider the set of germs of holomorphic mappings that satisfy be the vector space of germs of holomorphic mappings from neighbourhoods of a point into For define

where

Define

Then is a subspace of Theorem 2.1. Take Let coordinate system in a neighbourhood of

Then

be a local holomorphic if and only if

where Proof. By definition,

if and only if

Without loss of generality, we may assume germ H is given by its covergent series

The

285 Take Then Hence we have

can be written as

where

which gives

Hence by (12) and (15), the condition (11) implies Conversely, let us assume that and choose

such that

Then (15) and (16) yield

Hence

implies

Since (17) holds for all zero.

the corresponding coefficients must be

Set

and define a quotient space

The equivalence of the germ or simply a with a source and a target or Then define

is called a jet of order and denoted by

286 The linear structure on induces a linear structure on that is a complex vector space. That is, for H ,

such

We can prove the following result: Theorem 2.2. There is a complex structure on becomes a holomorphic vector bundle with fibers

such that it

where

Then

is the holomorphic cotangent space to X at

where is a local holomorphic coordinate system in a neighborhood of Several standard constructions of differentrial geometry fit into the language of jet bundles. For example the holomorphic cotangent space to X at is defined by

and the differential of a germ

at

is defined by

Note that for

is well-defined. Take if

the relation

We say that

The equivalence classes determined by with source and target The with target is denoted by

and

are

are called jets of order of a mapping

at

or

287

Theorem 2.3. Take coordinate system at

Let Then

be a local holomorphic if and only if

Proof. Without loss of generality, we may assume then (15) implies

If (26) holds,

which yields (25), i.e., at Conversely let us assume Hence (27) are satisfied. By (15), (27) is equivalent to the following:

which implies (26) as claimed. Now let

denote the set of all

Theorem 2.4.

with source

and target

Define

is a complex manifold of dimension

Proof. We introduce the natural projection

defined by If denotes a coordinate covering of X, then we define a topology on by prescribing a coordinate covering to have underlying open the set sets Now if denote the coordinate functions on rem 2.3 implies that we may define a coordinate system

on

by

then Theo-

288

where The Leibniz product formula together with the chain rule guarantee that a holomorphic change of local coordinates in will induce a holomorphic change of local coordinates in The fact that this change of local coordinates has a non-zero Jacobian determinant follows from the fact that the matrix is block upper triangular with the diagonal blocks given by symmetric powers of the Jacobian of the original coordinate change. Thus we have defined a complex structure on The total dimension of is given by

Usually and

is called the holomorphic tangent bundle of order is the holomorphic tangent bundle of X.

Now, we can define the following relation between

for from X into a section and a holomorphic mapping

of

and

If H is a holomorphic mapping is associated by

is defined by Write

where

According to (34), if phic coordinate system

is given by

is an open set on which we have the holomorthen this choice of coordinates

289

can be rewritten

where

if

which induces an isomorphism:

Thus for nations over

of

is a polynomial vector on with respect to whose coefficients are linear combipartial derivatives of with respect to

If X has a holomorphic quasi-coordinate system, i.e. a system of holomorphic functions on X such that never vanishes on X, then a trivialization

is well-defined, where

is the projection. by

Given a holomorphic mapping with the defined by the germ of at

will be used to denote the natural lifting of morphic mapping

to

we denote The notation

In general, a holo-

between complex manifolds X and Y induces a mapping

290

on

given by

We can describe the relation (35) by using the dual relation between T( X ) and Thus can be given as an element of

so that

If at

then in terms of a local holomorphic coordinates

Hence and is a vector bundle. In general, the jet manifolds are holomorphic fibre bundles over X, but for they are not vector bundles. However, can be fibered by the projection:

For simplicity, we look at

Using local coordinates on X so that jet coordinates may be expressed in the form (41), the fiber of (52) amounts to fixing and having free to vary over If, moreover,

the transform like a tensor in Hence the fibers of (52) are vector spaces whose associated vector bundle is 3. Nevanlinna type of theory

Let a complex manifold X of dimension morphic mapping

be given. We will study a holo-

291 Set

We note that (cf. [17])

Let

be a (1,1)-form on an open set of

and define the trace of

We can write

by

Lang [11] gave a neat expression for the trace:

For

Further, if

and

is a

if the integrals exist, where omit the subscript

define

on

write

dim

If

we will

292

Let be a meromorphic function on imity function of by

where

Fix

Define the prox-

and define the valence function of

where is the pole divisor of of is defined by

by

The Nevanlinna characteristic function

Let X be a compact Kähler manifold with the Kähler form Let be a holomorphic mapping. Then the characteristic function of relative to is defined by

If

is another Kähler form on X, then there are positive constants such that

and

Therefore, the order

of is defined independently of the choice of If is the complex projective space and if is the Fubini-Study Kähler form on put

Let

and

be real valued functions in

We write

if for where E is a Borel subset of finite measure. If is a monotone increasing function for then is differentiable at almost all points, and for

See Lemma 5.5.35 of Noguchi-Ochiai [14].

293

Lemma 3.1. Let X be a compact complex manifold of dimension and a Hermitian metric on X with Hermitian metric form Let be a holomorphic mapping. Let be a holomorphic 1-form on X and put

Then for

we have

Proof. Since X is compact, there exists a positive constant

for all holomorphic tangent vectors

such that

Put

Then

It follows from (58) that

From this, the concavity of the logarithm, (53)-(55) and noting that

we estimate

294

Lemma 3.1 is a generalization of Nevanlinna’s lemma on the logarithmic derivative. For see Lemma 6.1.29 of Noguchi-Ochiai [14]. In a manner similar to the proof of Lemma 6.1.5 of Noguchi-Ochiai [14], we can obtain Lemma 3.2. Let be a meromorphic function on morphic functions on such that and

and

holo-

Then

Let denote the field of meromorphic functions on X. The following fact is well known: Lemma 3.3. Let X be a projective algebraic manifold and let be a holomorphic mapping. Let be a system of generators of such that is defined. Then there are positive constants and such that

Proposition 3.4. Let X be a compact Kähler manifold with Kähler form Let be a holomorphic 1-form on X. Let be a holomorphic mapping. Define an entire function G by

Then we have a positive constant

Proof. Setting

then

such that

295

Therefore we have

where we used the following formula (see [17], p.188):

for a a and a with Since X is compact, there exists a positive constant such that Then, noticing we obtain

Let be the line bundle over X determined by a divisor D on X. Let (the first Chern class of [D]) be the curvature form of a metric in [D] and take a section such that the divisor equals to D and If then we have (cf. [17])

where

296 Since is positive definite and X is compact, there is a positive constant K such that is semi-positive definite, so that

Next, we shall deal with holomorphic mappings

where We assume that functions on and mappings from are defined in neighborhoods of in Let be a function on satisfying (i) is differentiable outside a thin analytic set; (ii) can be expressed locally as a difference of two plurisubharmonic functions. Then we have

where

is taken in the sense of currents, and where

Let D be an effective divisor on X such that We take a metric in the line bundle [D] and denote by the curvature form of the metric. Let be a global holomorphic section of [D] such that with Applying (67) to and using the Poincaré equation of currents (cf. [7]),

we obtain

where the valence function

of D restricts on

and where

297

Let be a many-valued meromorphic function on such that the modulus is one-valued. Applying (67) to and applying the Poincaré equation (cf. [7]),

we get a Jensen formula as follows:

where the valence functions yields

Let will be

and

restrict on

Thus (70)

be a reduced representative of an inhomogeneous form. Applying (67) to

where for the Fubini-Study form and (72), we can prove

on

and we obtain

By using (70)

According to the method of Vitter [19], we also obtain Lemma 3.5. If is a multiplicative meromorphic function on for

then

4. Degeneracy theorems for holomorphic mappings

We keep the natation from Section 2, and shall consider closed holomorphic 1-forms on X. Let be the universal covering space of X and

298

the projection. Fix a point

in

Define a holomorphic

mapping

Then

is independent of

Hence there exists uniquely a holomorphic mapping

such that We call I the canonical mapping attached to

Write

Then Now suppose that X has a holomorphic quasi-coordinate system We write

Then is a holomorphic quasi-coordinate system on By using the trivialization (43), then

but

299

so that Now suppose

We write

for

Hence to combinations over

Then

is a polynomial mapping with respect whose coefficients are linear of partial derivative of with respect to (cf. Ochiai [15],

Lemma 2.1). Let W be a connected open subset of and let holomorphic mapping. Let be any lift of

be a Put

Then we have

and by (80)

From now on, we assume that integers

Then we have

and

satisfy the condition:

300 A holomorphic mapping between complex manifolds M and N with dim M dim N is said to be nondegenerate if contains an open subset of N. Otherwise, it is degenerate. Lemma 4.1. Let X be a simply connected complex manifold of dimension Let be a holomorphic quasi-coordinate system on X. Let

be a closed holomorphic 1-form on X. Let

be the canonical mapping attached to Let be a holomorphic mapping. If I is degenerate at each point of then there exists such that

at each point of Proof. Put

Without loss of generality, we may assume Let be a holomorphic tangent vector at

and

Write

Then there exist a neighbourhood and a holomorphic mapping

where

We have

Take

of

and note that

a positive number such that

301

Write

By (86)-(88), we have

and

Therefore the Jacobian of I at

where

is of the form

Since I is degenerate at each point of

Thus there exists

Note that

Lemma 4.1 thus follows.

such that

then

302

Let be a system of closed holomorphic 1-forms on X. Then we call a holomorphic mapping degenerate with respect to if there exists such that

at every point of Lemma 4.2. Let X be a complex manifold of dimension be a system of closed holomorphic 1-form on X. Let

be the canonical mapping attached to Let a holomorphic mapping. If I is degenerate at every point of is degenerate with respect to

Let

be then

Proof. If one of

is zero at every point of

at for some system U of such that

we have nothing to do. So let us assume

Then there exists a holomorphic local coordinate defined on a simply connected open neighbourhood

Let

be the canonical mapping attached to Let be a simply connected open neighbourhood of in W such that Put Note that also is generate at every point of From Lemma 4.1, there exists such that

at every point of proved.

Since W is connected, the assertion has thus been

303 Theorem 4.3. Let X be a quasi projective algebraic manifold of dimension and be a system of closed regular rational 1-forms on X. Let be a holomorphic mapping which is not degenerate with respect to Put

Then for a rational function on X such that can be defined, the meromorphic function is algebraic over the field generated by

Proof. Fix There exists a non-singular Zariski open subset N of X and regular rational functions on N such that and that is a holomorphic quasi-coordinate system on N. Fix a simply connected open neighbourhood W of in such that Put Then the holomorphic mapping is not degenerate with respect to Let

be the canonical mapping attached to

Set

Then is a regular rational function on N, and I is a regular rational mapping. From Lemma 4.2, I is of maximal rank at some point of Taking W sufficiently small, we can assume that I is of maximal rank at every point of Let A be the Zariski closure of in

and B be the Zariski closure of I(A) in

the Zariski closure of

in

Then is B. Since W is

an irreducible analytic subset, A and B must be irreducible Zariski closed subsets. Since B is the Zariski closure of I(A), we have dim A dim B. Since is a Zariski dense open subset of A, where there exists a point such that Since

304 is of maximal rank at we see that the differential of at is injective, and so dim A dim B. Hence dim A = dim B and is of maximal rank. Now take any rational function on X such that can be defined. Then we can consider

as a rational function on

Then rank, putting

can be defined. Since

we know that

is of maximal

is algebraic over the field generated by

Thus there exists a relation of the form

where are polynomials of the functions A is the Zariski closure of we see that Hence (95) implies

From (85), we see that

where

Since

and on

Since

are polynomials of

Theorem 4.3 follows.

Theorem 4.4. Let X be a projective algebraic manifold of dimension Suppose X has a system of regular rational 1-forms such that the system of

is linearly independent. Then any holomorphic mapping degenerate with respect to

is

305 Proof. Assume, to the contrary, that the holomorphic mapping X is not degenerate with respect to Let be any system of generators of such that is defined. Define entire functions by

Then we have where are defined by (94). From the lemma of the logarithmic derivative of several complex variables (cf. [1], [19]), we obtain

for over the field generated by

From Theorem 4.3,

is algebraic

Hence Lemma 3.2 implies

where

Setting

from (97) and (98), we obtain

From Lemma 3.3 and Proposition 3.4 we have

where

are constants. Then (99) and (100) give

At least one of is not constant. Hence at least one of is not a rational function. Hence is a strictly increasing function with

Thus (101) is impossible.

306 To state some corollaries of Theorem 4.4, we recall that a holomorphic mapping into an algebraic variety is said to be algebraically degenerate in the case the image lies in a proper algebraic subvariety. Let X be an irreducible Zariski closed subset of an Abelian variety A. According to Ochiai [15], X is said to be in good position in A if and X satisfies the following conditions: (i) if is a non-zero regular rational 1-form on A, then the restriction is non-zero; (ii) if B is a connected algebraic subgroup of A such that B leaves X invariant, then B is either {0} or A. By Ueno [18], the condition (ii) is equivalent to saying that the proper subvariety X of A is of general type. If X is in good position in A, by Kawamata [9] (also see Ochiai [15] and Smyth [16]) there exists a system of regular rational 1-forms on A such that the restriction of the system of

onto is linearly independent. Let N be a projective algebraic manifold of dimension with irregularity Then we can construct a regular rational mapping (Albanese map) into a certain Abelian variety A such that A. Take integers and such that

is in good position in

Let be the dimension of X and let be the regular rational 1forms on A given above. Let be the resolution of the singularity of X. If is in the singular locus of X, the holomorphic mapping clearly is algebraically degenerate. If there exists a holomorphic mapping such that Since satisfies the condition in Theorem 4.4, is degenerate with respect to Therefore is algebraically degenerate. Hence we have Corollary 4.5. Let N be a projective algebraic manifold of dimension with irregularity Then any holomorphic mapping is algebraically degenerate. Corollary 4.6. Let N be a closed irreducible subset of an Abelian variety A and let be a holomorphic mapping. Then the image is contained in a translation of an Abelian subvariety of A.

307 Corollary 4.6 and Kawamata structure theorem [9] imply Corollary 4.7. Let N be a closed irreducible subset of an Abelian variety. If N is of general type, then any holomorphic mapping is algebraically degenerate. For the case see Ochiai [15], Kawamata [9] and Noguchi-Ochiai [14]. See also Green-Griffiths [6]. 5. Second Main Theorem type problems

Let X be a compact Kähler manifold of dimension Let D be an effective divisor on X. In a small neighborhood U of each point of X there is a holomorphic function such that Since the ring of holomorphic functions defined in a small neighborhood U of is a unique factorization domain, thus can be written as a product

where are irreducible, relatively prime in and unique up to multiplication by units. Then a divisor is defined on X by Let be the irreducible analytic hypersurfaces of X. Then on U, D and are formal linear combinations

Let denote the sheaf of germs of non-zero meromorphic functions whose zeros and poles are contained in D, and be the sheaf of germs of meromorphic closed 1-forms with In a small neighborhood U of each point of we can take a local coordinate system so that Then every global section is written in U as

where is an integer and residue of on by so that the residue is obtained (cf. Noguchi [12]).

does not contain the term

Define the

308 Let

Then

Since

be a holomorphic mapping and write

is a meromorphic function on For simplicity, we assume that knowing that the exceptional case can be treated. Set

may be a pole of

only if

and

then

Hence

By induction, we obtain

We fix once and henceforth a Kähler metric and associated form Lemma 5.1. There are constants and such that

on X.

Proof. Set

Let be the line bundles over X determined by and take a section such that Let be a finite open covering of X such that every is simply connected and each restriction of is trivial. Then each is given by a system of holomorphic functions in In each is written as

309 where is a holomorphic closed 1-form in so that

Take a metric

in each

and set

Then (107) yields

for

Let be the segment which connects 0 and Take points on L so that and the image of each closed line segment of L between and by is contained in some It follows from (108) that

We may assume that each closed 1-form is defined in a neighbourhood of the closure of Since X is compact, there is a positive constant K such that

for any vector field

in every

Set

310

Then

we get

Using the notation

where

Hence we have

Note that

where

and note that

Increasing K, and if necessary, we obtain Lemma 5.1 from the inequalities (109)-(111), (65) and (66). For a given holomorphic mapping a quantity satisfying

let us denote by

for all if is of finite order, and otherwise except for Borel subset in of finite measure.

belonging to a

311

Lemma 5.2. Define

by (105). Then

Proof. Note that

Suppose that

with Lemma 5.1 implies that of Biancofiore-Stoll [1], we have

If

is of finite order. Then

is of finite order. By Proposition 3.7

is of infinite order, then

See [1] or [19]. Now Lemma 5.2 follows from (113), (114), Lemma 5.1 and the inequality (55). Now (106) and Lemma 5.2 give

Let

Let X be a projective algebraic manifold N and D a hypersurface in N. (resp. be the sheaf of germs of holomorphic functions (resp. 1-

forms) over N. Let

denote the sheaf of germs of the form

with and where which is called the sheaf of germs of logarithmic 1-forms along D (cf. Deligne [4], Iitaka [8] and Noguchi [12]). Lemma 5.3. Let N be a projective algebraic manifold and D a hypersurface in N. Let be a holomorphic mapping and set

for

Then

312

Proof. By Proposition 1.2 of Noguchi [12], there are and such that

Setting

we get

By Lemma 5.2, we obtain

According to (103)-(106), we easily prove

Hence (117) follows. Theorem 5.4. Let N be an projective algebraic manifold and D a hypersurface in N. Suppose that there exists a system of rational closed l-forms such that are linearly independent (over where

Take integers

and

such that

Let be a holomorphic mapping which is non-degenerate with respect to i.e. the restriction is not degenerate with respect to Then there is a positive constant such that where

is a Kähler form on N.

313

Proof. Set

Using inductively Vitter’s generalization on Nevanlinna’s lemma on the logarithmic derivative in several complex variables, Lemma 5.2 and 5.3, we see that

Let

be a system of generators of the rational function field over such that all are defined. By Theorem 4.3, there are algebraic relations

where

for

are rational functions in

Hence Lemma 3.2 and (120) imply that this is a positive constant K such that for all

Now (119) follows from (122) and Lemma 3.3.

Remark 1. If

If

(119) is nothing but Noguchi’s inequality [12]:

(119) yields

but we cannot derive the inequality (123), where

Remark 2. Let be a complex torus and the natural flat Kähler form on N. Let and be effective divisors on N with no common component which are homologously equivalent. Then any

314

holomorphic mapping omitting is necessarily algebraically degenerate. The proof can be given according to Theorem 4.1 of Noguchi [12]. 6. Second Main Theorem type Problems (II)

In this section, we use some notations in Section 5 and consider a holomorphic mapping in a compact Kähler manifold X with a Kähler metric and the associated form Let where D is a hypersurface in X. By Weil [20], p. 101, or Noguchi [13], there is a multiplicative meromorphic function on X such that the divisor equals to Res and such that

where

is a holomorphic 1-form on X. We set

Then by Weil [20], p. 101, there is, respectively, a metric in each of such that both metrics have the same curvature form furthermore there are sections and such that and

We put

Lemma 6.1. Let the notation be as above. Assume that Res Then

Proof. Set

Then we have

Take a positive constant

so that

Supp

315

for every holomorphic tangent vector

Setting

we get

According to the proof of Lemma 3.1, we also obtain

for

Set and by (126),

Then g is a multiplicative meromorphic function on so that

On the other hand, where the valence functions restrict on (70), that Letting

Thus we see, taking into account

be a positive constant such that

We have

we complete the proof by combining (129) with Noting that (132), (133) and Lemma 3.5. Making use of Lemma 6.1 and Theorem 4.3 as in (115), Lemma 5.3 and Theorem 5.4, we have the following theorem: Theorem 6.2. Let N be an complex projective algebraic smooth variety and D an effective reduced divisor on N. Assume that there is a system in such that are linearly independent over Take integers and such that

Let be a holomorphic mapping such that Assume that is non-degenerate with respect to Let be a Kähler form on N. Then there is a positive constant depending only on and such that

where

restricts on

316

(134) implies

If

which is the inequality of the second main theorem type proved by Noguchi [13]. For (134) yields

7. Geometric notes

We keep the notation from Section 2. Set

and take

Take

Then

and define

acts on

by setting

writing

For

then

Thus we define an equivalent relation in

so that we have

the quotient

Here objects

is a weighted projective space (cf. Dolgachev [5]). For are less familiar.

the

317

We can define the

Given

on

and

by

we get

and define

In the coordinates (41),

Let for

denote the nonconstant jets - i.e., those with some - then this action preserves and we define

Here is the projectivized tangent bundle has been studied by Green-Griffiths [6], but are less familiar for and Note that is the fiber of Now we only consider the

Denote the quotient of the

An action of (41),

action on

defined by

by

on jets is defined like above such that in the coordinates

We define Obviously and is the fiber of We now define the sheaves of jet differentials on a complex manifold X. On the space we consider polynomials in the variables

318

Assigning to the weight we consider polynomials that are homogeneous of weight Equivalently, the polynomial should satisfy

By taking local coordinates on X and allowing the coefficients of to be holomorphic functions, we may define the sheaf of differentials on X of weight Thus according to Green-Griffiths [6], we can study the geometric theory on Acknowledgements The work of the first author was partially supported by NSFC of China and of the second author by a UPGC grant of Hong Kong. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Biancofiore, A. and Stoll, W., Another proof of the lemma of the logarithmic derivative in several complex variables, Ann. of Math. Stud., 100, pp. 29–45, Princeton University Press, Princeton, N.J., 1981. Bloch, A., Sur les systèms de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension, J. Math. Pure Appl. 5 (1926), 19–66. Bogomolov, F., Families of curves on a surface of general type, Soviet Math. Dokl. 236 (1977), 1294–1297. Deligne, P., Equations différentielles à points singuliers réguliers, Lecture Notes in Math., 163, Springer-Verlag, Berlin, 1970. Dolgachev, I., Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981), 34–71, Lecture Notes in Math., 956, Springer, Berlin, 1982. Green, M. and Griffiths, P., Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), pp. 41–74, Springer, New York-Berlin, 1980. Griffiths, P. and King, J., Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130 (1973), 145–220. Iitaka, S., Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), 525–544. Kawamata, Y., On Bloch’s conjecture, Invent. Math. 57 (1980), 97–100. Lang, S., Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no.2, 159–205. Lang, S. and Cherry, W., Topics in Nevanlinna Theory, Lecture Notes in Math., 1433, Springer Verlag, 1990. Noguchi, J., Holomorphic curves in algebraic varieties, Hiroshima Math. J. 7 (1977), 833–853. Noguchi, J., Lemma on logarithmic derivatives and holomorphic curves in algebraic varieties, Nagoya Math. J. 83 (1981), 213–233. Noguchi, J. and Ochiai, T., Geometric Function Theory in Several Variables, Translations of Mathematical Monographs, 80, AMS, Providence, 1990. Ochiai, T., On holomorphic curves in algebraic varieties with ample irregularity, Invent. Math. 43 (1977), 83–96. Smyth, B., Weakly ample Kähler manifolds and Euler numbers, Math. Ann. 224 (1976), 269–279.

319 17. Stoll, W., Value distribution on parabolic spaces, Lecture Notes in Math., 600, Springer-Verlag, Berlin-New York, 1977. 18. Ueno, K., Classification of algebraic varieties I, Compositio Math. 27 (1973), 277– 342. 19. Vitter, A., The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44 (1977), 89–104. 20. Weil, A., Introduction à l’étude des variétés kählériennes, (French) Publications de l’Institut de Mathmatique de l’Universit de Nancago, VI. Actualits Sci. Ind. no. 1267 Hermann, Paris 1958.

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NORMAL FAMILIES OF MEROMORPHIC MAPPINGS OF SEVERAL COMPLEX VARIABLES INTO THE COMPLEX PROJECTIVE SPACE

ZHEN-HAN TU School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, People’s Republic of China, [email protected]

Abstract. In this paper, we discuss normality criteria for families of holomorphic mappings and meromorphic mappings of several complex variables into the complex projective space related to Green’s and Nochka’s Picardtype theorems, and improve an earlier result of singular directions for holomorphic curves in the complex projective space. Some related topics will also be given here. Mathematics Subject Classification 2000: Primary 32A17, 32H25, 32H30; Secondary 30D35, 30D45. Key words and phrases: Holomorphic and meromorphic mappings, normal families, Picard-type theorems, singular directions and value distribution theory.

1. Introduction Let be the extended complex plane Picard’s theorem asserts that any meromorphic function on which omits three distinct points in must be constant. Let be a family of meromorphic functions defined on a domain D of the complex plane. is said to be normal on D if every sequence of functions of has a subsequence which converges uniformly on every compact subset of D with respect to the spherical metric to a meromorphic function or identically on D. Perhaps the most celebrated criterium for normality in one complex variable is the following Montel’s theorem: the collection of all meromorphic functions on a domain which omit three distinct fixed values in is a normal family on D.

© 2004 Kluwer Academic Publishers

322

The fact that Picard-type theorems and normality criteria were so intimately related led Bloch to the hypothesis that a family of meromorphic functions which have a property P in common on a domain D is normal on D if the property P forces a meromorphic function on the complex plane to be a constant. This hypothesis is called Bloch’s heuristic principle in complex function theory (see [1, 9, 28, 36]). Rubel [28] gave some counterexamples to Bloch’s heuristic principle. Although the principle is false in general, many authors proved normality criteria for families of meromorphic functions by starting from Picard-type theorems. Hence an interesting topic is to make the principle rigorous and to find its applications. Zalcman [36] gave a well-known heuristic principle in the theory of functions as follows: Zalcman’s Principle [36]. Write to denote the function defined on the domain Let P be a property of meromorphic (holomorphic) functions which satisfies the following three conditions: and then (i) If and is nonconstant, then (ii) If P. (iii) Let where and If spherically uniformly on compact subsets of Suppose that only if is constant. Then normal on D for each

then is

By Picard’s theorem and Zalcman’s principle we immediately have Montel’s theorem. Zalcman’s principle provides a highly satisfactory explication of the heuristic principle so far as properties formulated in terms of the values taken on or omitted by functions is concerned. In such cases, conditions (i) and (ii) will generally be satisfied trivially, while (iii) follows more or less routinely from Hurwitz’s theorem. There are many investigations in this field for one complex variable (see, e.g., [1, 28, 36, 37] and their references for related results). In the case of higher dimension, the notion of normal family has proved its importance in geometric function theory in several complex variables (see, e.g., [18, 19, 23, 35]). Bloch [4], Green [15] and Nochka [22] established some Picard-type theorems for holomorphic mappings into the complex N-dimensional projective space. Fujimoto [12] and Nochka [22] gave some normality criteria related to Green’s and Nochka’s Picard-type theorems in several complex variables by using various methods. Aladro and Krantz [1] proved a criterion for normality in several complex variables and for the first time implemented a Zalcman-type heuristic principle in this more general content. By modifying the heuristic principle obtained

323 by Aladro and Krantz [1], the present author [31, 32, 33] proved some normality criteria for families of holomorphic mappings and meromorphic mappings of several complex variables into related to Green’s and Nochka’s Picard-type theorems. The equivalence of normality to being uniformly Montel at a point is given in [32]. These topics will be discussed in Sections 2 and 3. In the Section 4 we will apply our normality criteria to improve an earlier result of singular directions for holomorphic curves in the complex projective space. Some related topics will be discussed in Section 5. 2. Normal Families of Holomorphic Mappings in Several Complex Variables For the general reference of this section, see [18, 23, 32, 33, 35]. Let be a complex N-dimensional projective space and let be the standard projective mapping. A subset H of is called a hyperplane if there is a N-dimensional linear subspace of such that If we write for the dual space of then there is such that Let be the set of Euclidean unit vectors in Then satisfy if and only if with and Let be hyperplanes in Let such that Define

which only depends on but does not depend on the choice of with When N = 1, is just the spherical distance between Definition 2.1. Define

Let

be hyperplanes in

where the product is taken over all We say the hyperplane family to be in general position if Let D be a domain in and vanishing identically. For a point

with in

a holomorphic function on D not we expand as

324 a compactly convergent series

in a neighborhood of where is either identically zero or a homogeneous polynomial of degree The number is said to be the zero multiplicity of at the point Set for and Let be a holomorphic mapping of into Then there exists a holomorphic mapping of into such that and on We call to be a reduced representation of on Let H be a hyperplane of and We say that the holomorphic mapping intersects the hyperplane H with multiplicity on if and the holomorphic function on has zero multiplicities at all the zeros of on while at least one zero has multiplicity We say that the holomorphic mapping intersects the hyperplane H with multiplicity on if or Hence we always have Let be a holomorphic mapping of a domain D in into and let H be a hyperplane of We say that the holomorphic mapping intersects the hyperplane H with multiplicity at least on D if intersects H with multiplicity at least in any contained in D. Bloch [4] and Green [15] gave the following Picard-type theorem: Theorem 2.2 [4, 15]. A holomorphic mapping 2N + 1 hyperplanes in general position in

that omits is a constant.

In 1983, Nochka [22] improved the above theorem and proved the following Cartan conjecture: Theorem 2.3 [22]. Suppose that hyperplanes given in general position in along with positive integers (some of them may be ). If

are

then there does not exist a nonconstant holomorphic mapping such that intersects with multiplicity at least Definition 2.4. A family of holomorphic mappings of a domain D in into is said to be normal on D if any sequence in contains

325 a subsequence which converges uniformly on compact subsets of D to a holomorphic mapping of D into and is said to be normal at a point in D if is normal on some neighborhood of in D. Suppose that are 2N + 1 hyperplanes given in general position in Then is a complete hyperbolic manifold and thus the family of holomorphic mappings of a domain in into is normal, see Theorems 1.5.8 and 1.8.9 in [23]. The proof can be obtained by elementary technique of complex hyperbolic spaces. This idea does not work in proving normal criteria related to Theorem 2.3. Using Cartan’s second main theorem for nondegenerate holomorphic curves [6] and the method in [9], Fujimoto [12] proved some normality criteria for a family of meromorphic mappings into Aladro and Krantz [1] proved a criterion for normality in several complex variables and for the first time implemented a Zalcman-type heuristic principle in this more general context. By modifying the heuristic principle obtained by Aladro and Krantz [1], the author [31, 32] gave the following results related to Theorem 2.3. Theorem 2.5 [31, 32]. Let be a family of holomorphic mappings of a domain D in into Suppose that for each there exist hyperplanes which may depend on in such that intersects with multiplicity at least where are fixed positive integers independent of and may be with

and Then

is a normal family on D.

By Theorem 2.5 we immediately have the following corollaries. Corollary 2.6 [31, 32]. Let domain D in into 2N + 1 hyperplanes

for

Then

be a family of holomorphic mappings of a Suppose that for each there exist in such that

and

is a normal family on D.

326

Corollary 2.7 [31, 32], Let be a family of holomorphic mappings of a domain D in into Suppose that are hyperplanes given in general position in along with positive integers such that

If each intersects with multiplicity at least is a normal family on D.

then

Definition 2.8. Let be a family of holomorphic mappings of a domain D in into is said to be uniformly Montel on D if for any there exist 2N + 1 hyperplanes (which may depend on ) located in in general position such that

for i = 1,2, ...,2N + 1 and

is said to be uniformly Montel at a point on some neighborhood of in D.

in D if

is uniformly Montel

For example, let be a family of holomorphic mappings of a domain D in into Obviously, is uniformly Montel on D if there exist 2N + 1 hyperplanes located in in general position with for any The author [32] proved the following necessary and sufficient Monteltype criterion for normality in several complex variables: Theorem 2.9 [32]. Let be a family of holomorphic mappings of a domain D in into Then is normal at a point in D if and only if is uniformly Montel at the point 3. On Families of Meromorphic Mappings in Several Complex Variables For the general reference of this section, see [12, 23, 33]. Let A be a non-empty open subset of a domain D in such that S = D \ A is an analytic set in D. Let be a holomorphic mapping. Let U be a non-empty connected open subset of D. A holomorphic mapping from U into is said to be a representation of on U if for all where is

327 the standard projective mapping. A holomorphic mapping is said to be a meromorphic mapping from D into if and only if for any there exists a representation of on some neighborhood of in D. For example, let be a holomorphic mapping of a domain D in into and at least one on D. Then, obviously, the holomorphic mapping from into is obviously a meromorphic mapping of D into Define the polydisc

for and be a meromorphic mapping from has a representation

with into on

Let Then always with

A representation of satisfying this condition is called a reduced representation of on Let be a meromorphic mapping of a domain D in into Then for any always has a reduced representation on some neighborhood of in D. We denote by the set of all points of indetermination of on D. Then is an analytic set in D with dim Obviously, a meromorphic mapping from D into is a holomorphic mapping from D into if and only if Definition 3.1. A sequence of meromorphic mappings from a domain D in into is said to converge meromorphically in D to a meromorphic mapping if and only if, for any each has a reduced representation

on some fixed neighborhood U of such that converges uniformly on compact subsets of U to a holomorphic function N + 1, in U with the property that

is a representation of

on U, where

We note the following basic fact: Let morphic mappings of a domain U in into two reduced representations

on U for some be a sequence of meroIf each has

328 on U such that of U to

and

converge uniformly on compact subsets

respectively, then there exists a nowhere zero holomorphic function on U with for see (3.2) in [12]. For a detailed discussion about meromorphic convergence, see [12]. Definition 3.2. Let be a family of meromorphic mappings of a domain D in into is said to be a meromorphically normal family on D if any sequence in has a meromorphically convergent subsequence on D. Definition 3.3. A sequence of meromorphic mappings from a domain D in into is said to be quasi-regular on D if and only if any has a neighborhood U with the property that converges compactly on U outside a nowhere dense analytic subset S of U, i.e., for any domain (the closure of G is a compact subset of U \ S), there is some such that for all and converges uniformly on G to a holomorphic mapping of G into Remark. Obviously, a meromorphically convergent sequence on D is always a quasi-regular sequence on D. But a quasi-regular sequence on D is not necessarily meromorphically convergent on D, see (3.4) in [12]. Definition 3.4. Let be a family of meromorphic mappings of a domain D in into is said to be a quasi-normal family on D if any sequence in has a subsequence quasi-regular on D. Let be a holomorphic function on a connected open neighborhood D of Then where the series converges uniformly to in an open neighborhood of and the term is either identically zero or a homogeneous polynomial of degree The number is said to be the zero-multiplicity of at Let be a meromorphic mapping from a domain D in into Take a hyperplane H in defined by

For

taking a reduced representation

on a neighborhood U of

we consider the holomorphic function

329

Then is determined independently of the choice of reduced representations, hence is well-defined on the totality of D. We note that always holds for any hyperplane H in and if for some hyperplane H in We say that a meromorphic mapping intersects H with multiplicity at least on D if and for all such that and that intersects H with multiplicity on D if or Fujimoto [12] introduced the notion of meromorphic convergence and gave the following result. Theorem 3.5 [12]. Let be a family of meromorphic mappings of a domain D in into and let be 2N +1 hyperplanes in located in general position such that for each and for any fixed compact subset K of D, the Lebesque areas of counting multiplicities for all in are bounded above. Then is a meromorphically normal family on D. By using Stoll’s normality criteria [29] for families of non-negative divisors on a domain in the author [33] proved the following results. Theorem 3.6 [33]. Let S be an analytic subset of a domain D in with dim Let be a holomorphic mapping from D – S into If there exist hyperplanes in in general position such that intersects with multiplicity at least on D \ S, where are positive integers and may be with then the holomorphic mapping from D \ S into extends to a holomorphic mapping from D into Corollary 3.7 [33]. Let be a meromorphic mapping from a domain D in into If there exist hyperplanes in in general position such that intersects with multiplicity at least on D, where are positive integers and may be with then is a holomorphic mapping from D into Corollary 3.7 played a key role in proving the following result. Theorem 3.8 [33]. Let be a family of meromorphic mappings of a domain D in into Suppose that for each there exist hyperplanes (which may depend on ) in with

330

such that for any fixed compact subset K of D, the Lebesque areas of

regardless of multiplicities for all

in

are fixed positive integers and may be is a quasi-normal family on D.

are bounded above, where with

Then

4. Singular Directions of Holomorphic Curves

In this section, we will apply our normality criteria to improve an earlier result of singular directions for holomorphic curves in the complex projective space. For the general reference of this section, see [30]. Bloch [4] and Green [15] gave the following Picard-type theorem: A nonconstant holomorphic mapping cannot omit 2N + 1 hyperplanes in general position in We say that a nonconstant holomorphic mapping has an asymptotic value in if there exists a continuous path: in satisfying such that exists. By applying techniques from normal families, the author [30] gave the following existence theorem of Julia directions of holomorphic curves related to Bloch and Green’s Picard-type theorem. Theorem 4.1 [30]. Suppose that is a transcendental holomorphic mapping of into and has an asymptotic value in Then there exists a ray such that in any open sector with vertex containing cannot omit 2N + 1 hyperplanes in general position in The ray in Theorem 4.1 is called a Julia direction of We must note that not every transcendental holomorphic mapping of into has a Julia direction. Thus the condition that has an asymptotic value in Theorem 4.1 cannot be removed. Now by applying Lemma 4 in [30] and Corollary 2.7 and modifying the proof of Theorem 1 in [30], we can easily give the following new result related to Nochka’s Picard-type theorem. Theorem 4.2. Suppose that hyperplanes in general position in along with positive integers (some of them may be ) such that

are given

331

If is a transcendental holomorphic mapping of into and has an asymptotic value in then there exists a ray such that in any open sector with vertex containing cannot intersect with multiplicity at least for all Theorem 4.2 is an improvement of Theorem 4.1. 5. Some Related Topics

Gu [16] showed that for an integer and a domain the family of meromorphic functions on D such that and on D is normal on D. Pang [24, 25] gave a much stronger version of Zalcman’s principle dealing with properties formulated in terms of values omitted by derivatives of a function, which immediately implies Gu’s result [16] by Hayman’s theorem [17]. Recently, C.A. Berenstein, D.C. Chang and B.Q. Li [2, 3] proved some uniqueness theorems related to shared values of entire functions and their partial differential polynomials on and the author [34] proved a uniqueness theorem for meromorphic mappings of several complex variables into for moving targets. A very interesting problem is to study the analogue of Pang [24, 25] and the generalization of Gu [16] to the case of several complex variables related to [2, 3] and [34]. In this paper, we have focused on Picard’s theorem and normal criteria in complex variables. It turns out that the ideas involved in this topic have a much wider applicability. For example, the ideas and techniques in normal families can be used to study complex dynamics (see [7, 11]), quasimeromorphic mappings (see [10, 20, 26]), hyperbolicity (see [5, 18, 19, 23]) and minimal surfaces (see [13, 14]). We omit these topics here. Acknowledgements Supported in part by NSFC-10371091. References 1. Aladro G. and Krantz S.G., A criterion for normality in J. Math. Anal. Appl. 161 (1991), 1–8. 2. Berenstein C.A., Chang D.C. and Li B.Q., On the shared values of entire functions and their partial differential polynomials in Forum Math. 8 (1996), 379–396. 3. Berenstein C.A., Chang D.C. and Li B.Q., The uniqueness problem and meromorphic solutions of partial differential equations, J. Analyse Math. 77 1999), 51–68. 4. Bloch A., Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires, Ann. Ecole. Norm. Sup. 43 (1926), 309–362. 5. Brody R., Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213–219.

332 6. Cartan H., Sur les zéros des combinaisons linéaires de p fonctions holomorphes données, Mathematica 7 (1933), 5–31. 7. Carleson L. and Gamelin T.W., Complex dynamics, Springer-Verlag, New York, 1993. 8. Chirka E., Complex analytic sets, in Several Complex Variables I (edited by A.G. Vitushkin), Encyclopaedia of Mathematical Sciences 7, 117–158, Springer-Verlag, Berlin, 1990. 9. Drasin D., Normal families and the Nevanlinna theory, Acta Math. 122 (1969), 231–263. 10. Eremenko A., Bloch radius, normal families, and quasiregular mappings, Proc. Amer. Math. Soc. 128 (2000), 557–560. 11. Fornaess J., Dynamics in several complex variables, CBMS 87, Amer. Math. Soc., Providence, 1996. 12. Fujimoto H., On families of meromorphic maps into the complex projective space, Nagoya Math. J. 54 (1974), 21–51. 13. Fujimoto H., On the number of exceptional values of the Gauss map of minimal surfaces, J. Math. Soc. Japan 40 (1988), 237–249. 14. Fujimoto H., Value distribution theory of the Gauss map of minimal surfaces in Friedr. Vieweg & Sohn, Braunschweig, 1993. 15. Green M., Holomorphic maps into complex projective space omitting hyperplanes, Trans. Amer. Math. Soc. 169 (1972), 89–103. 16. Gu Y., Un critère de normalité des familles de fonctions méromorphes, Sci. Sinica Special Issue 1 (1979), 267–274. (Chinese) 17. Hayman W.K., Picard values of meromorphic functions and their derivatives, Ann. of Math. 70 (1959), 9–42. 18. Kobayashi S., Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. 19. Lang S., Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. 20. Miniowitz R., Normal families of quasimeromorphic mappings, Proc. Amer. Math. Soc. 84 (1982), 35–43. 21. Montel P., Leçons sur les familles normales des fonctions analytiques et leurs applications, Gauthier-Villars, Paris, 1927. 22. Nochka E., On the theory of meromorphic functions, Soviet Math. Dokl. 27 (1983), 377–381. 23. Noguchi J. and Ochiai T., Geometric function theory in several complex variables, Translations Math. Monographs Vol. 80, Amer. Math. Soc., Providence, R.I., 1990. 24. Pang X., Bloch’s principle and normal criterion, Sci. China Ser. A 32 (1989), 782791. 25. Pang X., On normal criterion of meromorphic functions, Sci. China Ser. A 33 (1990), 521–527. 26. Rickman S., On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100–117. 27. Rickman S., Quasiregular mappings, Springer-Verlag, Berlin, 1993. 28. Rubel L.A., Four counterexamples to Bloch’s principle, Proc. Amer. Math. Soc. 98 (1986), 257–260. 29. Stoll W., Normal families of non-negative divisors, Math Z. 84 (1964), 154–218. 30. Tu Z., On the Julia directions of the value distribution of holomorphic curves in Kodai Math. J. 19 (1996), No.1, 1–6. 31. Tu Z., Normality criterions for families of holomorphic mappings into Geometric Complex Analysis (edited by Junjiro Noguchi et al.), World Scientific Publishing Co., 1996, 623–627. 32. Tu Z., Normality criteria for families of holomorphic mappings of several complex variables into Proc. Amer. Math. Soc. 127 (1999), No. 4, 1039–1049. 33. Tu Z., On meromorphically normal families of meromorphic mappings of several

333 complex variables into J. Math. Anal. Appl. 267 (2002), 1–19. 34. Tu Z., Uniqueness problem of meromorphic mappings in several complex variables for moving targets, Tôhoku Math. J. 54 (2002), 567–579. 35. Wu H., Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. 36. Zalcman L., A heuristic principle in complex function theory, Amer. Math. Monthly 82 (1975), 813–817. 37. Zalcman L., Normal families, new perspectives, Bull. Amer. Math. Soc. 35 (1998), No. 3, 215–230.