276 60 6MB
English Pages 280 Year 1993
Yang Lo
Value Distribution Theory
SpringerVerlag Berlin Heidelberg GmbH
Yang Lo Institute of Mathematics Academia Sinica The People's Republic of China
Revised edition of the original Chinese edition published by Science Press Beijing 1982 as the 9th volume in the Series in Pure and Applied M athematics.
Distribution rights throughout the world, excluding the People's Republic of China, granted to SpringerVerlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Mathematics Subject Classification (1991): 30D35,30D30 ISBN 9783662029176 DOI 10.1007/9783662029152
ISBN 9783662029152 (eBook)
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© SpringerVerlag Berlin Heidelberg 1993 Originally published by SpringerVerlag Berlin Heidelberg New York 1993 Softcover reprint of the hardcover 1st edition 1993
Typesetting: Science Press, Beijing. The People's Republic of China 41/3140543210 Printed on acidfree paper
Introduction
It is well known that solving certain theoretical or practical problems often depends on exploring the behavior of the roots of an equation such as
J(z)
= a,
(1)
where J(z) is an entire or meromorphic function and a is a complex value. It is especially important to investigate the number n(r, J = a) of the roots of (1) and their distribution in a disk Izl ~ r, each root being counted with its multiplicity. It was the research on such topics that raised the curtain on the theory of value distribution of entire or meromorphic functions. In the last century, the famous mathematician E. Picard obtained the pathbreaking result: Any nonconstant entire function J(z) must take every finite complex value infinitely many times, with at most one exception. Later, E. Borel, by introducing the concept of the order of an entire function, gave the above result a more precise formulation as follows. An entire function J(z) of order A( 0 < A < (0) satisfies
1' logn(r, J = a) 1m logr
r>oo
=\
1\
for every finite complex value a, with at most one exception. This result, generally known as the PicardBorel theorem, lay the foundation for the theory of value distribution and since then has been the source of many research papers on this subject. At the beginning of this century, P. Montel introduced the concept of normality for a family of holomorphic or meromorphic functions in a region, which in some sense corresponds to the concept of compactness for this family, and established the Montel Criterion. This theorem connects
vi
Introduction
the normality of a family with the value assignment of every function of tliis family: Let:F be a family of holomorphic functions defined in a region D, and a, b two distinct finite complex values. If none of the functions in :F takes either a or b, then the family :F is normal in D. Later, by applying the Montel Criterion to an arbitrary transcendental entire function J(z), G. Julia proved that there exists a ray J : argz = 80 such that in any angular domain largz  801 < e, e > 0, the function J(z) takes every finite complex value, except for at most one value. Such a ray is known as a Julia direction of J(z), and it symbolizes the beginning of the research on the theory of angular distribution. It was R. Nevanlinna who made the decisive contribution to the development of the theory of value distributIon. Before "hIm, the prInc1pal object and tool of the theory were the class of entire functions and the maximum modulus, respectively. It was Nevanlinna who elevated the theory of meromorphic functions to a new level by introducing the characteristic function T(r, J) for a meromorphic function J(z) in a domain Izl < R(R ~ 00,0 < r < R), as an efficient tool. His theory can be summed up as follows. Let J(z) be a meromorphic function in the finite plane and a a complex value. Then the equality
N(r, J holds as r
+
= a) + mer, J = a) + 0(1) = T(r, J)
(2)
00, where
N(r,J
r
= a) =
and 1
10
n(t,J = a) dt t
r
21r
1
mer, J = a) = 211" 10 log+ IJ(rei9) _ al dB. Moreover, if aj{j = 1, ... , q) are q distinct finite complex numbers, then the inequality q
(q  2)T(r, J)
O} of the deficient values of a transcendental meromorphic function J(z) in the finite plane is at most countable and
L6(a,f) ~ 2,
(4)
where the sum is taken over all the deficient values. In this volume we intend to present a systematic and comprehensive survey of the theory of value distribution of meromorphic functions, including both the classical results as well as more recent achievements, especially the developments in the past two decades. Chapters 1 and 2 afford a brief introduction to Nevanlinna theory and the normal family, respectively, though the last section of Chapter 1 is devoted to a recent important theorem due to Osgood and Steinmetz, which improves the deficiency relation (4) as
L 6(a(z), f) ~ 2, where the sum is taken over all meromorphic functions a(z) with T(r, a(z)) = o(T(r,f)) as r t 00. In Chapter 3 we prove that every meromorphic function J(z) of order ..\(0 < ..\ < (0) in the finite plane must have a Borel direction arg z = eo such that for any positive number c and any complex number a we have 1' logn(r, 1m
r+oo
eo, c, J = logr
a)
=\
A,
except for at most two values of a, where the symbol n(r, eo, c, J = a) denotes the number of zeros of J(z)  a in the region (Izl ~ r) n (Iargzeol ~ c), each zero being counted with its multiplicity. In Chapter 4 we discuss the value distribution of a meromorphic function together with its derivatives. In this connection Hayman obtained an important inequality, where the characteristic function T(r, f) can be bounded by just two counting functions N(r, J = 0) and N(r, J(k) = 1) with k ~ 1. Corresponding to Hayman's inequality, we prove an alternative theorem: If J(z) is meromorphic in Izl < 1 with J(z) =1= 0 and
viii
Introduction
f(k)(z) =F 1 for a positive integer k, then either If(z)1 < 1 or If(z)1 > Ck
> 0 uniformly in Izl < 1/32, where
Ck does not depend on
f. Then
Gu Yongxing's criterion for normality follows immediately. Moreover, the author obtained the following general result: Let:F be a family of meromorphic functions in a region D and k a positive integer. If for each function fez) in :F, neither fez) nor f(k)(z) has a fixed point in D, then :F is normal. Finally, we shall also discuss a precise estimate of the total deficiency of f(k)(z) due to the author. In Chapter 5, we first establish the following theorem on the distribution of Borel directions: If A is a positive number and E is a nonempty closed set of real numbers (mod 211"), then there exists a meromorphic function fez) of order A such that all of its Borel directions constitute exactly the set {argz = 919 E E}. Next, we give a simple proof to an important result of H. Milloux which asserts that for an entire function fez) of order A(O < A < 00), every Borel direction of order A of f'(z) is also a Borel direction of order A of fez) itself. Then it is natural to ask if the inverse of Milloux's theorem is true or not. In this connection, Zhang Qingde and the author proved that if fez) is a meromorphic function of order A(O < A < 00) in the finite plane, then there exists a direction argz = 90(0 ~ 90 < 211") such that for any positive number e, any positive integer k and any two finite complex numbers a and b with b =F 0, we have lim log{n(r,90 ,e,f = a) + n(r,90 ,e,f(k) = b)} = A. logr
r+oo
Consequently, if the meromorphic function f (z) has a finite exceptional value, then all its Borel directions will be conserved under the derivation. Chapter 6 deals with the relationship between the number of deficient values and the number of Borel directions, due to Zhang Guanghou and the author. Given any meromorphic function of a finite positive order in the finite plane, the total number of its deficient values does not exceed the total number of its Borel directions. Moreover, if the function is entire, then the total number of its finite deficient values does not exceed half the total number of its Borel directions. In Chapter 7 we discuss Baernstein's function T* and the spread relation. As applications we present Fuchs' theorem, the ellipse theorem and a solution of the deficiency problem in the case of lower order less than one.
Introduction
ix
Most results in Chapters 4, 5, 6 and 7 were obtained in the past two decades and have not, as yet, been included in any book. On the other hand, Chapters 1, 2 and 3 provide the requisite knowledge of Nevanlinna theory, normal family and Borel direction for the reading of the successive chapters. Therefore, this volume is selfcontained and should be useful to both researchers and graduate students. The contents of this volume are the result of a series of lectures by the author at the Graduate School of Chinese Academy of Sciences several years ago, and most recently at the University of Notre Dame. The author wishes to thank Professors D. Drasin, G.Frank, W.H.J. Fuchs, W.K. Hayman, W. Stoll, A. Weitsman, H. Wu and C.C Yang for discussions and encouragement. Special thanks go to Professor Jiang Jiahe and Madam K. Weltin who read the manuscripts most carefully and provided many valuable comments. Finally, the author acknowledges the partial support provided by the National Science Foundation of China.
Lo Yang
Contents
Chapter 1 Essentials of Nevanlinna Theory ................................ 1 1.1 1.2 1.3 1.4 1.5
The PoissonJensen Formula .................................. 1 Characteristic Functions and the First Fundamental Theorem .. 5 The Second Fundamental Theorem ........................... 13 Applications of the Second Fundamental Theorem ............ 25 Generalizations of the Second Fundamental Theorem ......... 34
Chapter 2 Normal Families ................................................ 42 2.1 Normal Families of Holomorphic Functions ................... 42 2.2 Montel's Criterion ........................................... 47 2.3 Montel Cycle, Normal Families of Meromorphic Functions .... 54
Chapter 3 Borel Directions ................................................ 58 3.1 3.2 3.3 3.4
Preliminaries ................................................ 58 Fundamental Theorems ...................................... 65 Filling Disks and Borel Directions ............................ 75 Properties of Borel Directions ................................ 85
Chapter 4 Value Distribution of Meromorphic Functions Together with Their Derivatives ............................... 93 4.1 Comparison Between Growths of T(r, 1) and T(r, /') ......... 93
xii
Contents
4.2 Modular Distribution of Meromorphic FUnctions Together with Their Derivatives .............................. 99 4.3 An Inequality of Hayman ................................... 108 4.4 A General Criterion for Normality .......................... 112 4.5 Total Deficiency of Meromorphic Derivatives ................ 121 4.6 A New Method for Proving Normality ...................... 131
Chapter 5 Recent Studies on Borel Directions ........................... 138 5.1 Distribution of Borel Directions ............................. 138 5.2 Common Borel Directions of a Meromorphic Function and Its Derivatives .......................................... 148 5.3 Angular Distribution of Meromorphic Functions Together with Their Derivatives ............................. 163
Chapter 6 Deficient Values and Borel Directions of Meromorphic Functions .................................... 172 6.1 Precise Order and Three Lemmas ........................... 172 6.2 Distribution of Borel Directions of Meromorphic Functions with Deficient Values ....................................... 184 6.3 Deficient Values and Borel Directions of Meromorphic Functions .................................. 190 6.4 Deficient Values, Borel Directions and the Order of Entire FUnctions ......................................... 197
Chapter 7 The Spread Relation and Its Applications .................... 216 7.1 7.2 7.3 7.4 7.5
Sequence of P6lya Peaks and Its Existence .................. 216 The T* Function ............................................ 221 The Spread Relation ........................................ 231 Applications ofthe Spread Relation ......................... 240 The Deficiency Problem .................................... 246
Bibliography .................................................. 259 Index .......................................................... 269
Chapter 1
Essentials of N evanlinna Theory
In 1925, R. Nevanlinna[l] established two fundamental theorems; in one stroke he initiated the modern research on the theory of value distribution, and laid down the foundation for its development ever since. Therefore, the first chapter will be devoted to a brief introduction to Nevanlinna theoryl), and the last section of the chapter, as an illustration of the development, will discuss an important theorem by Osgood[l] and Steinmetz[l].
1.1
The PoissonJensen Formula
1.1.1 The PoissonJensen formula. In Nevanlinna theory, the following PoissonJensen formula plays a very important role. Theorem 1.1 Suppose f«() is meromorphic in 1(1 ~ R (0 < R < 00) and that aJ.l. (JL = 1,2,···, M) and bv (v = 1,2,···, N) are the zeros and poles of f«() in 1(1 < R, respectively. If z = rei8 is a point in 1(1 < R, distinct from aJ.l. and bv , then
~log IR(zaJ.l.)I_~l IR(zbv)1 + L..J R2 L..J og R2 b . J.I.=l

aJ.l.z
v=l

(1.1.1)
vZ
1) cf. Nevanlinna [1, 2, 3], Hayman [2], Tsuji [1], Goldberg & Ostrowski [1] and Ozawa [1].
2
Chapter 1
Essentials of Nevanlinna Theory
Proof. First of all, we consider the special case where f(() has neither zero nor pole on 1(1 :S R. Then log f( () is regular on 1(1 :S R. Thus 1. logf(O) = 2
1
I(I=R
7l"Z
d( = 2 1 1211" logf(Ret'P)dc.p. . logf(();:."
7r
(1.1.2)
0
Taking the real part of both sides in (1.1.2), we prove (1.1.1) in the case R((  z) of z = O. In the general case, set w = R2 _ z( , which maps 1(1 :S R onto Iwl :S 1 and ( = z to w = O. Its inverse is ( = R(Rw+z)/(R+zw). Letting F(w) = f(R(RW + z)/(R + zw)), we see that F(w) is meromorphic on Iwl :S 1 and has neither zero nor pole there. According to (1.1.2), we have logF(O)
1 = .
1,
2m,
Iwl=l
dw logF(w). w
Since
dw d( w = d(logw) = (z we deduce log f(z) Therefore, on 1(1
1
= 2 . 7r~
1 I(I=R
zd(
+ R2 
(R2  IzI2)d( z( = (R2  z()((  z)'
log f(() (R2
R2 lzl2 _()(( ) d(.  z  z
(1.1.3)
= R, we have ( = Rei'P, d( = iRei'Pdc.p, and
(R2  z()((  z)
= (R2 
Rrei('P 9))(Rei'P  re i9 )
= Rei'P { R2 
2Rr cos( c.p  0)
+ r2 }.
Substituting these quantities into (1.1.3) and taking the real parts, we obtain loglf(z)1
1
= 27r
1211" 0
.
loglf(Ret'P)IR2

2R
R2  r2 ( 0) r cos c.p 
+ r 2dc.p.
(1.1.4)
Next, when f(() has a finite number of zeros and poles on 1(1 = R and none in 1(1 < R, we consider the domain De consisting of all points of 1(1 :S R of distance greater than c from every pole and zero of f((). For small c, the boundary of De is denoted by reo The function log f(()(R2 lzI2) (R2  z()((  z)
3
1.1 The PoissonJensen Formula
is holomorphic in De, except for the point ( = z, which is a pole with residue log fez). Thus
Since the length of every small arc is less than 211"e and the integrand on it is O(log lie), the corresponding integral tends to zero with e. Thus we also obtain (1.1.4) in this case. Finally, when fee) has zeros and poles in 1(1 < R, denoted by a,.,.(/L = 1,2, ... ,M) and b/l(v = 1,2, ... ,N), respectively, the function
is meromorphic on 1(1 :::; R and has neither zero nor pole in therefore, according to the above result, we have 1 1271" . log Ig(z)1 = 2 log Ig(Re' 0, a contradiction. Therefore we have limv>oorv = 00, implying Eo C U~l[rv,r~], hence mes Eo ~ L:~l(r~ rv) = L:~11/(T(rv)). Since rv E Eo(v = 1,2,·· .), we have
°
T(rv) ~ T(r~_d ~ 2T(rvl) ~ ... ~ 2v 1T(r1) ~ 2v 1. So we obtain finally 1
00
mes Eo <  1 = 2.  '"" L...J 2v
v=1
If Eo n [r~,oo) = 0 for some ii = 0,1,2,···, then Eo C u~=drv,r~], and therefore mes Eo ~ 2 in view of the above argument. (2) We shall change the variable and reduce this case to (1). In fact, by setting
1 p=logR ,r=ReP , r
Po = log R
1
ro '
then T1 (p) = T( R  e P ) is defined in Po ~ P < 00. According to the case (1), the linear measure of the set E of the values p(~ Po) such that the inequality
Tl (p + Tl~P))
~ 2T1 (p)
holds does not exceed 2. If the set Eo of r corresponds to E, then
r
R dr
lEo 
r
=
r dp ~ 2.
lE
23
1.3 The Second Fundamental Theorem
When rE.Eo, we have T(r')
< 2T(r), where
1 1 log Rr' = log Rr
1
+ T(r)·
This gives r'
=R 
1
(R  r)e T(r)
= r + (R 
1
r){ 1  e T(r)}.
Noting that 1
®1
1
>  eT(r) ,
eT(r)
we obtain
r' ~ r
Rr
+ ::::;c:eT(r) .
When R  r2 < (R  rl)/e 2, it is clear that
l
r2
rl
dt
R  rl
 R = log R >2.  t  r2
Thus some r can be found in (rt, r2) such that rEEo, hence
T (r
+ Rr) eT(r) < 2T(r).
0
1.3.5 Error terms in the second fundamental theorem. We now discuss the error terms in the second fundamental theorem by using Lemma 1.3 and Lemma 1.4. Theorem 1.6 Suppose fez) is meromorphic in the finite plane and nondegenerate into a constant, and S(r, I) is expressed by (1.3.4) of Theorem 1.4 or (1.3.9) of Theorem 1.5. If the order of fez) is finite, we have S(r, I) = O(1ogr), r + 00. (1.3.25) If the order of fez) is infinite, we have
S(r, I) = O(1og(rT(r, I))), except on a set E with finite linear measure.
r
+ 00,
(1.3.26)
24
Chapter 1
Essentials of Nevanlinna Theory
Prool. We consider only the error terms in Theorem 1.5 since those in Theorem 1.4 are even simpler. If q
ip(z) =
II (f(z) 
av),
v=1
then
S(r,/) = m(r,
~) + m(r, ~) + 0(1).
When the order oX of I (z) is finite, we have
T(r, /) < r,X+1, so
r > ro,
q
T(r,ip) ~ LT(r,lav ) ~ qT(r,/) +0(1). v=1
Choosing p = 2r in Lemma 1.3, we can see that
I'
m(r, 7) = O(logr),
m(r, ~) = O(1ogr), I
r
+ 00.
Thus
S(r, /) = O(1ogr),
r
+ 00.
When the order of I(z) is infinite, there exists ro such that T(ro, /) ~ 1 and T(ro,ip) ~ 1. Denoting by E1 and E2 the exceptional sets illustrated in Lemma 1.4 which correspond to T(r, /) and T(r, ip), respectively, we have mes Ej ~ 2(j = 1,2). When rE(E1 U E 2), choosing p = r + l/(T(r, /), we have
f'
m(r, 7) = m(r,
o{ 10g(rT(r, /)},
~) = o{ 10g(rT(r, I»},
r
+ 00.
Hence
S(r, /) =
o{ 10g(rT(r, /)},
r
+ 00,
except on a set with linear measure less than or equal to 4.
0
25
1.4 Applications of the Second Fundamental Theorem
If av is finite and f(O) =I av , then
m(r, f
~ aJ =
T(r,f  av )  N(r, f
::; T(r, f)
~ aJ + log If(O)l_ avl
+ log+ lavl + log 2
N(r, f
~ aJ + log If(O)l_ avl·
Therefore Theorem 1.5 can be expressed as follows. Theorem 1.5' Suppose f(z) is meromorphic in the finite plane and nondegenerate into a constant. If av(ZI = 1,2,···, q) are q(~ 3) distinct complex numbers (one of them may be infinity), then q 1 (q  2)T(r, f) < N(r, f _ aJ  Nl(r) + S(r, f),
?;
where Nl(r) is expressed by (1.3.3) and S(r,f) has the properties formulated in Theorem 1.6.
1.4 Applications of the Second Fundamental Theorem 1.4.1 The PicardBorel theorem. theorem, we note the following lemma.
Before introducing the Picard
Lemma 1.5 If f{z) is a transcendental meromorphic function in the finite planel ) , then we have
11· m T(r, f) = r:::;OO
log r
00.
( 1.4.1 )
Proof. Suppose the conclusion is not true. Then there exist a large positive integer M and a sequence rv tending to infinity such that limv+oo(T(rv, f))/(logrv) < M. By the first fundamental theorem of Nevanlinna, we have
lim N(rv, f) v+oo logrv
< M,
l) A meromorphic function in the finite plane, nondegenerate into a rational function, is said to be a transcendental meromorphic function.
26
Chapter 1
Essentials of Nevanlinna Theory
Thus both the number of zeros and the number of poles of fez) in the finite plane are less than M. Constructing a rational function R( z) which has the same zeros and poles with the same multiplicities as fez), we have fez) =
eg(z)
R(z)
,
where g( z) is entire. Since R( z) is rational, we have
T(r, R) = O(logr). Thus so
( f)
10gM r", R
f)
~ 2r" 2 + _ r" T ( 2r", R
r"
r"
(
)
= 0 logr" .
Therefore we have Reg(z)
=log I~~:~ 1= O(1ogr,,)
for every point z on Izl = r". By the generalized Liouville theorem (See Titchmarsh [1, 8687]), g(z) must be a constant. Consequently, fez) is rational, which contradicts the hypothesis of Lemma 1.5. Now let us prove the following Picard theorem. Theorem 1.7 If fez) is transcendental meromorphic in the finite plane, then fez) takes every complex number infinitely many times, except for at most two values.
Suppose that the conclusion of Theorem 1. 7 is not true. Then Proof. there exist three distinct complex numbers a,,(v = 1,2,3) such that fez) takes every a" only finitely many times. Thus N(r, a,,)
= O(1ogr),
(v
= 1,2,3).
By defining 9 () z =
fez) fez) 
al
a3
.
a2  a3
(1.4.2)
,
a2  al
it is clear that
N(r,~) + N(r, g: 1) + N(r, g) =
O(1ogr).
27
1.4 Applications of the Second Fundamental Theorem
Then, applying Theorem 1.4 and Theorem 1.6 to g(z), we have T(r, g) = O(log(rT(r,g»), except on a set with linear measure less than or equal to 4. Thus lim T(r, g) r~ logr
< 00.
By Lemma 1.5, g(z) is rational, and therefore so is fez). This contradicts the hypothesis of Theorem 1.7. 0
Definition 1.7 Let fez) be meromorphic in the finite plane and let a be a complex number. a is called an exceptional value of fez) in the sense of Picard, if fez)  a has no zeros. By Theorem 1.7, the number of exceptional values of a transcendental meromorphic function in the sense of Picard is at most equal to 2. Clearly, the upper bound 2 is precise. For instance, eZ has two Picard exceptional values, D and 00. Moreover, eZ  a has infinitely many zeros for any finite nonzero complex number a.
Definition 1.8 Let fez) be a meromorphic function of order A(D < A < 00) in the finite plane. A complex number a is called an exceptional value of fez) in the sense of Borel, if li logn(r,f = a) m r ..... oo logr
,
< 1\.
(1.4.3)
Lemma 1.6 If fez) is meromorphic and of order A(D < A < 00) in the finite plane, then a necessary and sufficient condition that fez) have an exceptional value a in the sense of Borel is
1. 10gN(r,f = a) logr
1m r ..... oo
Proof. n(r,f
From = a) = a) ::; n(r,f 1 2 og
1t 2r
r
dt
1 og
,
< 1\.
 ::;  12 N (2r,f
= a),
r
~
1,
28
Chapter 1
Essentials of Nevanlinna Theory
and N(r,f
= a) =
l
N(ro,f
= a)
r
n(t,f = a)dt
~
t
~
(f
n r,
)1
r
= a og, ~
we can see that
Ii logN(r,f m r ..... oo log r
= a)
=
1. logn(r,f = a) 1m . r ..... oo log r
Thus the conclusion of Lemma 1.6 follows from Definition 1.8.
0
Now let us prove the following Borel theorem by use of the second fundamental theorem of Nevanlinna. Theorem 1.8 If fez) is merom orphic and of order A(O < A < 00) in the finite plane, then fez) has at most two exceptional values in the sense of Borel. Proof. Suppose the conclusion is not true. Then there are three distinct exceptional values all(v = 1,2,3) in the sense of Borel. By Lemma 1.6, we have 1· logN(r,f = all), 1m < A, V = 1, 2,3. r ..... oo logr
We can assume that the all(v = 1,2,3) are 0, 1, 00, for otherwise, it would be sufficient to make the transformation (1.4.2). Applying Theorems 1.4 and 1.6 to fez), we have 3
T(r, f)
< "E N(r, f
= all)
+ O(logr).
11=1
Thus
Ii logT(r, f) m logr
r ..... oo
, oc hm T( r, f)
f~a)
= 1
_. N(r, hm T( r, f) r>oc
.
(1.4.4)
It is easy to see 0 ::; c5 (a, f) ::; 1.
Definition 1.10 A complex number a is called a deficient value of f(z), if its deficiency is positive. A deficient value is also called an exceptional value in the sense of Nevanlinna. Theorem 1.9 If f(z) is transcendental meromorphic in the finite plane, then the set of deficient values of f (z) is at most countable, and
L c5(a, f) ::; 2.
(1.4.5)
Proof. Suppose a ll (lI = 1,2"", q) are distinct complex numbers. Theorem 1.5 yields
~
~
· { m(r, f aJ } < 1' 2T(r, f) 11m L...J 1m r>oc 11=1 T(r, f)  r>oc T(r, f)
+
l' S(r, f) 1m. r>oc T(r, f)
Thus
Consequently, for each positive integer j, the number of deficient values with deficiencies larger than 1/j is less than 2j. Since
E{a: c5(a,f)
> O} =
oc
1
j=l
J
UE{a: c5(a,f) > "7}'
30
Chapter 1
Essentials of Nevan1inna Theory
the set of deficient values is at most countable. Moreover, for any finite q, we have E~=1 6(all , f) ~ 2, so that the sum of all the deficiencies does not exceed 2. 0 The upper bound 2 may be attained. For instance, the exponential function eZ satisfies
When fez) is a transcendental entire function, it is clear that 6(00, f) = 1. Thus
L 6(a, f) ~ 1.
(1.4.6)
a:f:OO
Obviously, Theorem 1.9 implies Theorem 1.7. Remark. The deficient value, though actually introduced by Collingwood, is usually called the exceptional value in the sense of Nevanlinna, because it was introduced on the basis of Theorem 1.5, a generalization of the second fundamental theorem of Nevanlinna (Theorem 1.4). In Theorems 1.7 and 1.9, we assume that fez) is transcendental. When fez) is rational, the corresponding conclusions1} can be obtained by a similar method. As a matter of fact, if fez) = P(z)jQ(z), then
f' pI Q' m(r, 7) ~ m(r, p) + m(r, Q) = 0,
r
+
00,
so S(r, f) = 0(1). The rest of the proof is the same. Furthermore the following conclusion can be obtained by a direct computation. If fez) is rational, then fez) takes any complex number, except for at most one value. Moreover, fez) has a unique deficient value.
1.4.3 Multiple value. Now we derive another form of the second fundamental theorem of Nevanlinna. 1} The conclusion of Theorem 1.7 should be revised by saying that /(z) can take every complex value, except for at most two values.
31
1.4 Applications of the Second Fundamental Theorem
Let I(z) be meromorphic in Izl < R(5: 00) and a be a finite complex number. For < r < R, denote by n(r, I = a), sometimes by n(r, 1/(Ja)) or n(r, a), the number of zeros of I(z)  a in Izl 5: r, each zero being counted only once. Moreover, let
°
n(O, I = a) = {
0,
if 1(0) =I a,
1,
if 1(0) = a.
Introduce the notation
(
)l
N r,l=a =
rn (t,l=a)n(o,l=a) dt+n(O,I=a ) logr, o t
which is sometimes denoted by N(r, 1/(J  a)) or N(r, a), and called the reduced counting function of I(z) a. Similarly we can define n(r, 1= 00) (sometimes denoted by n(r, j) or n(r, 00)) and N(r, I = 00) (sometimes denoted by N(r,j) or N(r, 00)). If a,,(lI = 1,2,···, q) are distinct complex numbers, then we have from Theorem 1.5'
where 8 1 (r, j) has the properties formulated in Theorem 1.6. Now let us discuss the properties of
N 1 (r) = 2N(r, j)  N(r, I')
+ N(r, ;/).
Since
2N(r, j) _ N(r, I') =
r
10
nl(t, 1= 00)  nl(O, 1= 00) dt t
+nl(O, 1= 00) logr, where nl(t,1 = 00) = 2n(t,j)  n(t,I'), it is a counting function of multiple poles. If I (z) has a pole of order k at Zo, then nl (t, I = 00) is counted k 1 times there. When a is a finite complex number and I (z)  a has a zero of order k at z~, then n(t, 1/1') is also counted k  1 times at z~. Thus
32
Chapter 1
Essentials of Nevanlinna Theory
is a counting function of multiple values (finite or infinite), where
nI(t) = nI(t,f = 00) +n(t, ;,), so q
?;
1 N(r, f _ a)  N I (r) ::;
q
?;
1 N(r, f  a)·
Therefore the second fundamental theorem of Nevanlinna can be written as
?; N(r, f _1 a) + SI(r, f). q
(q  2)T(r, f)
0 is at most countable and (1.4.8) {6(a, f) + 8(a, f)} ::; e(a, f) ::; 2.
L
L
a
a
Definition 1.12 a is called a complete multiple value of fez), if all the zeros of f (z)  a (or all the poles of f (z), when a = 00) are multiple. Corollary 1. Under the hypothesis of Theorem 1.10, the number of complete multiple values of fez) does not exceed 4.
1.4 Applications of the Second Fundamental Theorem
33
In fact, if a is a complete multiple value, then
e(a,j) = 1 lim N(r,a) > 1 lim N(r,a) > !. r+oo T(r, j) r+oo N(r, a)  2
(1.4.9)
Thus (1.4.8) yields the required conclusion.
Corollary 2. If f(z) is a tmnscendental entire junction, then J(z) has at most two complete multiple values. In fact, we have
L
e(a,J) ~ 1.
(1.4.10)
a:/:oo
Comparing (1.4.10) with (1.4.9), the conclusion of Corollary 2 is immediate. The upper bounds of Corollaries 1 and 2 are precise. For instance, the Weierstrass elliptic function B(z) is meromorphic in the finite plane and satisfies
B'(Z)2 = {B(z)  a1 }{ B(z)  a2}{ B(z)  a3},
(1.4.11)
where all a2, and a3 are distinct finite complex numbers. Obviously B'(z) = a when B(z) = av (II = 1,2,3). Thus the av (II = 1,2,3) are complete multiple values of B(z). Moreover, 00 is another complete multiple value of B(z), since B(z) has only poles of order 2 by (1.4.11). It is easy to see that sinz is an entire function with two complete multiple values 1 and 1.
Corollary 3. Let J(z) be tmnscendental merom orphic in the finite plane. Suppose av (II = 1,2"" ,p) are distinct complex numbers and Iv (II = 1,2"" ,p) are integers greater than 1. If all the zeros of J(z)  av (II = 1,2"" ,p) have multiplicities ~ lv, then we have (1.4.12)
34
Chapter I
Essentials of Nevanlinna Theory
1.5 Generalizations of the Second Fundamental Theorem In this section, we discuss some generalizations of the second fundamental theorem by replacing complex numbers with "small" meromorphic functions.
1.5.1 A special result. R. Nevanlinna himself obtained a generalization involving three small functions. Theorem 1.11 III(z) and tt'1I(Z) (1/ = 1,2,3) are meromorphic functions in the finite plane such that
T(r, tt'1I)
= o{T(r, In, 1/ = 1,2,3,
then we have 3
{I  o(lnT(r, f)
~') points of {aIL}' each being in a range < ~, and then the concentric disk S" with radius ~"h/n must also contain ~"'(> ~") points of {aIL}' each being in a range < ~. Continuing this procedure, we finally obtain a concentric disk S* having radius ~*h/n and containing ~*(> ~) points of {aIL}' each being in a range < ~. This contradicts the above definition.
60
Chapter 3 Borel Directions
For every C v (ll = 1,2"" ,p), let r I' denote the concentric disk with radius 2>"vh/n. Clearly, the number of rv's (ll = 1,2, ... ,p) does not exceed n and the total sum of the radii is equal to 2h. Then, nothing remains but to prove that (3.1.1) is false outside U~=lr v' In fact, if zoE ~=l r v and 1 '$ >"0 '$ n, then the disk S = {z : Iz  zol < >"oh/n} does not meet Cv , whenever >"v ;::: >"0. Thus each of the points of {a v } in S is in a range < >"0, so their number does not exceed >"0  1. Let us rearrange al' (1£ = 1,2"", n) according to their distance from zo, i.e. Izo  all '$ Izo  a21 '$ .... Then the distance between Zo and al' must be ;::: I£h / n (1£ = 1, 2, ... , n). Consequently we have
IIn Izo 
1'=1
h 2h nh (h)n h)n ...  = . n! > ( . n n nne
al'I ;:::  . 
o
3.1.2 Spherical distance. In order to introduce the Borel direction of a meromorphic function, we need to use the concept of spherical distance and its properties. We take the real and imaginary axes of the complex plane as two axes of a rectangular coordinate system in space. The third axis is perpendicular to the complex plane. The Riemann Sphere with center at (0, 0, 1/2) and radius 1/2 is tangent to the complex plane. Denote by N the north pole (0, 0, 1). Given a point z in the complex plane, the point of intersection W of the segment zN and S is called the spherical image of z. In particular, the spherical image of infinity is N. Definition 3.1 Given two complex numbers Zl and Z2, the distance between their spherical images WI and W2 is called the spherical distance of Zl and Z2, denoted by IZl' z21·
°
By definition, we have '$ IZl, z21 '$ 1. By Definition 3.1 and elementary computations it is easy to verify that the spherical image W of z = x + iy is
and hence Iz,ool =
1 1;
(1 + Iz12)2
(3.1.2)
61
3.1 Preliminaries
the spherical distance of two finite complex numbers
ZI
and
Z2
is (3.1.3)
The following properties will be used later on: (1) If z is a complex number satisfying (2) If ZI and
Z2
Iz,ool > d> 0, then Izl < lid.
are two finite complex numbers, then (3.1.4)
In fact, we have from (3.1.3) log
1 (1 + IZI12)1/2(1 + IZ212)1/2 = log ...!.......:.=.:......;.=,....:.....::..:.....:IZl, z21 IZI  z21 = log(l
1 + IZI12)1/2 + log(l + IZ212)1/2 + log .,.....,.
IZI  z21
Definition 3.2 Let ao be a complex number and r(~ 1) a positive number. {z: Iz, aol < r} is called a spherical disk with center ao and radius r. Using a proof similar to that of Theorem 3..l.r we can prove Theorem 3.1'.
Theorem 3.1' Let ap' (1£ = 1,2,···, n) be n complex numbers and h a positive number. Then the points which satisfy the inequality
can be covered by a collection of spherical disks whose number does not exceed n and the sum of whose radii does not exceed 2h.
62
Chapter 3 Borel Directions
3.1.3 Theorem of bounded type. In order to derive the existence of a Borel direction, we shall use the second fundamental theorem of Nevanlinna with an appropriate estimate of the error terms. Theorem 3.2 Let fez) be meromorphic in Izl ~ R. If f(O) and 1'(0) i= 0, then we have
T(r, I)
i= 0,1,00
< 2{N(R, 0) + N(R, 1) + N(R, oo)} + 191 + 4log+ If(O)1 1 R +2 log Rlf'(O)1 + 12 log Rr'
(3.1.5) whenever 0
< r < R.
Proof. Starting from the Nevanlinna second fundamental theorem (Theorem 1.4) we have T(r, I)
f' f' < N(r, 0) + N(r, 1) + N(r, 00) + m(r, 7) + m(r, f 1)
I
I
+ log f(O)(J(O)  1) + log 2 f'(O) . In a way similar to the proof of Theorem 2.3, we deduce by using Lemma 1.3 T(r, I)
< N(r, 0) + N(r, 1) + N(r, 00) + 12 + 22 log 2 1 1 +4Iog+  + 6log+   + 8log+ p r pr
+8Iog+ T(p, I) + 21og+ If(O)1 + log+
1f'~0)1'
(3.1.6)
If T(r, I) < 1 holds uniformly in (0, R), then (3.1.5) is already proved. Otherwise, there exists ro, 0 ~ ro < R with T(ro, I) = 1. Using Lemma
1.4 to ro
< r < R, there exists T in
[r, 7R8+ r] such that
RT ) T ( T + eT(T, I)' f < 2T(T, I). Choose
, RT T = T + eTC T, ff
63
3.1 Preliminaries
Clearly 0
1r
t
5
5
and 1
1
N(p, Zo, I ) ~ T{p, Zo, 1) a a = T{p, zo,
1
I  a) + log I/(zo) _ al
~ T(p, zo, f) + log+ la I + log 2 + log I/(zo)1 _ ai'
we deduce that 1 r n ( r+ 5' zo,
max {240 , 24010g( log 2 ' "10g2 og rkl and such that
is also sufficiently large.
81
3.3 Filling Disks and Borel Directions
Then, by Lemma 3.4, there exists a point Zl in 1 < Izi < 2rkl such that in r 1 : 1z  zll < 471" 1zll / qr, J(z) takes every complex number at least
times, except for those complex numbers contained in the union of two spherical disks each with radius e n1 • Choose rk'1 > 3rkl and rk2 sufficiently large such that
and such that
is also sufficiently large. Then, once more by Lemma 3.4 there exists a point Z2 in rk'1 < Izl < 2rk2 such that, in r 2 Iz  z21 < 471"lz21/q2, J{z) takes every complex number at least 1
n2
=
C*T{r k 2 ,
1)2 logrk2
times, except for those complex numbers contained in the union of two spherical disks each with radius e n2 • Continuing this procedure, we obtain a sequence of disks with the property formulated in Theorem 3.6. 0 Corollary. IJ J{z) is meromorphic in the finite plane satisJying (3.3.5), then there exists a ray J: arg z=Oo{O ~ 00 < 271") such that given an arbitrary positive number €, J{z) takes every complex number infinitely many times in the angular region larg z  00 1 < €, with two possible exceptions.
In fact, according to Theorem 3.6, J{z) has a sequence of filling disks rj : Iz  zjl < €jlzjl. The sequence {eiargzj, j = 1,2,···} has an accumulation point ei80 (O ~ 00 < 271"). We claim that the ray J: arg z = 00 has the desired property. Otherwise there is an angular region n: larg z  00 1 < € such that J{z) takes none of the three complex numbers all (v = 1,2,3) in n. Note that n contains a subsequence of (rj),
82
Chapter 3 Borel Directions
still to be denoted by (rj) for the brevity. When j is sufficiently large, we have 2e nj < min {lalL' avl}. Thus at least one of av (v = 1,2,3), 1~IJ.';ev9
say aI, belongs to neither of the two exceptional spherical disks each with radius e nj . Therefore fez) takes al in rj at least nj times, where limj+oo nj = 00. This contradicts the assumption that fez) does not take al inn. The rays with the property formulated in the corollary are called Julia directions of f (z). 3.3.3 Filling disks and Borel directions of order A of a meromorphic function. For meromorphic functions of finite and positive order, we can draw stronger conclusions. Theorem 3.7 If fez) is meromorphic and of order A (0 in the finite plane, then there exists a 1Jequence vi di1Jks
< A < 00)
such that J(z) takes every complex number at least IZjl~ej times in r j , except for those complex numbers contained in the union of two spherical , >'6disks each with radius e Zj ' 3, where limj.oo OJ = o.
The disks r j are called filling disks of order A. Proof. Since the order of fez) is A, there is a sequence of positive numbers rk tending to infinity and satisfying
lim logT(rk' J) = A. logrk
k+oo
Choose rkl sufficiently large such that 240 log rkl 2T( f) 12T(2, J) I } T (rkll f) :::: max { 240, log 2 ,1 1, , log 2 og rkl .and Buchthat
ql = logrkl is also sufficiently large. By Lemma 3.3, there exists a point Zl in 1 < Izl < rkl such that in the disk rl: Iz  zll < 47rlzll/logrkl' fez) takes
83
3.3 Filling Disks and Borel Directions
every complex number at least nl
= C· T(rkl' J) (logrk1)4
times, except for those complex numbers contained in the union of two spherical disks each with radius e n1 • Take rk'1 > 2rk1' and take rk2 sufficiently large such that T(rk2' J) ~ max { 240,
12T( 2rk' , J) } 240logrk 1 2, 12T(rk" J), 1 21 logrk2 og2 1 og
and such that q2 = log rk2 is also sufficiently large. Then, there is a point Z2 in rk'1 < Izl < rk2 such that in the disk r2: Iz  z21 < 47rlz21/1ogrk2' fez) takes every complex number at least n2 = C· T(rk2' J) (logrk2)4
times, except for those complex numbers contained in the union of two spherical disks each with radius e n2 • Continuing this procedure, we obtain a sequence of disks. Since • T( rk; ,I) nj = C (l )4 ogrk;
~e~;
> r k3·
>
~e~.
IZk; I
3,
this sequence of disks has the property formulated in Theorem 3.7. Let fez) be meromorphic in the finite plane, and let argz = eo (0 ::; eo < 27r) be a ray. For r > 0, e > 0 and an arbitrary complex number a, we denote by nCr, eo, e, f = a) or nCr, eo, e, 1/(1  a)) the number of zeros of fez)  a in the region (Izl ::; r) n (Iargz  eol ::; e), multiple zeros being counted with their multiplicities. When a = 00, we write nCr, eo, e, J). Theorem 3.8 If fez) is meromorphic and of order.x (0 < .x < 00) in the finite plane, then there exists a ray B: arg z = eo (0 ::; eo < 27r) such that, given any positive number e, the equality
1. logn(r, eo, e, 1m r+oo logr
f
= a)
=
, 1\
(3.3.6)
holds for every complex number a, with at most two possible exceptions.
According to Theorem 3.7, fez) has a sequence of filling disks Proof. of order .x, rj: Iz  zjl < ejlZjl, such that in every rj, fez) takes every
84
Chapter 3 Borel Directions
complex number at least IZj I,\e; times, except for those complex numbers contained in the union of two spherical disks Sj and Sj each with radius e 1Z; 1.\63 , where limj+ooh'j = O. Let ei80 (0 ~ 80 < 2n) be an accumulation point of {eiargz;, j = 1,2,,, .}. We claim that argz = 80 is the desired ray. In fact, if this assertion is not true, there exist an angular region no: largz  80 1 < eo and three distinct complex numbers av (v = 1,2,3) such that 1' logn(r, 80 , eo, 1m logr
f
= av )
r+oo
\ < A,
V
= 1,2,3.
(3.3.7)
n contains a
subsequence of (rj), still to be denoted by (rj). When j is sufficiently large, we have
Thus at least one of av (v = 1,2,3), say at, belongs to neither of Sj or Sj (j = 1,2", .), so that fez) takes al in rj at least IZjl,\c5j times. Hence lim _10_g_n....:(_r,_80_,_e_,_f_=_a_l=) > lim logn(rj, f = al) log r  j+oo 10g(21zj I)
r+oo
> lim 10g(lzjl'\c5;)
= A.
 j+oo 10g(21zjl) This inequality contradicts (3.3.7).
0
The ray in Theorem 3.8 is usually called a Borel direction of fez). It is also called a BorelValiron direction, since the proof of its existence is due to G. Valiron [2]. Much research was made into Borel directions of entire and meromorphic functions. For instance, M. Biernacki [1] proved the following theorem.
Theorem 3.8' If f(z) is meromorphic and of order A (0 < A < 00) in the finite plane, then there exists a ray B: argz = 80 (0 ~ 80 < 27r) such that, given any positive number e, the equality lim logn(r, 80 , e, logr
r+oo
f = II) = A,
(3.3.8)
85
3.4 Properties of Borel Directions
holds for every meromorphic function II(z) of order less than A (including all the complex numbers), with two possible exceptions at most.
A. Rauch [1] proceeded to generalize the Biernacki theorem. In order to introduce his result, we need the concepts of convergent and divergent classes of meromorphic functions. Let fez) be meromorphic and of order A (0 < A < 00) in the finite plane. If the integral oo T(r,J)d (3.3.9) r,\+l r
J
is finite, then fez) belongs to the convergent class of order A. Otherwise, fez) belongs to the divergent class of order A.
Theorem 3.8" If fez) is meromorphic in the finite plane and belongs to the divergent class of order A, then there exists a ray (B): argz = (Jo (0:::; (Jo < 27r) such that, given any positive number e, the equality
J
OO
nCr, (Jo, e, f
dr
= II) r'\+1 = 00
(3.3.10)
holds for every meromorphic function II (z) belonging to the convergent class of order A, with at most two possible exceptional functions.
Theorems 3.8' and 3.8" can be proved with the aid of Theorem 3.5. Those readers interested in these results may consult the original papers. We shall introduce some recent results on Borel directions in later chapters.
3.4 Properties of Borel Directions 3.4.1 One lemma. Before introducing the properties of Borel directions, we need to prove a lemma.
Lemma 3.5 Let fez) be meromorphic in the finite plane. Suppose rj (j = 1,2",,) are the moduli of the poles of fez) arranged by nondecreasing order and counted with their multiplicities. If u is a positive number, then the series
86
Chapter 3 Borel Directions
and the integrals /
oon(t,J)d to'+! t
an
d
are either simultaneously convergent or simultaneously divergent. Proof.
~~=
L...J
0'
j=jo rj
Using the Stieltjes integral, it is easy to see that
lR ~dn(t ro
f)
to"
When 'ErjO' is convergent,
= nCR, J) RO'
r
o
_ nero, J) 0' ro
+
lR ro
(m(t, J) dt. (3.4.1) to'+l
nt~~{) dt is also convergent.
Conversely, if
J) / 00 net to'~l dt converges, then we have
1> [00 net, J) dt > nCR, J) [00 ~ = nCR, J) to'+!
}R

} R to'+!
u RO'
for sufficiently large R. Combining this fact and (3.4.1), the series 'ErjO' is also convergent. Similarly, by using
lR ro
net, J) dt = t o'+ 1
lR ~dN(t to"
ro
f) = N(R, J) _ N(ro, J) RO' rO'
°
+
lR ro
uN(t, J) dt to'+ 1 '
/00 nt~~{) dt and /00 ~~t~[) dt are either simultaneously convergent or simultaneously divergent.
0
The following lemma can be proved by the same way.
°
Lemma 3.5' Let fez) be meromorphic in the finite plane, ~ 0o < 27r, 'fl> 0, and let a be a complex number. Suppose rj (0o, 'fl, f = a) (j =
1,2",,) are the moduli of the zeros of f(z)a in largzOol
~ 'fl, arranged
by nondecreasing order and counted with their multiplicities (When a = are the moduli of poles of f(z).), and suppose
00, rj
1
N(r, 0o, 'fl,
lor n(t, 0o, 'fl, f ~ a) 
)= f  a °
t
n(0,00 , 'fl, f
~ a) dt
87
3.4 Properties of Borel Directions
If q is positive, then the series
are either simultaneously convergent or simultaneously divergent. 3.4.2 Properties of meromorphic functions in an angle Theorem 3.9 Let f{z) be meromorphic in the angle largz  00 1 < TJ. If there are three distinct complex numbers ay (v = 1,2,3) and a positive number q such that the series
I)rk (00 , TJ, f
= ay)}O',
v = 1,2,3
(3.4.2)
are convergent, then the series
converges for any positive number e and every complex number a, except for a set of a with linear measure zero. Divide the angular domain largz  00 1 < TJ  e into equal Proof. small angles with magnitude not exceeding e / 4. The number of such small angles J satisfies
J~
[2( TJ
i e )] +1. 4
Choosing a sufficiently large positive number ro, now divide this same angular domain largz  00 1 < TJ  e by arcs of the circumferences of the circles Izl = ro, ro(l + ~), ro(l + ~)2,
....
Thus the region (Izl > ro) n (Iargz  00 1 < TJ  e) is divided into small quadrangles Ojl (j = 1,2"", Jj I = 1,2", .). For every Ojl, there are disks r jl and rjl such that Ojl C rjl C rj,
c
(Iargz  00 1 < TJ),
88
Chapter 3 Borel Directions
where the radius of rj, is twice that of r jl. For every rjolo' the number of other rj, having a nonempty intersection with rjolo' has a fixed upper bound. For example, we can take one hundred as this upper bound. From the convergence of (3.4.2) and Lemma 3.5', we have 00 /
n(r, (Jo, 'f/, f = aV)d ro+ 1 r < 00,
v = 1,2,3.
But Theorem 3.4 gives 3
f = a) < C{ L
n(Ojl'
f = av ) + log
n(rj"
v=1
in every
rj"
!!}
(3.4.3)
1 e e(ro(1+:4)I) 2
except for those complex numbers contained in a spherical e
0
I 
disk ejl with radius e (ro(1+:4) ) 2 • Setting J
ej = .U ejl, 3=1
and
e=
en ( U el),
k=1 l=k
it is clear that mese
= k+oo lim mes ( U el) = o. l=k
For any complex number a outside the set e, there exists a positive integer lo such that aE U~'o e,. Thus (3.4.3) holds for j = 1,2,··" J and a, when l ~ lo. Consequently L
J
L
L Ln(Ojl' l=lo j=1
f
= a)
J
3
< C{ L L Ln(rj" f
= av)
l=lo j=1 v=1 L
l
0
+JL (ro(1+~) )2}, l=lo so that 3
n(r, (Jo, 'f/ 
f
E:,
= a)
< C{ L n(2r, (Jo,
0
'f/,
f
= av )
v=1 Therefore 00 /
n(r, (Jo, 'f/  E:, f = a)dr < 00 ro+ 1
..:.'......:...:.':::,....:..:....!..
+ (2r)2}.
89
3.4 Properties of Borel Directions
for any complex number a outside the set e. The conclusion of Theorem 3.9 follows from this fact and Theorem 3.5'. 0 The following important result will be deduced from Theorem 3.9. Theorem 3.10 Let f(z) be meromorphic and of order A (0 < A < 00) in the finite plane. If f (z) has no Borel direction of order A in the angle (h < argz < O2 , then, for any small positive number 0:, there are three distinct complex numbers al/ (ll = 1,2,3) and a positive number T, T < A, such that 3
L n(r,
01
+ 0:,
O2

0:, f
= al/)
< rT.
(3.4.4)
1/=1
Proof. Since argz = cP is not a Borel direction of order A of f(z), for every value cP on [0 1 + 0:, O2  0:] there are three distinct complex numbers {3j(CP), a small positive number c:(cp) and a positive number T(cp), T(cp) < T such that n(r, cP  c:(cp), cp + c:(cp) , f
= {3j(CP)) < rT(cp) ,
j
= 1,2,3.
Thus we have
J
OO
n(t, cp  c:(cp), cp + c:(cp),
f
dt
=
{3j(CP)) tTl (cp)+l
2(h+1)(27'2'x) i~
00.
.
Therefore, we can find a sequence of disks rz : jzzzl < C"Izlzzj, argzz = 00 , such that J(z) takes every complex value at least jZlj,X2('x7l) times, except possibly for those values contained in two spherical disks each with radius 2jl . 0
Chapter 4
Value Distribution of Meromorphic Functions Together with Their Derivatives
The present chapter will be devoted to an important topic of the theory of value distribution, namely, the study of meromorphic functions together with their derivatives.
4.1 Comparison Between Growths of T(r, f) and T(r, 1') 4.1.1 Two lemmas. Let J(z) be meromorphic in the finite plane. It is easy to bound T(r, 1') in terms of T(r, I). However a reverse estimate, for example, an inequality of Chuang Chitai, is more difficult. In order to introduce this inequality, we need the following two lemmas. Lemma 4.1 Let J(z) be meromorphic in the finite plane and J(O) =f: 00. If Rand R' (R < R') are two positive numbers, then there exists a real number 00 such that
log+
R'+R
IJ(rei80 )1 :5 R' _ Rm(R', I) + n(R', I) log 4 + N(R', I)
(4.1.1)
Jor 0:5 r:5 R. ProoJ. Let z = rei8 , 0 :5 r :5 R, be an arbitrary point distinct from the poles of J(z). The PoissonJensen formula gives
r
I I
1 21r R'2 r2 log J(z) :5 27r 10 log IJ(R'e iIP ) R'2 _ 2R'rco:(cp _ 0) + r2dcp
+
I
R'2  b z
L IOgIR'( z _; )1, Ibl'l~R' p.
94
Chapter 4 Value Distribution of Meromorphic Functions
where bl' are the poles of J(z) on Izl ~ R'. Thus
R' + r
log+ IJ(z)1 ~ R' _ m(R', J)
+
r
L
2R'
IblLl~R'
R' + r , ~  R ' m(R ,J) + log  r
log I _ b I
z I' (2R,)n(R',/)
II
Iz  bl'I
(4.1.2)
.
IblLl~R'
thus
(4.1.3)
Since
r log In(R',f) II sin(O  'PI') IdO = n(R', J) ior log IsinOldO
io
o
1'=1
0
= n( R' , J)1f' log 2, there exists at least one real number 00 such that
II
I
n(R',f)
I
sin(Oo  'PI')
1
(4.1.4)
> 2n(R''/)·
It is obvious that z = rei90 (0 ~ r ~ R) is not a pole of J(z). Otherwise, one factor on the lefthand side of the above inequality is zero. Substituting (4.1.3) and (4.1.4) into (4.1.2), we have
log+ IJ(rei90 )1
R' +R
~ R' _ Rm(R', J) + log4n(R',/) + E log
~ ~: ~ ~m(R" for 0
~
r
~
R.
0
R'
Ibl'I
J) + n(R', J) log 4 + N(R', J)
4.1 Comparison Between Growths of T(r, 1) and
T(r, 1')
95
Lemma 4.2 Let f(z) be meromorphic in the finite plane. If R, R' and R" are three positive numbers and R < R' < R", then there exists a positive number p such that R ~ p ~ R' and
R" + R' log+lf(z)1 ~ R"_R,m(R",f) on Izl
8eR"
(4.1.5)
+ n(R",f) log R'R
= p.
Proof. Suppose bp.(JL = 1,2,··.,n(R",f)) are the poles of f(z) on Izl ~ R". According to the BoutrouxCartan theorem, the inequality n(R",J)
II
Iz 
bp.I ~
(R' _ R)n(R",J)
p.=1
(4.1.6)
4e
holds, except for those points in a group (r) of disks, where the total sum of their radii does not exceed (R'  R)/2. Since there is a circle Izl = p in the annulus R ~ Izl ~ R', not intersecting (r), (4.1.6) holds on Izl = p. From the PoissonJensen formula and (4.1.6), we have "
R + log+ If ()I z ~ R" _ p m (R ", f) + P
R" + R' ~ R" _ R' m(R", f) for any point z on Izl
= p.
I
Ibl'l~R"

Z
I
p.
8eR" R
+ n(R", f) log R' _
0
4.1.2 Comparison between T(r, f) and T(r, f'). the Chuang Chitai's inequality.
00,
"2
'L....J " log R" R ( _ bp.z b )
Now let us prove
Theorem 4.1 If f (z) is meromorphic in the finite plane and f (0) then
T(r, f)
< CrT(rr, f') + log+(rr) + 4 + log+ If(O)1
:f
(4.1.7)
for r > 1 and r > O. Proof. Write u = r 1/ 3, rl = ur, r2 = urI and r3 to Lemma 4.1, there is a real number 80 such that
= ur2.
According
96
Chapter 4 Value Distribution of Meromorphic Functions
for 0 ~ t ~ ri. By Lemma 4.2, a number p in [r, rl] can be found with the property (4.1.9)
on
Izl =
p.
If L is the union of the circumference Izl = p and the segment between the origin and pei6o , then its length does not exceed (211' + 1) p. Denoting by M the maximum modulus of I'(z) on L, we have
on
Izl
1/(0)1 + M(211' + l)p
I/(z)1
~
~ log+
1/(0)1 + log+ M + log+ p + log 811'.
= p. Thus
log+
I/(z)1
On the other hand, (4.1.8) and (4.1.9) give us log+ M
~ ~
r2 + rl m(r2' I') r2  rl
+ n(r2' 1') log
8er2 rl  r
+ N(r2' 1')
} { IOg~ rl r~ r + r2 + rl T(r3, I') logr2  rl
r2
= {
u
:e~: ++ 1 }T(r3, I')
log logu
u 1
= C~T(r3, I').
Therefore m(p, I)
< C~T(r3, /') + log+(rr) + 4 + log+ 1/(0)1,
and hence T(r, f) ~ T(p, f)
1. Then there exists a set M(K) whose upper logarithmic density is at most o(K) = min {(2e K 
1
1)1, (1 + e(K  1)) exp(e(1  K))}
such that lor every positive integer k, .
J!.~
,,':M(K)
T(r,J) (l(1e») ~ 3eK. T r,
II I is entire we can replace 3eK by 2eK in the last inequality.
4.2 Modular Distribution of Meromorphic Functions Together with Their Derivatives
99
The upper logarithmic density of a set has a definition similar to that of the lower logarithmic density, but lower limit is now replaced by upper limit. Thus as r + 00 on a set of positive lower logarithmic density, we have for every meromorphic function I (z) and every e > 0 1
T{r,f)
2  e < T{r, I') < 3e + e, and for every entire function
T{r,f)
1 e
< T{r, I') < 2e + e.
The constants 1/2 and 1 on the lefthand side are sharp. The quantities 2e and 3e on the righthand side are probably not sharp, but cannot at any rate be replaced by 1.
4.2 Modular Distribution of Meromorphic Functions Together with Their Derivatives 4.2.1 Generalization of the Nevanlinna fundamental lemma on In order to discuss the value distribution logarithmic derivative. of meromorphic functions together with their derivatives, we need the following useful lemma, which was given by Hiong Kinglai [2] as a generalization of the Nevanlinna lemma on the logarithmic derivative. Lemma 4.3 Let I{z) be meromorphic in 0,00, then lor every positive integer k,
Izl
Ck uniformly in Izl < 1/32.
Proof. The conclusion holds for Ck = 1 unless there are points z' and z" such that IJ{z')1 ~ 1, IJ{z")1 ::; 1, Iz'l < 1/32 and Iz"l < 1/32. Then because of the continuity of J, there is a point Zl such that (4.4.8) Let us show that (4.4.8) implies IJ{z)1 There are two mutually exclusive cases. Case 1. The inequality
> Ck uniformly in Izl < 1/32.
116
Chapter 4 Value Distribution of Meromorphic Functions
holds uniformly in Izl < 1/8. In this case we have Izl
']
j
P'
= 1,2""
where [xl denotes the greatest integer not exceeding x. Since 00
Lj
m'
00
L L  II>. = j=l L j=ll=l aJI
Lj
[
2j>. ]
Y
~
V>'
,J
and
00
1
L =2 < j=l J
00
2j >'
00
Lj
LL
m. J
00
>L
j=ll=l lajll>.e 
[
:s ] J
j=l (2 j )>.e
= 00
for any positive number e, the convergent exponent of the sequence {ajl' ajl, ... ,ajl : j = 1,2, ... ; l = 1,2, ... ,Lj} equals A. Setting ,
I
v
mj
if A is not an integer,
q = { [Al
AI
otherwise,
we construct the canonical product
IIl(Z)
=
IT IT (1 aZ, )mjemj{a:I+~(a:I)2+··+~(a:l)q}. j=ll=l
Jl
(5.1.7)
143
5.1 Distribution of Borel Directions
Moreover, set
(2 .+ 2j>'1)'8 e'i 3
bjl =
l,
j = 1,2,···,l = 1,2, .. ·,Lj.
We can see that 00
Li
00
Li
m' m' I:I:_3 .e 3=11=1 ~I: la~Le = 00 3 3
3=11=1
for any positive number e, so the convergent exponent of {bjl, bjl,"" bjl : ,
y
,
mi
j = 1,2"", 1 = 1,2"", Lj} is also equal to A. Construct the canonical product (5.1.8) Let us prove that the function
J{z) = II1{Z) II 2 {z)
(5.1.9)
meets the requirement for Theorem 5.1. First, the order of J{z) is equal to A since both II1{Z) and II2{z) have order A. Next, arg z = 80 , 80 E E, is a Borel direction of J{z). For suppose otherwise, then there is a positive number eo such that J{z) has a finite nonzero exceptional value ao in G : largz  80 1 < eo, i.e.,
n(r, 80 , eo, J = ao) < r'T for some r(O < r < A), whenever r is sufficiently large. Since 80 E E, there exists, for every sufficiently large integer j, a point ajl such that the disk K : Iz  ajd < 4 is contained in G. In the disk K, J(z) has only one zero ajl of order mj and one pole bjl of order mj; in addition, in K, the number of zeros of J(z)  ao does not exceed rJ with rj
= 2j
+ 4.
144
Chapter 5
Recent Studies on Borel Directions
Note that 1 )
T(2rj,l ao) < (2rj).\+1,log+log+ I/(O)l_ aol < (2rj).\+1, n(2rj, I
= ao) < (2rj)Hl,n(2rj, I = 00) < (2rj).\+1
for sufficiently large j . Let ('Y) be the union of all the disks having their centers at each aopoint and pole of I(z) in Izl < 2rj and each having radius 1
d = 8(2rj)'\+1. According to Lemma 5.2,
+log+ n(2rj, I = 00) + log 2 +log(8(2rj).\+1) ( 1 ao) +log+ log+ I/(O)aol +log+ T2rj,l~
(5A+6)log(2rj)+910g2 (5.1.10)
for any point z in the region
(Izl ~ rj) \ C'Y).
By our choice of d, there are two circumferences Izajz! = rl(l < rl < 2) and Iz  ajll = r2(3 < r2 < 4), not intersecting C'Y), so that Lemma 5.3 gives us log
'..1_') II(z)I'(z) ao I dlzl.
when q = 0, (5.1.13) when q
~
1.
Because (hEE and E is a closed set, there is an angle G 1 : largzlhl < 8 such that the Ljl (j ~ jo, 1 ~ I ~ Lj ) are not contained in G 1 whenever j is sufficiently large. Thus J(z) has neither zero nor pole in the region (Izl > r) n G 1 for every large number r. Suppose our assertion is false, i.e., suppose arg z = (h is a Borel direction of J(z). Then, according to Theorem 3.11, there exists a sequence of filling disks
lim IZkl
k+oo
= 00,
lim Ck
k+oo
= 0,
k
= 1,2"",
such that J(z) takes every complex value IZkl>'Ok times in rk, except for those values in the union of two spherical circles each with radius 2 k , where limk+oo 8k = 0. Let r k be the disk Iz  zkl < 2cklzkl, k = 1,2,···. Since r k is contained in larg z  lh I < 8/2 for k sufficiently large, we have from (5.1.13),
IJ'(z)1 J(z)
J1
when q
= 0,
when q
~
1,
whenever z E r k. For every sufficiently large integer k, there is a point zk in r k such that IJ(zk)1 ~ 1. Otherwise, IJ(z)1 > 1 in r k, and the unit disk would be covered by two spherical circles each with radius 2 k , since rk is a sequence of filling disks of J(z). Furthermore, for every small positive number
147
5.1 Distribution of Borel Directions
"I, there is a point z~ in
Otherwise, we would have
r k with
IJ(z)1
1/2, a necessary and sufficient condition for E to be the set of all Borel directions of an entire function of order A is that each element in E be a member of a chain of order A: ()I, ()2,"', ()n with ()j (mod 271') E E(j = 1,2"", n).
5.2 Common Borel Directions of a Meromorphic Function and Its Derivatives 5.2.1 On the Milloux theorem. For every meromorphic function f(z) of positive and finite order in the plane, Theorem 3.8 states that f(z) has at least one Borel direction, and that I' (z) also has a Borel direction since the order of f'(z), by Theorem 4.2, is also equal to A. In 1928, G. Valiron posed an interesting but difficult problem: Do a meromorphic function and its derivatives necessarily have a common Borel direction? After the partial results of A. Rauch [2] and Chuang Chitai [1], H. Milloux [3] established the following striking theorem.
Theorem 5.2 If f(z) is an entire function of order A (0 < A < 00), then every Borel direction of its derivative f'(z) is also a Borel direction of f(z). The original proof by Milloux is very long and complicated. Later on, Zhang Guanghou gave a simple proof and extended the Milloux theorem to the case of meromorphic functions having a Borel exceptional value 00. However, Zhang's handling of initial values remains complicated. Now let us establish a general theorem, from which Milloux's theorem and Zhang's theorem follow immediately. Its proof is direct and simple (See Yang Lo [2]).
149
5.2 Common Borel Directions of a Meromorphic Function
5.2.2 Fundamental theorem. Theorem 5.3 Suppose that J(z) is meromorphic and oj order ).(0 < ). < (0) in the finite plane, and takes infinity as a Borel exceptional value in larg zl < 'Yo. Let
rk: Iz lim
k ...... oc
ek
Rkl
= 0,
< ekRk,
Rk+1
> 2Rk, } (5.2.1)
= 1,2, ...
k
be a sequence oj filling disks oj order). oj f' (z) such that J' (z) takes every complex value at least R~e~ times in rk, except Jor those values in the union oj two spherical disks each with radius 8k on the Riemann sphere, where lim e~ = lim 8k = O. k ...... oc
k ...... oc
Let (./ _ (
I'k 
10gT(r,J)) ,  A. r?R!/2 log r sup
(5.2.2)
IJ (5.2.3)
then the regions (5.2.4) 1 'f/k
= 47re, ,
k = 1,2, ...
(5.2.5)
must contain a subsequence (Gkj) as filling regions oj order). Jor J(z), >.e"
i.e. J(z) takes every complex number at least Rkj kj times in Gkj' except Jor those numbers in the union oj two spherical disks each with the radius 8~. on the Riemann sphere, where limj ...... oc e%. 3
3
= limj ...... oc 8~J. = O.
Proof. Let us suppose otherwise, i.e., no subsequence of Gk is a sequence of filling regions for J(z). We shall prove that this is absurd. Since most of the inequalities in the following discussion are valid only for sufficiently large values of the index k, we shall assume throughout that k is sufficiently large.
150
Chapter 5
Recent Studies on Borel Directions
Since (rk) is a sequence of filling disks for f'(z), there exists a sequence of complex numbers {ak} such that (5.2.6) In the interval [R~'1Ie, R~+'1Ie], we take the points
r~,p
= R~'11e (1 + TJk)P,
P
= 0,1,2, ... ,P,
where P = [2TJk log Rk/ log(l+TJk)]+l, and [2TJk logRk/ log(l+TJk)] denotes the integral part of 2TJk log Rk/ log(l + TJk). Setting
and (5.2.7) it is easy to see that p
Gk C (
p
U Sk,p) C ( U S~,p) C Gk'
p=o
(5.2.8)
p=o
Since (Gk) does not contain any subsequence as filling regions of order A for f(z), there is a subsequence (Gkj) having the following properties: Fo) every positive integer j, there are three distinct complex numbers a"kj(l = 1,2,3) such that la,1 ,kj,a,2 ,kj l > 6(1 ~ II =f.: l2 ~ 3) and E?=1 n( Gkj , f = a"kj) < R~, where 6 and 7"1 (7"1 < A) are two positive numbers independent of j. In fact, we take two sequences of positive numbers e'J and 6j which tend to zero. If the preceding assertion is not true, then a subsequence (Gk,l) of (Gk) can be found such that all the complex numbers satisfy>.
"
ing the inequality n( Gk,I'/ = a) < Rk~el belong to the union of two spherical circles with radii 6~ on the Riemann sphere. Similarly, there is a subsequence (Gk,2) of (Gk,l) such that all the complex numbers satisfying the inequality n(Gk,2, f = a) < R >.k;e"2 belong to the union of two spherical circles with radii 6~. By continuing this procedure and taking the diagonal sequence (Gk,k), the complex numbers satisfying the inequality >.
"
n( G k,k, f = a) < Rk~e Ie belong to the union of two spherical circles with radii 6~. This means the subsequence (Gk,k) is a sequence of filling regions of order A for f(z), and we derive a contradiction.
151
5.2 Common Borel Directions of a Meromorphic Function
In the following we shall use (G k) instead of (G kj) for the sake of brevity. It is obvious that we can take 03,k = oo( k = 1,2, .. ·). Hence, for every k, there are three distinct complex numbers ol,k(l = 1,2,3) such that
03,k =
00,
and
3
L n( Gk, f = Ol,k) < R'F:, 1=1
where 8 and T1(T1 By putting
< A) are two positive numbers independent of k.
and
Gk,p(t) = hk(Tk,p + 4017kTk,pt) , we see that Gk,p(t) is meromorphic in It I < 1, and 3
Ln(ltl < I,Gk,p(t).= ~,k,p(t» < R'F:, 1=1
where the functions ~,k,p(t) = Ol,k  akTk,p  40ak17kTk,pt (l = 1,2,3) have no zeros or poles in It I < 1 with
ff
JJltl
R~+11" 2
>

(5.2.27)
(5.2.28) (5.2.29) and
Ig(O)1 = 1/(0)  bkl >
1
2'
(5.2.30)
(When 1(0) = 00, we note that limz+o/(z)z"Y = limz+og(z)z"Y = C"Y is a finite nonzero number.) Then it only remains to estimate the quantity oCt) in Lemma 5.2. If , = re i . ! > R 2 > e2(logRIc)2 R >.elcelc k k _
+ 00.
This yields the contradiction and we complete the proof of Theorem 5.3.
o 5.2.3 Corollaries. laries.
From Theorem 5.3, we can obtain a series of corol
Corollary 1. Let fez) be a meromorphic function of order A{O < A < 00) in the finite plane. Suppose B : argz = (Jo {O $ (Jo < 271') is a Borel direction of order A of f'{z) and fez) has 00 as a Borel exceptional value in largz  (Jol < /'0. Then there exists a sequence of positive numbers Rkj tending to infinity and a sequence of positive numbers 'TIki tending to zero such that 1171c .
( Rk .
;
3
1+171c ')
< Izl < 2Rki
3
n {Iargz 
(Jo I
< 'TIki) (j =
1,2, ... ) (5.2.57)
is a sequence of filling regions for both fez) and f'(z).
Without loss of generality, we may assume that (Jo = O. Since B : argz = 0 is a Borel direction of order A of f' (z), according to Theorem 3.11 there exists a sequence of filling disks of order A, r k : Iz  zkl lim ek = 0
k+oo
< ekzk, argzk = 0, Zk+1 > 2Zk, (k = 1,2,,, .),
162
Chapter 5
Recent Studies on Borel Directions
such that J'{z) takes every complex number a at least z;e~ times in r k, except for those numbers in the union of two spherical circles with radii Ok on the Riemann sphere, where lim e~ = lim Ok = O. k+oo
k+oo
Choose
and
where 13k is given by (5.2.2). It is obvious that every rk : Iz  Rkl < ekRk contains the corresponding disk r k. Thus (rk) is a sequence of filling disks of order .x of f'(z) and satisfies the conditions of Theorem 5.3. Setting 1/2 17k = 47re k , then
must contain a subsequence of filling regions for both J(z) and J'(z). Corollary 2. oj order.x oj J(z).
Hypotheses as in Corollary 1, B is a Borel direction
In fact, it follows from Corollary 1 that
j
is a sequence of filling regions of order
.x
of J(z), Le.
= 1,2,···
J(z) takes every
>.e"
complex number a at least R k . kj times, except for those numbers in the 3 union of two spherical disks each with radius o~.3 on the Riemann sphere, where limj+oo eZ.3 = limj+oo O~.3 = o. Without loss of generality, we may 00
assume that
L O~j is less than a preassigned positive number PO. j=1
>.e"
Consequently, the inequality n( G kj , J = a) > Rkj kj holds for all positive integers k and all complex numbers a, except for those a in the union of a sequence of disks the total sum of whose radii is less than PO. For such "normal" (Le., nonexceptional) numbers a and any positive
163
5.3 Angular Distribution of Meromorphic Functions
number e, we have
\
f  "') 0 .. ,L< 1. logn(r ,o,
A> 1m  r+oo
00 .lim ej IZj+11 > 2lzjl, argzj
< A < 00), there
= 0,
= 8o,
j
= 1,2"",
such that
n(rj, J = a) > IZjl~1/i
(5.3.10)
169
5.3 Angular Distribution of Meromorphic FUnctions
holds for every complex value a, except for some values in the union of two )..,,'.
spherical disks Sj and S'J each with radius e 1zjl 3, where limj~ e 1zjl'7",
(5.3.19)
then (5.3.11), (5.3.13), (5.3.16), (5.3.17) (5.3.18) and (5.3.19), taken together, give us f(zj)  aj a  aj I> e 1 5Iz·I'7" Ibj(64cjIZjl)k' bj (64cjlzjl)k 2 3
(5.3.20)
Clearly, there is a complex number aE(SjUS'J) which satisfies not only (5.3.19) but also (5.3.20). Comparing (5.3.10), (5.3.14), (5.3.15) and (5.3.20), we obtain
< Ck{ IZjrr log IZjl + log IZjl + IZjIT}. This inequality contradicts the fact that T < A, since limj+oo "Ij = 0, and IZjl,xflj
the proof is complete.
0
As a consequence of Theorem 5.5, we have the follwing theorem and corollary: Theorem 5.6 If fez) is meromorphic and of order A (0 < A < 00) in the finite plane, then there exists a direction argz = 90(0 ~ 90 < 271") such that for any positive number C and any two finite complex numbers a and b(b # 0), we have lim log{n(r,90,c,f = a) +n(r,90,c,f(k) r+oo logr
= b)} = A.
Corollary. Let fez) be meromorphic and of order A (0 < A < 00) in the finite plane. If argz = 90(0 ~ 90 < 271") is a Borel direction of fez), and fez) has a finite Borel exceptional value in the angle largz  901 < coCco > 0), then arg Z = 90 is a common Borel direction of fez) and all its derivatives. Theorems 5.5 and 5.6 are due to Yang Lo and Zhang Qingde [1].
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
In Chapter 1, we have seen that while Borel direction are basic to angular distribution theory, the concept of a deficient value is important in modular distribution theory, where the definition of a deficient value depends only on the modulii of the points at which the function takes on this value. It would therefore not appear initially to be the case that there is any relationship between Borel directions and deficient values. However, Yang Lo and Zhang Guanghou have shown that if a meromorphie ftmetion of finite positive order has a deficient value, then the distribution of its Borel directions has to follow a certain rule. In the general case, the deficient values and Borel directions are closely related to each other in number.
6.1 Precise Order and Three Lemmas 6.1.1 Precise order. In the following section, we shall need a very important tool, viz., precise order. This was introduced originally by G. Valiron. Definition 6.1 Let f(z) be meromorphic in the finite plane and A a finite positive number. A nonnegative continuous function A( r) in [0,00) is called a precise order ofT(r,J) or f(z), if (1) lim A(r) = A, r+oo
(2) A'(r) exists everywhere in [0,00) except on a countable set of points, and limr + oo r,AI (r) log r = 0,
173
6.1 Precise Order and Three Lemmas
(3) r~(r) ~ T(r, I) always holds for sufficiently large r, and the equality is true for a sequence (rj) of positive numbers tending to infinity. r~(r) is called a type function ofT(r, I) or f(z).
Theorem 6.1 Let f(z) be merom orphic and of finite positive order ~ in the finite plane. Then f(z) has a precise order. Proof.
Let us consider two different cases as follows.
(i) T(r, I)
> r~
holds for a sequence of values r tending to infinity.
It is clear that the function
cp (r ) = max z;?:r
log+ T(x, I) log x
(6.1.1)
is continuous, nonnegative and nonincreasing in [0, 00), and that lim cp( r) = ~; moreover, the set M of values r statisfying cp(r) = (log+ T(r,
is unbounded, and T(r, I)
> r~
r+oo
1))/ logr
for every rEM.
Choose r1 and t1 such that r1 integer with h > 1 + r1 and cp(t1) Consider1)
> eee, r1 < cp(r1)'
EM, and t1 is the minimum
and
Y2(X) = cp(x),
x
~
h.
Since Y1(t) = cp(r1) > cp(t1) = Y2(td and limz +oo Y1(X) = 00, the curves Y = Y1 (x) and Y = Y2 (x) must intersect for x > h. Let U1 be the abscissa of the first point of intersection. Let r2 = mini M
n rUb oo)} and repeat the above procedure.
rj  rj1
~
Uj1  rj1
~
tj1  rj1 > 1,
1) We denote loglog a and logloglog a by log2a and log3a respectively.
Since
174
Chapter 6
we have limj+oo rj =
Deficient Values and Borel Directions of Meromorphic Functions
00.
Define
cp(rl), cp(rl) log3 r + log3 h, A(r) =
cp(r2), cp(r2) log3 r + log3 t2,
It is clear that the function A( r) is continuous and nonnegative in [0, (0), and that N(r) exists everywhere except at tj and Uj. Moreover, in every small interval, either A' (r) 0 or A' (r) = 1/ (r log r log2 r). Thus lim rA/(r)logr = o. r+oo
=
When r ~ rl, we have A(r) ~ cp(r) ~ (log+ T(r, I))/logr. On the other hand,
'(r,0) _ cp (0) r, _ log+ T(rj, I) , j = 1,2,···.
A
logrj
Since A(r) is nonincreasing, we have limr + oo A(r)
= limr+oo cp(r) = A.
(ii) T(r, J) ::::; r~ holds for sufficiently large r. If there is a sequence (rj) of positive numbers tending to infinity such that T(rj,1) = r;, then A(r) A is a precise order of J(z). Therefore we
=
only need to consider the case in which T(r, I)
< r~ holds for r
~ ro
> eee. The function
1jJ(r) = max log+T(x,1) ro~:r:~r
log x
(6.1.2)
is continuous and nondecreasing. The set of values r with 1jJ(r) = (logr)1 x (log+ T(r, I)) is unooundoo. There exists a positive and sufficiently large number rl > ro such that if 81 denotes the maximum abscissa of the points of intersection of the curves and
Y2(X) = 1jJ(x),
175
6.1 Precise Order and Three Lemmas
then we have ro
< 81 < r1 and L n [ro, 81l '" 0. Setting
we see that ro ::; t1 ::; 81 ::; r1. Furthermore, there exists r2 > r1 + 1 such that if 82 is the maximum abscissa of the points of intersection of the curves
and
Y2(X) = 1/J(x), then we have r1
< 82 and L n [r1, 82l '" 0. Let
and U1 the point on have
[S1,
r1l with ~
+ log3 U1 log3 r1
= 1/J(t2). Then we
Define
According to the definition of tj, Uj and 1/J(r), the function nuous and nonnegative in [0,00). Moreover we have
~(r)
is conti
~(r) ~ 1/J(r) ~ log+ T(r, f) logr and
~(t.) 3
= 1/J(t.) = 10g+T(tj,f). 3 logtj
Finally ~'(r) exists everywhere except at the points Sj and Uj, and lim {r r+oo N(r)logr} = o.
176
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
6.1.2. Three lemmas. Now we establish three lemmas which playa very important role in the present Chapter. Lemma 6.1 1 ) Let J(z) be merom orphic and oj order A (0 < A < +00) in the finite plane and let all(v = 1,2,··· ,p) be p(1 ~ p < 00) distinct complex numbers. Suppose 6(all , J) = 611 > 0 (v = 1,2,··· ,p) and 6 = min1$v$p 6v . If A(r) is a precise order of J(z) and U(r) = r~(r), then there exists a sequence oj positive numbers Rj(j = 1,2,···) tending to infinity such that, for every sufficiently large j and v = 1,2,··· ,p, the set Ejll of values c,o(O ~ c,o < 27r) with log {
IJ(Rjei~) _ alii> 2:+4 U(Rj), .
when all '"
00,
(6.1.3)
6
log IJ(Rje ZIP ) I > 2~+4 U(Rj),
when all =
00
satisfies mes Ejll
> K(6,p, A) > 0,
(6.1.4)
where K(6,p, A) is a positive constant depending only on 6, p and A, e.g.,
67r K(6,p,A) = (
(6.1.5)
)
5 + log 25:e
4.\+2
log3
Proof. According to the definition of A(r), we can choose a sequence of positive numbers r j tending to infinity such that lim T(rj, J) = 1. U(rj)
j+oo
Let us consider the function 1/(J(z)  all), v = 1,2,··· ,p, (when all =
00, 1/(J(z)  all) should be replaced by J(z)). Applying the BoutrouxCartan theorem for every fixed j, to the poles bill (l = 1,2,···, n(3rj, J = all)) of 1/(J(z)  all) in the disk Izl ~ 3rj, we have n(3rj,!=a,,)
II l=l
Iz 
bllli
>
h
(_rj)n
(3 . J
r"
)
a" ,
v = 1,2,···,p,
P
1) Although Lemma 6.1 can also be derived from the spread relation (Theorem 7.4 of Chapter 7), the proof here is much simpler, and it does not require the precise lower bound established by the spread relation.
177
6.1 Precise Order and Three Lemmas
except on at most n(3rj, f = av ) disks (y)v the total sum of whose radii does not exceed 2ehrj/p. Now let us set h = 1/5e and (y) = ~=l(y)v' In the annulus rj ::; Izl ::; 2rj, there is a cride Izl = Rj, not intersecting (1'), on which an arbitrary point z satisfies
Iog If(z)1_ avl K(8,p, >.) =
For the value cp in
hence Elv
1 ~~ > 0, ( 5 + og pe)4A+2 log ~
= 1,2,,,, ,po
E!Jv' we have
c Ejv , and the proof is complete.
D
Let f (z) be meromorphic on 1z1 ~ R( < (0). If
Lemma 6.2
N
1/
= n(R,f = 0) +n(R,f = 1) +n(R,f = (0)
and the shortest distance between the origin 0 and these N points equals d(O < d < 1), then we have
T(
r,
f)
1) When 1(0)
'TrIA. Choose the number b such that 1 < b < 2 and J(b) '" 00. Now we construct the transformation 1
1
1
l'
zk  bk (=
(6.2.2)
zk +bk
which maps the angle D : 'Pmo + a < arg z < 'Pmo+1  a onto the unit disk '(I < 1. The function J(z) becomes g(() = J(z((». Let z be an arbitrary point on L j • Its image ( must satisfy 1.f£.
1(1
1
1 1
'P
1
4 IZlkbkCOS}2 Izlke'kb k  { 1 k = 2 1 1 2 1.f£. 1 Izlke'k + bk Izlk + 21zlkb k cos I£. + bk k
By noting that
I'PI:::; 'Pmo+1 aj:::; 'Pmo+1 8a =
k(i 
~a),
186
Chapter 6
Deficient Values and Borel Directions of Meromorphic FUnctions
we have
Thus the image lj of Lj under the transformation (6.2.2) is located in 1(1 ~ rj, where (6.2.3) The inverse of (6.2.2) is
z = z( ()
()k = b( 11 + _ ( .
Because there is no Borel direction between Bmo and Bmo+1' we can find three distinct complex numbers f3v(lI = 1,2,3) and a number r(O < r < 'x) such that 3
L n{ D(R), f
=
f3v} < R\
R > ~,
v=1
where D(R) denotes the region (Izl ~ R)n(IPmo +a ~ arg z ~ IPmo+1a). Moreover, r can be chosen so that 1 < kr < k'x. When
1(1 ~ r, we have Izl ~ b2 k /(I r)k, hence
Thus
1L 1 3
ro v=1
for any e
> O.
n(r,g = f3v){l r)kT1+€dr
1 and
(kr 1 + e)
l
r
ro
= (1 
3
~ N(t,g = ,8,,)(1 t)kT2+ e dt
,,=1 3
ro)kT1+ e ~ N(ro,g
= ,8,,) 
(1 _ r)kT1+ e
,,=1
we obtain
1,,=1L 1 3
N(r,g = ,8,,)(1 r)kT2+e:dr
< +00.
ro
Furthermore, the second fundamental theorem of Nevanlinna yields
1
1 T(r,g)(l
r)kT2+e:dr
< +00.
(6.2.4)
ro
This means the order of g«) in On the other hand, put
1(1 < 1 does not exceed kT 
1 +r·
Nj=n ( T,g=oo),
1.
lr· 4
hj='.
Let ('Y)j denote the family of all the exceptional disks with centers at a pole of g«) in 1(1 ~ (1 +rj)/2 and the same radius hj/(nj + 1). Then the total sum of the radii of the disks in ('Y) j is less than hj. There exists a point (2 on lj (hereinafter we omit the lower index j of (2 and (1 for brevity), outside the disks in ("t)j, since the images (1 and (1 of the extreme points Rjei ( tp""o+l +aj) and R j ei (tp""O+l aj) of Lj under (6.2.2) satisfy
,
1
1 (
1
1
2R~ b'k ~
R}
. tp""O+l aj k 
e'
. tp""o+l aj
+ R}b'k(e'
k
. tp""O+l +aj ) e' k .tp""o+l +aj
+ e'
k
2
)
+ b'k (6.2.5)
1 1 80: 4R~bk cos , k ;::: 1 1 (Rjk + b'k)2
;:::
1 Rk j 2
> 4hj.
188
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
Then the PoissonJensen formula gives
1+rj +r'
(1+rj)2 b r    .. 2 'm(TL,g)+.E IOg I1';rj II I  2   rj II 2«(2  bll )
loglg«(2)1~ l~rj
. ~ r which contradicts the hypothesis r < >.. Therefore, the magnitude of the acute angle between Bmo and Bmo+1 does not exceed 71"/>'. Finally, when J(z) has only one Borel direction B l : arg z = 1/2, the above discussion
6.2 Distribution of Borel Directions of Meromorphic Functions with Deficient Values
can be repeated in the angle larg zl < 1r  a, where k1r > 1r / A. A contradiction is again derived. 0
1r 
189
a = k/21r and
6.2.2 Discussion. Corollary. Let J(z) be meromorphic and oj order A(O < A < (0) in the finite plane, and suppose J(z) has a deficient value. IJ A is larger than 1/2, then there are at least two Borel directions with the acute angle between them not exceeding 1r / A. It is easy to see that this Corollary is equivalent to Theorem 6.2 and implies the following result of Valiron and Cartwright: An entire function of finite order A > 1/2 has at least two Borel directions. In the statement of Theorem 6.2, we can replace 'a deficient value' of the function J(z) by a 'Borel exceptional value' to obtain a similar theorem. Theorem 6.3 Let J(z) be meromorphic and oj order A(O < A < (0) in the finite plane, and suppose J(z) has a Borel exceptional value ao (finite or infinite). When J(z) has more than one Borel direction, there exist two Borel directions with an acute angle between them which is less than or equal to 1r/A. When J(z) has only one Borel direction, then A ~ 1/2. In fact, the condition that J(z) has a deficient value is used only in (6.1.9) of Lemma 6.1. When J(z) takes ao as a Borel exceptional value, we may assume that ao = 00. Thus
for some T,O < T inequality and
< A, and for sufficiently large integers
we have m(Rj, J)
j. From this
1
> 4U (rj).
The proofs of Lemma 6.1 and Theorem 6.2 can be completed as before.
190
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
Furthermore, the condition that f(z) have a deficient value or a Borel exceptional value in Theorems 6.2 and 6.3, can be replaced by a corresponding condition for its derivative. More precisely, we have the following theorem:
Theorem 6.4
Let f(z) be meromorphic and of order A (0
< A
1/2, an entire function of order ,x having an infinite number of deficient values. Later, A. Weitsman [1 J extended Pfluger's result to meromorphic functions. Here, we shall exphasize the relation between the number of deficient values and the number of Borel directions (Yang Lo and Zhang Guanghou
[3]). Theorem 6.5 Let fez) be meromorphic and of order,x in the finite plane, where 0 < ,x < 00. If p denotes the number of deficient values of fez) and q the number of Borel directions of fez), then we have p ::::; q. According to Theorem 3.8, fez) has at least one Borel direction. Thus q ~ 1. When q = 00, the conclusion of Theorem 6.5 is obvious. When 1 ::::; q < 00, let us suppose Theorem 6.5 is not true, i.e. p ~ q+ 1, and derive a contradiction. Take q+1 deficient values all (1I = 1,2,···, q+1) of fez) and let o(all , J) = Oil. We may assume that the all (1I = 1,2,···, q+1) are all finite. Otherwise we can consider the function l/(J(z)  ao) (ao is not a deficient value of fez)) instead of fez). Clearly fez) and l/(J(z)ao) have the same number of deficient values and the same Borel directions. Proof.
Denote by Bm: arg z = jv
and I
R~e 1 og If(z) _ al'I::; jv ,
It = 1,2, ... ,v  1, v
+ 1, ... ,q + 1,
(6.3.11)
for every z E L)~) and two arbitrary small positive numbers TJ and e. Moreover, (Rjl) is a subsequence of (Rj), (Rjv+!) is a subsequence of (Rjv) (v = 1,2,··· ,q 1), and mes {'P : Rjveilf' E
A)~)  L)~)} < !K(8, q + 1, X), q
where A)~) denotes the arc {Rjveilf' : 'Pm" < 'P < 'Pm,,+!}. Thus, in every angle G v (v = 1,2, ... ,q), the set of values 'P for which log
.1 > R~e, If(Rjqe'lf')  aq+!1 3q
(6.3.13)
has measure less than or equal to
The q Borel directions Bm : arg z = 'Pm(m = 1,2,···, q) divide the plane into q contiguous angular sectors, and therefore each of the q angles
195
6.3 Deficient Values and Borel Directions of Meromorphic Functions
coincides with one of the G~s (ll
= 1,2","
q
q). Thus
q
( UG U( U Bm) v)
v=l
m=l
covers the plane. Consequently the set of values 'P(O ~ 'P (6.3.13) has measure
~
q
L
mes {'P: Rjqei'{J E A;~)  L;~)}
< 211")
satisfying
< K(8,q + I,A).
v=l
However, since (Rjq) is a subsequence of (Rj), it follows from (6.3.1) that the set of values 'P(O ~ 'P < 211") satisfying (6.3.13) must have measure greater than K(8, q + 1, A). With this contradiction our proof is complete. D
6.3.2 Complement. The inequality p ~ q in Theorem 6.5 is sharp in the following sense: Given a positive integer n, there is a meromorphic function fez) of finite positive order, such that the number of its deficient values and the number of its Borel directions are both equal to n. In fact, when n = 1, the function n
fez)
= II (1 :2)
(6.3.14)
k=l
provides such an example. Following the calculation in Lemma 7.9 in the next chapter, we can derive that the order of fez) equals 1/2 and '8
log If(re~
)1 =
(
()  11") 1 11" cos 2 r2
1 + o(r2)
(6.3.15)
in the angle c < arg z < 211"  c for every positive number c. Thus fez) has no finite deficient value by Lemma 6.1 and (6.3.15). On the other hand, f (z) has at least one Borel direction, according to Theorem 3.8. However, (6.3.15) indicates that fez) cannot take any direction other than the positive real axis as a Borel direction. Therefore, p = q = 1. When n = 2, the function e Z affords us a good example. It is clear that e Z has order 1, two deficient values, zero and infinity, and two Borel directions, the positive and negative imaginary axes. Thus p = q = 2.
196
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
When n > 2, we consider the function
J(z)
=
(6.3.16)
where
and
n
1
2Z2) _ nn 00 (_1)k z nk J_1 ( 1 1 n n 22 k=O n 2k k! ~+k+
L
r( 
1)
.
By a calculation, we can see that J(z) has order n/2, exactly n deficient values e(2k+1)1ri/n (k = 0,1,···, n  1) and exactly n Borel directions argz = 2klr/n (k = 0,1,··· ,n 1). Consequently p = q = n. For meromorphic functions of higher order , we can obtain a more precise result.
Theorem 6.6 Let J(z) be merom orphic and oj order A (0 < A < 00) in the finite plane, p the number oj its deficient values and q the number oj its Borel directions. IJ A > p/2, then p ~ q  1. Theorem 6.6 clearly holds when q = 00. When q < 00, let ProoJ. us assume that Theorem 6.6 is not true, i.e., assume p > q  1. Then we can choose q deficient values av (lI = 1,2,···, q) with 8(av , J) > o. Set 8 = minI:5v:5q 8(a v , J) and denote by Bm : arg z = 'Pm(m = 1,2,···, q, o ~ 'PI < 'P2 < ... < 'Pq < 271", 'Pq+1 = 271" + 'PI), the q Borel directions of J(z). By a discussion similar to the proof of Theorem 6.5, there are q distinct angles G v : 'Pmv < arg z < 'Pm v +1 (ll = 1, 2, ... , q) corresponding to q deficient values av. For every G v , Lemma 6.3 gives us a sequence of positive numbers Rjv tending to infinity and a sequence of curves L}~). According to the proof of Lemma 6.3, the magnitude of G v does not exceed 71" / A, i.e., 71" 'Pmv+I  'Pmv ~ :X' 1I = 1,2,···, q.
197
6.4 Deficient Values, Borel Directions and the Order of Entire Functions
Thus
~
q
E(CPm.,+1  CPm..) ::; q.
~
X ::; p. X < 2~.
v=1
Since the finite plane is divided into q angles Gv(v = 1,2"", q) by the q Borel directions Bm(m = 1,2, ... , q), we have q
E(CPm.,+1  CPm..) = 2~. v=1
This contradiction proves that p does not exceed q  1.
0
6.4 Deficient Values, Borel Directions and the Order of Entire Functions 6.4.1 Several lemmas. In order to obtain further results for entire functions, we need to establish several lemmas.
Lemma 6.4 Let fez) be an entire function of order ~ (0 < ~ < 00) with no Borel direction in the angle CPl < arg z < CP2 (0 ::; CPl < CP2 ::; 2~ + CPl), and ~(r) its precise order with U(r) = r~(r). Suppose there are positive numbers 'f/ and K', a finite complex number ao, and a sequence of positive numbers Rj tending to infinity such that mes E{ cP : CPl
< cP < CP2, log If(Rjeicp )  aol < 'f/U(Rj)} > K'. (6.4.1)
Then, for any two positive numbers a and Q with a
< .
(K'
mm 32'
CP2  CPl)
20
an
d
1
.(r) and A(r) is a precise order of fez). Among the q angles 'Pm < arg z < 'Pm+1 (m = 1,2,···, q), there exists at least one angle, say GI : 'Pml < arg z < 'Pml+1, and a sequence Rjl of R j tending to infinity, with the property that the set of values 'P satisfying 'Pml
< 'P < 'Pml +1
and
1 log If(Rjleicp )
6 _
all> 2>'+4 U(Rjl),
has measure greater than K/q. Taking a, 0 < a < k/20q, and applying Lemma 6.4 to fez), aI, Gl, the sequence (Rjl) and positive numbers K' = K/q and Q = 2, we have log
1
. If(Rjle'CP) 
all
on the arc {Rjle icp : 'Pml
>
6 2>'+4 { 4(5 + 4 log ~)
r
Nl
U(Rjl)
(6.4.22)
+ lOa < 'P < 'Pml+1 lOa}, where h=
K 80(2e
+ 1)(11" + a)q
.
In the angle Gl, the set of values 'P satisfying at least one of the following inequalities has measure less than or equal to 20a < K / q: log
If(Rjle~) _ avl
> 2>.:4 U(Rjl),
j
> jo, v = 2,3,···, [~] + 1,
Thus, among the q  1 angles 'Pm < arg z < 'Pm+1 (m = 1,2,···, ml 1, ml + 1, ... , q), we can find G 2 : 'Pm2 < arg z < 'Pm2+1 and a subsequence Rj2 of Rjl tending to infinity, such that the set of values 'P satisfying 'Pm2
< 'P < 'Pm2+1 and
209
6.4 Deficient Values, Borel Directions and the Order of Entire Functions
has measure larger than K / q. Similarly, applying Lemma 6.4 to f(z), a2, G2, (Rj2), and K' we have
= K/q, (6.4.23)
on {R j2 ei 't' : 'Pm2
+ lOa < 'P < 'Pm2+1
 lOa}.
Now we claim that G 1 and G 2 cannot be adjacent angles. In fact, let us assume otherwise, say 'Pm2 = 'Pml+l. Then we can take two angles Gi : 'Pml < arg z < 'Pml +1  c and G~ : 'Pm2 + c < argz < 'Pm2+b where c = min(K/(2q),7I"/(4A)). Applying Lemma 6.6 to f(z), two distinct finite complex numbers al and a2, the angles Gi and G~, 'fJ = 8/2>.+4 and K' = K/2q, we see that the magnitude of the angle between any side of Gi and any side of G~ is not less than 71"/ A, i.e.,
This contradicts the choice of c. In G 1 U G2, the set of values 'P satisfying one of the inequalities
has measure less than or equal to 40a < 2K/ q. Among q  2 angles 'Pm < arg z < 'Pm+1(m :I ml,m2), there exists at least one, say G 3 : 'Pm3 < arg z < 'Pm3+b having a similar property with a subsequence (Rj3) of (Rj2) and the complex number a3. For the same reason, G3 can not be a neighbor of G 1 or G 2 • Following this procedure on, we can obtain [q/2] + 1 angles G v : 'Pmv < arg z < 'Pmv+1(lI = 1,2"", [q/2] + 1), corresponding to [q/2] + 1 deficient values a v (lI = 1,2"", [q/2] + 1), and a subsequence (R j ,[q/2)+1) tending to infinity such that the inequality
(6.4.24)
210
Chapter 6
Deficient Values and Borel Directions of Meromorphic Functions
holds on the arc {Rj ,[q/2)+!ei '),
and >.(r) is a precise order of J(z) (K{8,p', >.) is defined in (6.3.2).). Without loss of generality, we may suppose that
For otherwise, it is sufficient ot rearrange the order of av(v = 1,2,··· ,p'). Now applying Lemma 6.6 to J(z), G v and G v+!(1 ~ v ~ p', Gpl+! = Gd, a v and av+l, with
and
K'
. ('Pm+1 = 1::;m::;q mIn 2
we have 'Pm v+l

'Pm v+1 ~
11"
'X,
p'
~:)'Pmv+1
 CPmv+!)
v
'Pm) ,
= 1,2,··· ,p',
~ p'~ ~ 211",
v=1 so that p'
L:(CPmv+1  'Pmv+!) v=1
p'
+ L:('Pmv+!  'PmJ > 211". v=1
(6.4.28)
213
6.4 Deficient Values, Borel Directions and the Order of Entire Functions
But the lefthand side of (6.4.28) is obviously equal to 211". This contradiction proves Theorem 6.8 for A > 1/2. When A S 1/2, let us prove p = O. If fez) has a finite deficient value all then Lemma 6.1 yields a sequence (Rj) such that the two sets of values cp(t) rT  tT'
t
to},
B
= {cp : log If(rei'Pl ~ to}.
Then we have
mesA = A(to), mesB
= A(to 
0),
A
c B.
Choose a set E on [11", 11"] with
mesE = 2(J,
AcE c B.
We assert that E meets the requirement of Lemma 7.3. In fact, if there is another set F with mesF = 2(J, then we have
223
7.2 The T* Function
£
log If(rei'P)ldcp =
£
{log If(rei'P)I to}dcp + 20to
~ [1r1r max{O, log If(rei'P)I to}dcp + 20to = =
L
{log If(rei'P) I  to}dcp + 20to
L
log l{rei'P)ldcp.
This proves Lemma 7.3. D
If J(z) is meromorphic and not identical to zero in the finite plane, then the function T*(z) defined by (7.2.2) is continuous on H = {z : Imz ~ 0, z =I a}. Theorem 7.2
00,
°
Proof. According to Lemma 7.3, for each point z = re i8 , < r < ~ 0 < 71", there exists a set E in [71",71"] with mesE = 20 for which
°
the supremum in (7.2.1) is attained, i.e.,
m*(z) =  1
h
271" E
.
log If(re''P)ldcp.
Write f{z) = h(z)/h(z), where h{z) and h(z) are entire functions and have no common zero. Thus
h
T*{z) =  1
271" E
.
log If{re''P)ldcp + N(r, J)
1 JE[lh(rei'P)1 (1) log h(rei'P) dcp + N r, h .
= 271"
Since
we have 1 T*{z) = 2
71"
h E
.
log Ih{re''P)ldcp + 1 271"
1
CE
.
log Ih {re''P Idcp + CU), (7.2.3)
where CE denotes the complement of E in [71",71"]. Now we distinguish between two mutually exclusive cases. (i) h{z) and h{z) have no zeros in the finite plane. Taking two finite positive numbers Rl and R2 with Rl < R2, it is obvious that loglh{z)1 and loglh{z)1 are two continuous functions on Rl ~ Izl ~ R2. Let Zl = rle i81
224
Chapter 7
The Spread Relation and Its Applications
and Z2 = r2ei82, where RI ~ rl ~ r2 ~ R2. Corresponding to ZI, there exists a set EI with mes EI = 2(h such that T*(ZI)
1 = 2 1r
1 EI
.
1 log Ih(rIet'P)ldcp + 2 1r
1
GEl
.
log Ih(rIet'P)ldcp + C.
Choose a suitable small positive number 8. When IZ2  zll < 8, the set EI can be combined with or taken off by a small set e to produce a set E2 with mes E2 = 202. We consider only the case when E2 = EI U e, since the other case is similar. By the definition,
Thus T*(ZI)  T*(Z2)
~ 2.. { (log Ih(rIei'P)Ilog Ih(r2 ei'P)I)dcp 21r lEI
Since log Ih(z)1 and log Ih(z)1 are uniformly continuous on RI ~ Izi ~ R2 and Ir2  rll < 8, mes e = 102  OIl ~
IZ2  zll 1r 1r RI . '2 < 2R l 8
for IZ2  zll < 8, we obtain T*(ZI)  T*(Z2) < c. Similarly, we have T*(Z2)  T*(ZI) < c. This proves the continuity of T*(z) for this case.
(ii) h(z) or h(z) has zeros in the finite plane. For a sufficiently small positive number 8 and RI < Izi < R2, we define
(A6 log If(z)l)
1 2 1 6 111' log If(z + teiW)ltdwdt. = 8 1r
0
11'
225
7.2 The T* Function
Define A6log Ih(z)1 and A6log Ih(z)1 similarly. The subharmonicity of log Ih(z)1 and log Ih(z)1 implies that A6log 1!k(z)1 ~ log 1!k(z)l,
k
= 1,2.
For the simplicity of notations, we write Ul = log Ihl, U2 U = Ul  U2, T*(z,u) = T*(z). Thus (7.2.3) becomes
T*(z,u)  C(u) =  1
271"
1 . E
ul(re'C(J)dcp + 1
271"
= log 1121,
1 .
u2(re'C(J)dcp,
CE
(7.2.3)'
which yields
For a point re i8 , Rl mesE' = 2(}. Thus
< r < R2 ,
0 ::; ()
0,
lc o
foal X(t + e)dt + lalbl X(t  e)dt + lblcX(t + e)dt bl = fo cX(t + e)dt  lbl X(t + e)dt + l X(t  e)dt o al al
X(t + e)X(t  e)dt ::;
Choose 6 > 0 such that 0
0
< e < 6 implies that
Then
foc X(t + e)X(t 
e)dt ::;
foc X(t + e)dt 
e.
(7.2.7)
Similarly, using
f21r
Jc
X(t + e)X(t  e)dt ::;
l
a2
X(t + e)dt + 1~ X(t  e)dt + 121r X(t + e)dt,
c
a2
~
X(t + e)dt  e,
0< e < 6.
we obtain
f21r
Jc
X(t + e)X(t  e)dt ::;
f21r
Jc
which combines with (7.2.7), yields, for 0
< e < 6,
mes(Ee n Ee) = fo21r X(t + e)X(t  e)dt ::; fo21r X(t + e)dt  2e = mesE  2e.
0
(7.2.8)
228
Chapter 7
The Spread Relation and Its Applications
If f(z) is meromorphic and not identical to zero in the finite plane, then T*(z), as given by (7.2.2), is subharmonic on H = {z: Imz ~ O,z i O}. Theorem 7.3
Proof. Set f(z) = h(z)/ h(z), where h(z) and h(z) are entire functions and have no common zeros. Write u(z) = log If(z)1 and Uk(Z) = log Ifk(Z)I(k = 1,2). For a real number 'Ij; and fixed rand p, 0 < p < r, we define larg zl Then
r + pei'IjJ
Note that r('Ij;) have
7r
< 2'
= r( 'Ij; )eio:('IjJ).
= r( 'Ij;) and a('Ij;) = a( 'Ij;). For any function v we
J:7r v(reicp + pei'IjJ)d'lj; = fo7r {v (re icp + pei(cp+'IjJ») + v(reicp + pei(cp'ljJ»)}d'lj; = fo7r {v(r('Ij;)ei(Cp+O:('IjJ») + v(r('Ij;)ei(CPO:('IjJ»)}d'lj;.
(7.2.9)
Fix re i8 with rl < r < r2, 0 < () < 7r. As in (7.2.3), there is a set E with mesE = 2() such that
T*(re t'8 ,u)  C(u)
= 1
27r
1 ' E
ul(retCP)dc.p + 1 27r
1 ' F
u2(retCP )dc.p,
(7.2.10)
where F = CEo Since Ul and U2 are subharmonic, we have for sufficiently small p and k = 1,2,
r
uk(reicp ) S ~ {uk(r('Ij;)ei(Cp+O:('IjJ») 27r 10
+ uk(r('Ij;)ei(CPO:('IjJ»)}d'lj;.
Substituting these inequalities into (7.2.10) and reversing the order of integration yields:
229
7.2 The T· Function
For this E, let 8 be as in Lemma 7.4. We assume that p is small enough so that 0 ~ a('Ij;) < 8 and 0 + a('Ij;) ~ 11" for 0 ~ 'Ij; ~ 11". Fix such a 'Ij;. By Lemma 7.4, we can choose a set C such that (i) C C (Ea (1/J) U E_ a (1/J)  (Ea (1/J) n E_ a (1/J) , and
= (Ea (1/J) n E_ a (1/J) U C, then
(ii) if A
mesA
Let B
= mesE 
= (Ea (1/J) U E_ a (1/J)

2a('Ij;)
= 2(0 
a('Ij;».
(7.2.12)
C. Then (7.2.13)
Since mesA + mesB = mes(A n B)
+ mes(A U B)
= mes(Ea (1/J) n E_ a (1/J) + mes(Ea (1/J) U E_ a (1/J) = mesEa (1/J) + mesE_ a (1/J) = 40, it follows that mesB = 2(0 + a('Ij;».
(7.2.14)
By (7.2.12), for every integrable function 9 we have
L
g( cp )dcp +
t
g( cp )dcp
f
g(cp)dcp +
= [ J AUB
g(cp)dcp
JAnB
= f J Ea(.p)UE_a(.p)
=
h
g( rp )drp +
Ea(.p)
(7.2.15)
f
g(cp)dcp +
g(cp)dcp
J Ea(.p)nE_a(.p)
h
g( cp )drp.
E_a(.p)
Similarly,
1
CA
g(cp)dcp +
1
CB
g(rp)drp
= f
g(rp)drp +
JFa(.p)
From (7.2.15) and (7.2.16), we obtain
[ JF_a(.p)
g(rp)drp.
(7.2.16)
230
Chapter 7
The Spread Relation and Its Applications
(7.2.17) According to (7.2.3)', for r(1/J) and 2(8  a(1/J)), there is a set E' with measure 2(8  a(1/J)) such that
T*(r(1/J)e i (Oa(1/J» , u)  C(u) (7.2.18) and (7.2.19) where the maximum of the righthand side of (7.2.19) is taken over all the measurable sets E of [0, 27r) with mesE = 2(8  a(1/J)). From (7.2.12), (7.2.19) and (7.2.18), we obtain 1 2 7r
1 A
.
ul(r(1/J)e t 'P)dcp + 1
27r
1
CA
= T*(r(1/J)e i (Oa(1/J)),u) 
.
u2(r(1/J)e l 'P)dcp
C(u).
(7.2.20)
Similarly, we have
(7.2.21)
231
7.3 The Spread Relation
Substituting (7.2.20) and (7.2.21) into (7.2.17) yields
~
T*(r('I/J)e i(8a(1/J»,u)  C(u) +T*(r('I/J)ei(8+ a(1/J»,u)  C(u).
Comparing this inequality with (7.2.11), we obtain finally
T*(re i8 ,u)
~
21
I'll' T*(rei8 + pe 1/J,u)d'I/J. i
11'
11'
7.3 The Spread Relation 7.3.1 Definition. Now we return to the problem of describing precisely the concept of deficiency, mentioned at the beginning of this chapter. To investigate this problem, it is necessary to introduce an important concept due to A.Edrei [3], viz., the spread O'(a) of a complex number a. Definition 7.3 Let fez) be meromorphic and of finite lower order J.t in the finite plane. Fix a sequence (rj) of P6lya peaks of order J.t of fez), and let A(r) be a positive function with
A(r) = o(T(r, f)),
r
(7.3.1)
+ 00.
For every r, define sets of arguments EA(r, a)
c (11',11')
by
when a#
00,
when a =
00,
(7.3.2)
and let O'A(a) = lim mesEA(rj, a). j+oo
(7.3.3)
232
Chapter 7
The Spread Relation and Its Applications
Then (T(a) = inf (TA(a),
(7.3.4)
A
where the infimum is taken aver all functions A(r) satisfying (7.3.1). u(a} is called the· spread of a with respect to fez). Below, we shall need the following integral which can be calculated by the residue theorem.
(Xl t T pet, 1, fJ)dt = s~n TfJ ,
Jo
(7.3.5)
SInT7r
where 0 < T < 1 and P(t,r,fJ) = (l/,Tr)· (r sin fJ)/(r2
+ 2trcosfJ + t 2).
7.3.2 Spread relation. The following theorem was conjectured by A. Edrei [3] in 1965 and proved by A. Baernstein [1] in 1973. Theorem 7.4 Let fez) be a meromorphic function of lower order JL(O < JL < 00). If a is a deficient value of fez) with deficiency 6(a, f), then (T(a) as given by Definition 7.3 satisfies
(T(a)
~ min {27r, ; arcsin V6(~!)}.
(7.3.6)
(7.3.6) is called the spread relation. Proof.
First, we assume that 4 . V6(a,!) < 27r. o < arCSIn
JL
2
(7.3.7)
We can also assume, without loss of generality, that
f(O)
= 1,
a = 00.
(7.3.8)
The general case is reduced to this special case by the following considerations: Suppose a =/: 00. Let 1
g(z) and define h( z) by
= fez)  a'
233
7.3 The Spread Relation
where k is an integer, it follows that
e is a nonzero constant and h(O) =
T(r,1) '" T(r,g) '" T(r, h),
r
t
1. Since J.L
> 0,
00.
Thus /, 9 and h have the same lower order and the same P6lya peaks. With a selfexplanatory notation, we have
and
where Al(r) = A(r)  klogr log lei. As r A(r) = o(T(r,I))
t
00,
we have
¢:::::::>
A(r) = o(T(r,g))
¢:::::::>
Al(r) = o(T(r, h)).
Hence
u(a, I) = u(oo,g),
u(oo, I) = u(oo, h).
It is also true that 6(a, I) = 6(00, g), 6(00,1) = 6(00,h). The above facts show that there is no harm in assuming (7.3.8).
Setting 2 . V6(00,1) 'Y = arCSIn , 7rJ.L 2
then 0
< 'Y < 1. Let A(r) be a positive function satisfying (7.3.1) and put
The spread relation is equivalent to the inequality
lim j+oo
Let
Uj ~
27r'Y.
(7.3.9)
234
Chapter 7
The Spread Relation and Its Applications
We then have
~
1.
T*(rje"2tO';) + A(rj).
Dividing by T(rj) and remembering that T*(z) ~ T(lzl, 1), we find that 1 
T*(re"2tO';) lim J =1. j+oo T(rj, 1)
(7.3.10)
On the other hand, define
v(z) = {
0,
z =0,
T*(z'Y),
Z
f=
(7.3.11) 0, Imz ~ O.
It follows from Theorem 7.3 that v(z) is subharmonic in 1m z > 0 and continuous on Imz 2 0, except at the origin where it is not defined. Since 1(0) = 1, it is clear that putting T*(O) = v(O) = 0 will remove this discontinuity. Now let us observe the half disk
DR
= {z = rei(J : 0 < r < R,
0 < () < 11"}.
With the boundary values of v(z), we construct the Poisson integral h(z) which is harmonic and majorizes v(z) in DR, i.e.
h(z)
=
jR v(t)A(t,r,(),R)dt + hr v(ReiO,
B>O,
A+B:::;7r,
cos1£A=18(a,j)=u
and cos 1£B
= 1  8(b, j) = v.
Thus inquality (7.4.6) can be obtained from Lemma 7.5. When u:::; cos1£7r, we have (4/1£)arcsiny'8(a,f)/2 ~ 27r. Hence (4/1£) arcsiny'8(b,f)/2 = 0, i.e. v = 1. Similarly, if v :::; cOS1£7r, then u = 1. D
Theorem 7.7 states that the point (u, v) cannot be located inside the ellipse u 2 + v 2  2uvcos1£7r = sin2 1£7r. Therefore it is called the ellipse theorem, which is due to Edrei and Fuchs [3]. A series of important consequences can be obtained from the ellipse theorem. For instance, we have the following corollaries: Corollary 1. Let f(z) be meromorphic and of lower order 1£(:::; 1/2) in the finite plane. If a is a deficient value of f(z) with 8( a, j) ~ 1 cos 1£7r, then a is the unique deficient value of f(z). In particular, a meromorphic function of lower order zero can have one at most deficient value.
245
7.4 Applications of the Spread Relation
In fact, if f(z) has another deficient value b, then from u 8( a, J) ~ cos J.L7r and Theorem 7.7, we have v = 1.
1
Corollary 2. Let J(z) be a transcendental entire function of finite order. If its lower order J.L satisfies 0 ~ J.L ~ 1, then
1 when 0 "'2 < I/. 6(a1,f) ~ 6(a2,f) ~ .... We define quantities dj(j = 1,2, ... ) by 6(aj,f) = (1 cosJL7r)dj = (2sin2 JL27r)dj. Thus
o ::; dj < 1,
j = 1,2, ... ,v(f).
Consider the function
. JLS7r
x =