JOURNAL OF APPLIED FUNCTIONAL ANALYSIS 7 (2012).pdf

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Volume 7, Numbers 1-2

January-April 2012

ISSN:1559-1948 (PRINT), 1559-1956 (ONLINE) EUDOXUS PRESS,LLC

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS

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SCOPE AND PRICES OF

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS A quartely international publication of EUDOXUS PRESS,LLC ISSN:1559-1948(PRINT),1559-1956(ONLINE) Editor in Chief: George Anastassiou Department of Mathematical Sciences The University of Memphis Memphis, TN 38152,USA E mail: [email protected] Assistant to the Editor:Dr.Razvan Mezei,Lander University,SC 29649, USA.

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The purpose of the "Journal of Applied Functional Analysis"(JAFA) is to publish high quality original research articles, survey articles and book reviews from all subareas of Applied Functional Analysis in the broadest form plus from its applications and its connections to other topics of Mathematical Sciences. A sample list of connected mathematical areas with this publication includes but is not restricted to: Approximation Theory, Inequalities, Probability in Analysis, Wavelet Theory, Neural Networks, Fractional Analysis, Applied Functional Analysis and Applications, Signal Theory, Computational Real and Complex Analysis and Measure Theory, Sampling Theory, Semigroups of Operators, Positive Operators, ODEs, PDEs, Difference Equations, Rearrangements, Numerical Functional Analysis, Integral equations, Optimization Theory of all kinds, Operator Theory, Control Theory, Banach Spaces, Evolution Equations, Information Theory, Numerical Analysis, Stochastics, Applied Fourier Analysis, Matrix Theory, Mathematical Physics, Mathematical Geophysics, Fluid Dynamics, Quantum Theory. Interpolation in all forms, Computer Aided Geometric Design, Algorithms, Fuzzyness, Learning Theory, Splines, Mathematical Biology, Nonlinear Functional Analysis, Variational Inequalities, Nonlinear Ergodic Theory, Functional Equations, Function Spaces, Harmonic Analysis, Extrapolation Theory, Fourier Analysis, Inverse Problems, Operator Equations, Image Processing, Nonlinear Operators, Stochastic Processes, Mathematical Finance and Economics, Special Functions, Quadrature, Orthogonal Polynomials, Asymptotics, Symbolic and Umbral Calculus, Integral and Discrete Transforms, Chaos and Bifurcation, Nonlinear Dynamics, Solid Mechanics, Functional Calculus, Chebyshev Systems. Also are included combinations of the above topics. Working with Applied Functional Analysis Methods has become a main trend in recent years, so we can understand better and deeper and solve important problems of our real and scientific world. JAFA is a peer-reviewed International Quarterly Journal published by Eudoxus Press,LLC. We are calling for high quality papers for possible publication. The contributor should submit the contribution to the EDITOR in CHIEF in TEX or LATEX double spaced and ten point type size, also in PDF format. Article should be sent ONLY by E-MAIL [See: Instructions to Contributors] Journal of Applied Functional Analysis(JAFA)

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Journal of Applied Functional Analysis Editorial Board Associate Editors Editor in-Chief: George A.Anastassiou Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA 901-678-3144 office 901-678-2482 secretary 901-751-3553 home 901-678-2480 Fax [email protected] Approximation Theory,Inequalities,Probability, Wavelet,Neural Networks,Fractional Calculus

23) Nikolaos B.Karayiannis Department of Electrical and Computer Engineering N308 Engineering Building 1 University of Houston Houston,Texas 77204-4005 USA Tel (713) 743-4436 Fax (713) 743-4444 [email protected] [email protected] Neural Network Models, Learning Neuro-Fuzzy Systems.

Associate Editors:

24) Theodore Kilgore Department of Mathematics Auburn University 221 Parker Hall, Auburn University Alabama 36849,USA Tel (334) 844-4620 Fax (334) 844-6555 [email protected] Real Analysis,Approximation Theory, Computational Algorithms.

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Tel.(+33)(0)3.20.43.42.18 Fax (+33)(0)3.20.43.43.02 [email protected] Approximation Theory, Functional Analysis, Operator Theory. 4) Erik J.Balder Mathematical Institute Universiteit Utrecht P.O.Box 80 010 3508 TA UTRECHT The Netherlands Tel.+31 30 2531458 Fax+31 30 2518394 [email protected] Control Theory, Optimization, Convex Analysis, Measure Theory, Applications to Mathematical Economics and Decision Theory. 5) Carlo Bardaro Dipartimento di Matematica e Informatica Universita di Perugia Via Vanvitelli 1 06123 Perugia, ITALY TEL+390755853822 +390755855034 FAX+390755855024 E-mail [email protected] Web site: http://www.unipg.it/~bardaro/ Functional Analysis and Approximation Theory, Signal Analysis, Measure Theory, Real Analysis. 6) Heinrich Begehr Freie Universitaet Berlin I. Mathematisches Institut, FU Berlin, Arnimallee 3,D 14195 Berlin Germany, Tel. +49-30-83875436, office +49-30-83875374, Secretary Fax +49-30-83875403 [email protected] Complex and Functional Analytic Methods in PDEs, Complex Analysis, History of Mathematics. 7) Fernando Bombal Departamento de Analisis Matematico Universidad Complutense Plaza de Ciencias,3 28040 Madrid, SPAIN Tel. +34 91 394 5020 Fax +34 91 394 4726 [email protected]

Mathematical Institute Academy of Sciences of Czech Republic Zitna 25 CZ-115 67 Praha 1 Czech Republic Tel +420 222 090 743 Fax +420 222 211 638 [email protected] Function spaces,Real Analysis,Harmonic Analysis,Interpolation and Extrapolation Theory,Fourier Analysis. 28) V. Lakshmikantham Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901 e-mail: [email protected] Ordinary and Partial Differential Equations, Hybrid Systems, Nonlinear Analysis 29) Peter M.Maass Center for Industrial Mathematics Universitaet Bremen Bibliotheksstr.1, MZH 2250, 28359 Bremen Germany Tel +49 421 218 9497 Fax +49 421 218 9562 [email protected] Inverse problems,Wavelet Analysis and Operator Equations,Signal and Image Processing. 30) Julian Musielak Faculty of Mathematics and Computer Science Adam Mickiewicz University Ul.Umultowska 87 61-614 Poznan Poland Tel (48-61) 829 54 71 Fax (48-61) 829 53 15 [email protected] Functional Analysis, Function Spaces, Approximation Theory,Nonlinear Operators. 31) Gaston M. N'Guerekata Department of Mathematics Morgan State University Baltimore, MD 21251, USA tel:: 1-443-885-4373 Fax 1-443-885-8216 Gaston.N'[email protected] Nonlinear Evolution Equations, Abstract Harmonic Analysis, Fractional Differential Equations,

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Operators on Banach spaces, Tensor products of Banach spaces, Polymeasures, Function spaces. 8) Michele Campiti Department of Mathematics "E.De Giorgi" University of Lecce P.O. Box 193 Lecce,ITALY Tel. +39 0832 297 432 Fax +39 0832 297 594 [email protected] Approximation Theory, Semigroup Theory, Evolution problems, Differential Operators. 9)Domenico Candeloro Dipartimento di Matematica e Informatica Universita degli Studi di Perugia Via Vanvitelli 1 06123 Perugia ITALY Tel +39(0)75 5855038 +39(0)75 5853822, +39(0)744 492936 Fax +39(0)75 5855024 [email protected] Functional Analysis, Function spaces, Measure and Integration Theory in Riesz spaces. 10) Pietro Cerone School of Computer Science and Mathematics, Faculty of Science, Engineering and Technology, Victoria University P.O.14428,MCMC Melbourne,VIC 8001,AUSTRALIA Tel +613 9688 4689 Fax +613 9688 4050 [email protected] Approximations, Inequalities, Measure/Information Theory, Numerical Analysis, Special Functions. 11)Michael Maurice Dodson Department of Mathematics University of York, York YO10 5DD, UK Tel +44 1904 433098 Fax +44 1904 433071 [email protected] Harmonic Analysis and Applications to Signal Theory,Number Theory and Dynamical Systems.

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32) Vassilis Papanicolaou Department of Mathematics National Technical University of Athens Zografou campus, 157 80 Athens, Greece tel:: +30(210) 772 1722 Fax +30(210) 772 1775 [email protected] Partial Differential Equations, Probability. 33) Pier Luigi Papini Dipartimento di Matematica Piazza di Porta S.Donato 5 40126 Bologna ITALY Fax +39(0)51 582528 [email protected] Functional Analysis, Banach spaces, Approximation Theory. 34) Svetlozar T.Rachev Chair of Econometrics,Statistics and Mathematical Finance School of Economics and Business Engineering University of Karlsruhe Kollegium am Schloss, Bau II,20.12, R210 Postfach 6980, D-76128, Karlsruhe,GERMANY. Tel +49-721-608-7535, +49-721-608-2042(s) Fax +49-721-608-3811 [email protected] Second Affiliation: Dept.of Statistics and Applied Probability University of California at Santa Barbara [email protected] Probability,Stochastic Processes and Statistics,Financial Mathematics, Mathematical Economics. 35) Paolo Emilio Ricci Department of Mathematics Rome University "La Sapienza" P.le A.Moro,2-00185 Rome,ITALY Tel ++3906-49913201 office ++3906-87136448 home Fax ++3906-44701007 [email protected] [email protected] Special Functions, Integral and Discrete

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12) Sever S.Dragomir School of Computer Science and Mathematics, Victoria University, PO Box 14428, Melbourne City, MC 8001,AUSTRALIA Tel. +61 3 9688 4437 Fax +61 3 9688 4050 [email protected] Inequalities,Functional Analysis, Numerical Analysis, Approximations, Information Theory, Stochastics. 13) Paulo J.S.G.Ferreira Department of Electronica e Telecomunicacoes/IEETA Universidade de Aveiro 3810-193 Aveiro PORTUGAL Tel +351-234-370-503 Fax +351-234-370-545 [email protected] Sampling and Signal Theory, Approximations, Applied Fourier Analysis, Wavelet, Matrix Theory. 14) Gisele Ruiz Goldstein Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA. Tel 901-678-2513 Fax 901-678-2480 [email protected] PDEs, Mathematical Physics, Mathematical Geophysics. 15) Jerome A.Goldstein Department of Mathematical Sciences The University of Memphis Memphis,TN 38152,USA Tel 901-678-2484 Fax 901-678-2480 [email protected] PDEs,Semigroups of Operators, Fluid Dynamics,Quantum Theory. 16) Heiner Gonska Institute of Mathematics University of Duisburg-Essen Lotharstrasse 65 D-47048 Duisburg Germany Tel +49 203 379 3542 Fax +49 203 379 1845 [email protected] Approximation and Interpolation Theory,

Transforms, Symbolic and Umbral Calculus, ODE, PDE,Asymptotics, Quadrature, Matrix Analysis. 36) Silvia Romanelli Dipartimento di Matematica Universita' di Bari Via E.Orabona 4 70125 Bari, ITALY. Tel (INT 0039)-080-544-2668 office 080-524-4476 home 340-6644186 mobile Fax -080-596-3612 Dept. [email protected] PDEs and Applications to Biology and Finance, Semigroups of Operators. 37) Boris Shekhtman Department of Mathematics University of South Florida Tampa, FL 33620,USA Tel 813-974-9710 [email protected] Approximation Theory, Banach spaces, Classical Analysis. 38) Rudolf Stens Lehrstuhl A fur Mathematik RWTH Aachen 52056 Aachen Germany Tel ++49 241 8094532 Fax ++49 241 8092212 [email protected] Approximation Theory, Fourier Analysis, Harmonic Analysis, Sampling Theory. 39) Juan J.Trujillo University of La Laguna Departamento de Analisis Matematico C/Astr.Fco.Sanchez s/n 38271.LaLaguna.Tenerife. SPAIN Tel/Fax 34-922-318209 [email protected] Fractional: Differential EquationsOperatorsFourier Transforms, Special functions, Approximations,and Applications. 40) Tamaz Vashakmadze I.Vekua Institute of Applied Mathematics Tbilisi State University, 2 University St. , 380043,Tbilisi, 43, GEORGIA. Tel (+99532) 30 30 40 office

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,13-25 ,COPYRIGHT 2012 EUDOXUS PRESS,LLC

MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS A. SAN ANTOL´IN AND R. A. ZALIK Abstract. We introduce the notions of matrix–valued wavelet set and matrix– valued multiresolution analysis (A-MMRA) associated with a fixed dilation given by an expansive linear map A : Rd → Rd , d ≥ 1 such that A(Zd ) ⊂ Zd , in a matrix–valued function space L2 (Rd , Cn×n ), n ≥ 1. These are generalizations of the corresponding notions defined by Xia and Suter in 1996 for the case where d = 1 and A is the dyadic dilation. We show several properties of orthonormal sequences of translates by integers of matrix–valued functions, focusing on those related to the structure of A-MMRA’s and their connection with matrix–valued wavelet sets. Further, we present a strategy for constructing matrix–valued wavelet sets from a given A-MMRA and, in addition, we characterize those matrix–valued wavelet sets which may be built from an A-MMRA.

1. Introduction Given a fixed expansive linear map A : Rd → Rd , d ≥ 1, such that A(Zd ) ⊂ Zd , we introduce the notion of matrix–valued wavelet and matrix–valued multiresoltuion analysis associated to A in a matrix–valued function space L2 (Rd , Cn×n ), n ≥ 1. A linear map A is said to be expansive if all (complex) eigenvalues of A have modulus greater than 1. The subject of this paper is the study of such wavelets and multiresolution analyses. Our starting point is the paper by Xia and Suter [22] where the notion of matrix–valued wavelet and matrix–valued multiresolution analysis have been introduced and studied for the case of d = 1 and dyadic dilations. Subsequently, and in this particular context, there appeared several papers related to matrix–valued multiresolution analyses and matrix–valued wavelets and their construction, e.g. [25], [1], [23], [28]. The notion of matrix–valued multiresolution analysis and matrix–valued wavelets when d = 1 and A may be any arbitrary integer dilation were introduced in [6], where a necessary and sufficient condition for the existence of matrix–valued wavelets and an algorithm for constructing compactly supported matrix–valued wavelets associated with an integer dilation factor m are presented. For the case m = 4 see [4]. Relaxing requirements, the articles [21], [5], [11], [8] study biorthogonal matrix– valued wavelets where d = 1 and A is the dyadic dilation. Since matrix–valued function spaces L2 (Rd , Cn×n ) are related to video imaging, we generalize results in [27] to these spaces with the purpose of showing that the ideas developed there for scalar–valued wavelets and multiresolution analysis fit perfectly in this context. That is our motivation for writing this article. Key words and phrases. matrix–valued function spaces, Fourier transform, multiresolution analysis, wavelet set. 2010 Mathematics Subject Classification: 42C40. 1

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A. SAN ANTOL´IN AND R. A. ZALIK

This work is organized as follows. In Section 2 we present the definitions and notation that will be used. Section 3 contains several properties of orthonormal sequences of integer translates of a function in a matrix–valued space L2 (Rd , Cn×n ), focusing on those related to multiresolution analyses and their connection with wavelet sets. Section 4 is devoted to the study of matrix–valued wavelet sets in a signal space L2 (Rd , Cn×n ), associated with a dilation given by an expansive linear map A. In addition, as a method for constructing these matrix–valued wavelet sets we introduce the notion of vector–valued multiresolution analysis associated with an expansive linear map A (A-MMRA). Futher, we study the structure of A-MMRA’s, present a strategy for constructing matrix–valued wavelet sets and characterize those sets constructed from a given A-MMRA. Our results are given in the context of Fourier space.

2. Notation and basic definitions The sets of integers, real and complex numbers will be denoted by Z, R and C respectively. The d–fold product of the interval [0, 1) with itself will be denoted Td . Thus Td := Rd /Zd , d ≥ 1. Unless otherwise indicated, In , n ≥ 1, will denote the n × n identity matrix and 0n will denote the n × n null matrix. Given an n × n, n ≥ 1, complex matrix M , aml ∈ C will denote the element on the m-th row and the l-th column of M . The complex vector space of all n × n complex matrices M will be denoted by Mn (C). Recall that a matrix M ∈ Mn (C) is said to be unitary if M M ∗ = In where M ∗ is the transpose of the complex conjugate of M . Let l2 (N, Cn×n ) := {M = {Mk }k∈N ⊂ Mn (C) : kMk = (

n X X

2

|aml (k)| )1/2 < ∞}.

m,l=1 k∈N

The space l2 (Zd , Cn×n ) is similarly defined. All functions considered in this paper will be assumed to be measurable. Given d, n ≥ 1, by L2 (Rd , Cn×n ) we will denote the space   f11 (x) f12 (x) · · · f1n (x)  f21 (x) f22 (x) · · · f2n (x)   : fml ∈ L2 (Rd ), m, l = 1, ..., n}. {f (x) =    ··· fn1 (x) fn2 (x) · · · fnn (x) We will also write f (x) = (fml (x))m,l=1,...,n . The spaces Lp (E, Cn×n ), 1 ≤ p < ∞, where E is a measurable set in Rd are defined similarly by replacing Rd and 2 by the E and p respectively. If we write f ∈ L2 (Td , Cn×n ) we will also mean that f is defined on the whole space Rd as a Zd -periodic matrix–valued function. Given f ∈ L2 (Rd , Cn×n ), kf k, will denote the Frobenius norm defined by (see [22]) (1)

kfk := (

n Z X

m,l=1

Rd

|fml (x)|2 dx)1/2 .

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MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

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R The integral of a matrix–valued function f , Rd f (x)dx, is defined by Z Z f (x)dx := ( fml (x)dx)m,l=1,...,n . Rd

Rd

The Fourier transform of a matrix–valued function f will be denoted by fb. For f ∈ L1 (Rd , Cn×n ) ∩ L2 (Rd , Cn×n ) Z b f (t) := f (x)e−2πix·t dx. Rd

For two matrix-valued functions f , g ∈ L2 (Rd , Cn×n ), Z (2) hf , gi := f (x)g∗ (x)dx Rd

and [f , g](t) :=

X

f (t + k)g∗ (t + k).

k∈Zn

Note that h·, ·i is matrix–valued and therefore it is not an inner product. It has the following properties: (a) For every f , g ∈ L2 (Rd , Cn×n ), hf , gi = hg, f i∗ ; (b) For every f , g, h ∈ L2 (Rd , Cn×n ) and every M1 , M2 ∈ Mn (C), hM1 f + M2 h, gi = M1 hf , gi + M2 hh, gi . Moreover, the scalar Plancherel formula implies that also in the matrix–valued case D E b . f, g hf , gi = b It is also readily seen that

kfk = (trace hf , f i)1/2 . Given an invertible map M : Rn → Rn , for every j ∈ Z and k ∈ Zd the dilation d n×n 2 operator DM ) by j and the translation operator Tk are defined on L (R , C j/2

j DM j f (t) := dM f (M t)

and Tk f (t) := f (t + k),

where dM = | det M |. A set S ⊂ L2 (Rd , Cn×n ) is called shift-invariant if f ∈ S implies that Tk f ∈ S for every k ∈ Zn . Let F ⊂ L2 (Rd , Cn×n ), then T(F) := {Tk f : f ∈ F, k ∈ Zn } 2

d

and S(F) := spanT(F),

n×n

where the closure is in L (R , C ). If F = {f1 , ..., fm } then S(F) is called a finitely generated shift-invariant space or FSI and the functions fl , l = 1, ..., m are called the generators of S(F). In this case we will also use the symbols T(f1 , ..., fm ) and S(f1 , ..., fm ) to denote T(F) and S(F) respectively. If F contains a single element, then S(F) is called a principal shift-invariant space or PSI. Two functions f , g ∈ L2 (Rd , Cn×n ) are said to be orthogonal if hf , gi = 0n . Further, let V, W be two closed subspaces in L2 (Rd , Cn×n ) such that W ⊂ V , then the orthogonal complement of W in V is the closed subspace defined by W ⊥ = {g ∈ V : hg, f i = 0n

15

∀f ∈ W }.

A. SAN ANTOL´IN AND R. A. ZALIK

4

d n×n 2 A sequence {fk }∞ ) is called an orthonormal set in L2 (Rd , Cn×n ) k=1 ⊂ L (R , C if  In if k = l (3) hfk , fl i = 0n if k 6= l. d n×n 2 Given a closed subspace S in L2 (Rd , Cn×n ), a sequence {fk }∞ ) k=1 ⊂ L (R , C is called an orthonormal basis for S if it satisfies (3), and moreover, for any g ∈ S n×n 2 there exists an unique sequence of constant matrices {Hk }∞ ) such k=1 ∈ l (N, C that ∞ X g(x) = Hk fk (x), for x ∈ Rd k=1

where, for each x, Hk fk (x) is the product of the n × n matrices Hk and fk (x), and the convergence for the infinite sum is in the sense of the norm k·k defined by (1). It readily follows that for every k = 1, 2, . . . , (4)

Hk = hg, fk i ,

k{Hk }∞ k=1 k = kgk .

and

Given a set of matrix–valued functions F = {f1 , ..., fm } in L2 (Rd , Cn×n ), its Gramian matrix will be denoted by G[f1 , ..., fm ](t) or GF (t) and defined as follows: m GF (t) := ([b fl , b fj ](t)) . l,j=1

3. Orthonormal bases of translates

In this section we show several properties on orthonormal sequences of integral translates of functions in a matrix–valued function space L2 (Rd , Cn×n ). We focus on those properties closely related to matrix–valued wavelets and matrix–valued multiresoltion analyses, concepts that will be discussed in the next section. Most of the properties presented here are well known in the scalar–valued function space context (cf. e.g. [27]). The following lemma generalizes a result in [22]. Lemma 1. Let F = {f1 , ..., fm } ⊂ L2 (Rd , Cn×n ). Then T(F) is an orthonormal sequence in L2 (Rd , Cn×n ) if and only if GF (t) = Inm a.e. Proof. Let us prove the necessity. By the orthonormality of T(F), given j, p ∈ {1, ..., m} and k ∈ Zd we have Z (5) fj (x)fp∗ (x − k)dx = δ(j, p)δ(k, 0)In , Rd

where δ(α, β) = 1 if α = β and δ(α, β) = 1 if α 6= β. By Plancherel’s formula, Z ∗ δ(j, p)δ(k, 0)In = fbj (t)fbp (t)e2πik dt Rd XZ ∗ fbj (t)fbp (t)e2πik dt = k∈Zd

(6)

=

Z

[−1/2,1/2]d +k

[−1/2,1/2]d

X

k∈Z

d

[b fj , b fp ](t)e2πik dt,

∀k ∈ Zd .

This implies that [b fj , b fp ](t) = δ(j, p)In a.e. on Rd , whence the assertion follows. Conversely, note that the orthonormality of T(F) follows immediately from GF (t) = Inm a.e., (5) and (6). 

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MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

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Lemma 2. Let F = {f1 , ..., fm } ⊂ L2 (Rd , Cn×n ) and assume that T(F) is an orthonormal basis of a closed subspace S ⊂ L2 (Rd , Cn×n ). Then, a matrix–valued function g ∈ L2 (Rd , Cn×n ) belongs to S if and only if there are Zd -periodic functions Hj ∈ L2 (Td , Cn×n ), j = 1, ..., m, such that b(t) = g

(7) and

m X j=1

Hj (t)fbj (t)

k g k2 =

(8)

m X

a.e. on Rd ,

k Hj k 2

j=1

Proof. Suppose that g ∈ S, then we may represent it in terms of the orthonormal basis T(F) as (9)

g(x) =

m X X

Hj,k fj (x − k),

j=1 k∈Zd

where {Hj,k }k∈Zd ∈ l2 (Zd , Cn×n ), j ∈ {1, ..., m}, and the convergence of the sum is in the sense of the norm k·k defined P by (1). Thus, taking the Fourier transform in (9) we obtain (7) with Hj (t) = k∈Zd Hj,k e−2πik·t . From (9) and (4) we deduce that k g k=k {Hj,k }j=1,...,m,k∈Zd k. Since k Hj k= k {Hj,k }k∈Zd k, equation (8) follows. P −2πik·t Conversely, assume that (7) holds. Since Hj (t) = with k∈Zd Hj,k e n×n 2 d Hj,k ∈ l (Z , C ), we deduce that (9) is satisfied in the sense of convergence in norm, and therefore g ∈ S.  Lemma 3. Let F = {f1 , ..., fm } and G = {g1 , ..., gp } be in L2 (Rd , Cn×n ). Assume that T(G) and T(F) are orthonormal sequences in L2 (Rd , Cn×n ), and that there are Zd -periodic functions Hl,j ∈ L2 (Td , Cn×n ), j = 1, ..., m, l = 1, ..., p, such that gbl (t) =

(10) Then (11)

m X

m X j=1

Hl,j (t)fbj (t)

a.e. on Rd

a.e. on Rd

Hl,j (t)H∗r,j (t) = In δ(l, r)

l = 1, ..., p.

l, r ∈ {1, ..., p}.

j=1

Proof. Since both sequences are orthonormal, given l, r ∈ {1, ..., p}, (3) yields In δ(l, r) = =

cr ](t) = [ [gbl , g

m X m X j=1 q=1

m X j=1

Hl,j fbj ,

m X j=1

Hr,j fbj ](t)

Hl,j (t)[fbj , fbq ](t)H∗r,q (t) =

m X

Hl,j (t)H∗r,j (t)

a.e. on Rd .

j=1

 We are now ready to prove

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A. SAN ANTOL´IN AND R. A. ZALIK

6

Proposition 1. Let p ≤ m and let S be a closed subspace of L2 (Rd , Cn×n ). Let F = {f1 , ..., fm } and G = {g1 , ..., gp } be such that T(F) and T(G) belong to S. If T(F) is an orthonormal basis for S, then T(G) is an orthonormal sequence in S np,nm if and only if there exists a matrix Q(t) := (hq,r (t))q,r=1 where hq,r ∈ L2 (Td ), which satisfies Q(t)Q∗ (t) = Inp a.e. t on Rd and also, T

(gb1 (t), ..., gbp (t)) = Q(t)(bf1 (t), ..., bfm (t))

(12)

T

a.e. on Rd .

The Zd -periodic matrix

np,nm

Q(t) = (hq,r (t))q,r=1

will be called a transition matrix from the sequence T(F) to the sequence T(G). Proof. To prove the necessity we proceed as follows: Since T(F) is an orthonormal basis for S and T(G) ⊂ S, Lemma 2 tells us that there are Hl,j ∈ L2 (Td , Cn×n ), j = 1, ..., m and l = 1, ..., p, such that m X (13) gbl (t) = Hl,j (t)fbj (t) a.e. on Rd l = 1, ..., p. j=1

p,m

Let Q(t) be the np × nm block matrix Q(t) := (Hl,j (t))l,j=1 , and for q = 1, . . . np, let vq (t) = (hq,1 (t), ..., hq,nm (t)), q = 1, ..., np, be the q-th row of Q(t). (Note that every hq,r belongs to L2 (Td )). Then, (11) implies that the vectors {vq (t) : q ∈ {1, ..., np}} are orthonormal a.e.(t). Thus, setting Q(t) := (hq,r (t))np,nm q,r=1 we conclude that Q(t)Q∗ (t) = Inp . Finally, note that (13) readily implies (12). To prove the sufficiency, for any l ∈ {1, ..., p} and j ∈ {1, ..., m} let Hl,j in ln,jn

L2 (Td , Cn×n ) be defined by Hl,j := (hq,r )q=(l−1)n+1,r=(j−1)n+1 . Then (12) yields (13). In addition, the assumption Q(t)Q∗ (t) = Inp a.e. on Rd implies that m X

a.e. on Rd

Hl,j (t)H∗b,j (t) = In δ(l, b)

l, b ∈ {1, ..., p}.

j=1

We complete the proof by showing that the Gramian associated to G is the unitary matrix a.e. on Rd and applying Lemma 1. For l ∈ {1, ..., p} and b ∈ {1, ..., m} we have: m m X X [gbl , gbb ](t) = [ Hl,j fbj , Hb,j fbj ](t) j=1

=

m X m X j=1 q=1

d

j=1

Hl,j (t)[fbj , fbq ](t)H∗b,q (t) =

a.e. on R , and the assertion follows.

m X

Hl,j (t)H∗b,j (t) = In δ(l, b)

j=1



Proposition 2. Assume that F = {f1 , ..., fm } and G = {g1 , ..., gp } are functions in L2 (Rd , Cn×n ). If T(F) and T(G) are orthonormal bases of the same closed subspace S ⊂ L2 (Rd , Cn×n ), then m = p. Proof. By the symmetry in the notation we may assume, without loss of generality, that p > m. Since T(F) is an orthonormal basis for S and T(G) ⊂ S, we infer from Proposition 1 that there exists an np × nm matrix Q(t) such that Q(t)Q∗ (t) = Inp a.e. This means that the np vectors defined by the rows of the matrix Q(t)

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MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

7

are orthonormal in the complex vector space Cnm . Since nm < np, we get a contradiction.  Proposition 3. Asssume that the matrix–valued functions F = {f1 , ..., fm }, G = {g1 , ..., gm } ⊂ L2 (Rd , Cn×n ) are such that T(F) and T(G) are orthonormal sequences in a closed subspace S ⊂ L2 (Rd , Cn×n ). If T(F) is an orthonormal basis for S and there exists a matrix nm Q(t) := (hl,j (t))l,j=1 , where hl,j ∈ L2 (Td ), such that Q(t) is unitary a.e. (t) on Rd and (12) holds, then T(G) is an orthonormal basis for S. Proof. According to Proposition 1, T(G) is an orthonormal sequence in S. Thus it suffices to show that S = spanT(G). The hypotheses imply that we only need to check that S ⊂ T(G). Let h ∈ S then, by Lemma 2 there exist Hj ∈ L2 (Td , Cn×n ), j = 1, ..., m, such that T b h(t) = (H1 (t), ..., Hm (t))(fb1 (t), ..., fc m (t))

a.e. on Rd .

Thus, by (12)

b h(t)

=

(H1 (t), ..., Hm (t))Q∗ (t)(gc1 (t), ..., gc m (t))

T

T

(L1 (t), ..., Lm (t))(gc1 (t), ..., gcm (t)) a.e. on Rd , where Lj (t) = (v(j−1)n+1 (t), ..., vjn (t)) is the n × nm matrix such that vl is the l-th column vector of the matrix (H1 (t), ..., Hm (t))Q∗ (t). Observe that for every =

j ∈ {1, ..., m} the entries of the matrix Lj are Zd -periodic functions. Applying the Minkowski and H¨ older inequalities, we conclude that Lj ∈ L2 (Td , Cn×n ), and the conclusion follows by another application of Lemma 2.  A straightforward consequence of the preceding propositions is the following. Corollary 1. Let F = {f1 , ..., fm }, G = {g1 , ..., gm } ⊂ L2 (Rd , Cn×n ) such that T(F) and T(G) are orthonormal sequences in a closed subspace S ⊂ L2 (Rd , Cn×n ). If T(F) is an orthonormal basis for S, then T(G) is an orthonormal basis for S. nm

Proof. By Proposition 1, there exists a matrix Q(t) := (hq,r (t))q,r=1 where hq,r ∈ L2 (Td ), which satisfies Q(t)Q∗ (t) = Inp a.e. (t) on Rd and also (12) holds. Thus, the proof is finished by Proposition 3.  4. Wavelets and Multiresolution analysis In what follows we will assume that A is an expansive linear map A : Rd → Rd such that A(Zd ) ⊂ Zd . Here and further we use the same notation for a linear map on Rd and its matrix with respect to the canonical base. In this section we introduce the notions of matrix–valued wavelet set and matrix– valued multiresolution analysis (A-MMRA) associated with a dilation given by a fixed map A as above in a signal space L2 (Rd , Cn×n ), d, n ≥ 1. These definitions generalize the matrix–valued wavelet and matrix–valued multiresolution analysis notions defined in [22] when d = 1 and A is the dyadic dilation. We study the structure of an A-MMRA, present a strategy to construct matrix–valued wavelet sets associated with a fixed dilation A and characterize the matrix–valued wavelet sets constructed from a given A-MMRA.

19

8

A. SAN ANTOL´IN AND R. A. ZALIK

Given an expansive linear map A, a matrix-valued wavelet set associated with A is a finite set of functions {Ψ1 , ..., Ψs } ⊂ L2 (Rd , Cn×n ) such that the system {DjA Tk Ψr : r = 1, 2..., s, j ∈ Z, k ∈ Zd }, is an orthonomal basis for L2 (Rd , Cn×n ). A general method for constructing matrix–valued wavelet sets on L2 (Rd , Cn×n ) is related to the concept of matrix–valued multiresolution analysis associated with A (A-MMRA): Given an expansive linear map A as above, we define an A-MMRA as a sequence of closed subspaces Vj , j ∈ Z, of L2 (Rd , Cn×n ) that satisfies the following conditions: (i) For every j ∈ Z, Vj ⊂ Vj+1 ; (ii) For every j ∈ Z, f (x) ∈ Vj if and only if f (Ax) ∈ Vj+1 ; (iii) ∪j∈Z Vj = L2 (Rd , Cn×n ); (iv) There exists a function Φ ∈ V0 , called a scaling function, such that { Tk Φ(x) : k ∈ Zn } is an orthonormal basis for V0 . To construct a matrix–valued wavelet set associated with a dilation map A from an A-MMRA with scaling function Φ, we denote by Wj the orthogonal complement of Vj in Vj+1 . Thus, by condition (i), we have Vj+1 = Wj ⊕ Vj . Moreover, condition (iii) implies that L2 (Rd , Cn×n ) = ⊕j∈Z Wj . Observe that by condition (ii) we have (14)

∀j ∈ Z,

f (·) ∈ W0 ⇔ f (Aj ·) ∈ Wj .

Thus, to find a matrix–valued wavelet set from an A-MMRA, it will suffice to construct a set of functions {Ψ1 , ..., Ψs } ⊂ L2 (Rd , Cn×n ) such that the system {Tk Ψr : r = 1, 2..., s, k ∈ Zd }, is an orthonomal basis for W0 , for then {DjA Tk Ψr : r = 1, 2..., s, k ∈ Zd }, is an orthonormal basis of Wj . We now focus on how to construct orthonormal bases of integer translates for the subspaces V1 and W0 . For this purpose we study the structure of the subspaces Vj and Wj . Let us recall that if A : Rd → Rd is an expansive linear map such that A(Zd ) ⊂ Zd , then the quotient group Zd /A(Zd ) is well defined. We will denote by ∆A ⊂ Zd a full collection of representatives of the cosets of Zd /A(Zd ). There are exactly dA A −1 cosets (see [10] and [24, p. 109]). Let ∆A = {qi }di=0 where q0 = 0. Note that, if l ∈ {0, 1, 2, .....}, then l = adA + i, where a ∈ {0, 1, 2, .....} and i ∈ {0, 1, ...dA − 1}. We have: Theorem 1. Let A : Rd → Rd be an expansive linear map such that A(Zd ) ⊂ Zd . Let F = {f0 , ..., fm−1 } be a set of functions in L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of a closed subspace V of L2 (Rd , Cn×n ), and let U = {f ∈ L2 (Rd , Cn×n ) : f (A−1 ·) ∈ V }. If 1/2

gl := dA fa (Ax + qi ),

l ∈ {0, ..., mdA − 1}

20

MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

9

then T(g0 , ..., gmdA −1 ) is an orthonormal basis of U . Moreover, any set of functions G in L2 (Rd , Cn×n ) such that T(G) is an orthonormal basis of U has exactly mdA functions. Proof. Since T(F) is an orthonormal sequence, a trivial change of variables shows A −1 that T(g0 , ..., gmdA −1 ) is an orthogonal sequence. Further, since ∆A = {qi }di=0 is d d a full collection of representatives of the cosets of Z /A(Z ), given a ∈ {0, ..., m− 1} and k ∈ Zd we have that there exist unique l ∈ {0, ..., m − 1} and r ∈ Zd such that DA Tk fa = Tr gl . Thus T(g0 , ..., gmdA −1 ) is an orthonormal basis of U . Since the set {g0 , ..., gmdA −1 } has exactly mdA functions, Proposition 2 implies that every other set of functions G in L2 (Rd , Cn×n ) such that T(G) is an orthonormal basis of U has exactly mdA functions.  Theorem 1 yields Theorem 2. Let Φ ∈ L2 (Rd , Cn×n ) be a scaling function in an A-MMRA, {Vj : j ∈ Z}. If (15)

1/2

Θi := dA Φ(Ax + qi ),

i = 0, 1, ..., dA − 1,

then T(Θ0 , ..., ΘdA −1 ) is an orthonormal basis of V1 . Using Theorem 1 we can deduce some properties of the subspaces Vj . We have the following. Theorem 3. Let {Vj : j ∈ Z} be an A-MMRA. Then (a) If j > 0, then there exists a finite set F ⊂ L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of Vj . (b) If j ≥ 0, then any set F ⊂ L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of Vj has exactly djA functions. (c) If j < 0, then there is no set of functions F ⊂ L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of Vj . (d) If j 6= 0, then there is no function f ∈ L2 (Rd , Cn×n ) such that T(f ) is an orthonormal basis of Vj . Proof. To prove (a), let Φ be a scaling function in the A-MMRA. According to Theorem 2, there exists a set of exactly dA functions, F1 , such that T(F1 ) is an orthonormal basis of V1 . Thus for j ≥ 0 the existence of a set of exactly djA functions, Fj , such that T(Fj ) is an orthonormal basis of Vj follows by repeated application of Theorem 1. From (a), for j ≥ 0 the set Fj has exactly djA functions; thus (b) follows from Proposition 2. We now prove (c). Let m := d−j A . By repeated application of Theorem 1 we conclude that there are functions f0 , . . . fm−1 such that T (f0 , . . . fm−1 ) is a basis of V0 . Since A is expansive, we know that dA > 1; thus m > 1, which is a contradiction of (b). Finally, if j < 0 (d) follows from (c), whereas if j > 0, (d) follows from (b).  The following two corollaries are immediate consequences of Theorem 3. Corollary 2. Let {Vj : j ∈ Z} be an A-MMRA, let s > 0 and Uj := Vj+s . Then {Uj : j ∈ Z} is a sequence of closed subspaces in L2 (Rd , Cn×n ) satisfying the conditions (i), (ii), (iii) in the definition of A-MMRA, and also, there exists a set

21

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A. SAN ANTOL´IN AND R. A. ZALIK

of functions F ⊂ L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of U0 and it has exactly dsA functions. Corollary 3. Let {Vj : j ∈ Z} be an A-MMRA. Then Vj is a proper subset of Vj+1 for every j ∈ Z. Proof. Assume that there is j ∈ Z such that Vj = Vj+1 , then by the definition of A-MMRA we have Vj = Vj+s for every s ∈ Z. Thus, in particular V0 = V1 and which is impossible by the condition (b) in Theorem 3.  We have the following characterization of matrix–valued wavelet sets constructed from an A-MMRA: Theorem 4. Let Φ ∈ L2 (Rd , Cn×n ) be a scaling function in an A-MMRA, {Vj : j ∈ Z}, and let Θ0 , ..., ΘdA −1 ∈ L2 (Rd , Cn×n ) be such that T(Θ0 , ...ΘdA −1 ) is an orthonormal basis of V1 . The following propositions are equivalent: (a) {Ψ1 , ..., ΨdA −1 } is a matrix–valued wavelet set constructed from the given A-MMRA. (b) There is an n dA × n dA matrix Q(t) of Zd -periodic functions and unitary a.e. on Rd such that b b 1 (t), ..., Ψ b dA −1 (t))T := Q(t)(Θ b 0 (t), Θ b 1 (t), ..., Θ b dA −1 (t))T a.e. on Rd . (Φ(t), Ψ

Proof. Let us prove (a) ⇒ (b). The condition (a) means that T(Ψ1 , ..., ΨdA −1 ) is an orthonormal basis of W0 where W0 is defined as the orthogonal complement of V0 in V1 . Further, since T(Φ) is an orthonormal basis of V0 then T(Φ, Ψ1 , ..., ΨdA −1 ) is an orthonormal basis of V1 . Thus the conditions (b) follows from Proposition 1. We now prove (b) ⇒ (a). According to Proposition 1, we know that T(Φ, Ψ1 , ..., ΨdA −1 ) is an orthonormal sequence in V1 , and further, by Proposition 3, we know that T(Φ, Ψ1 , ..., ΨdA −1 ) is an orthonormal basis of V1 . Thus, since T(Φ) is an orthonormal basis of V0 and V1 = W0 ⊕ V0 then T(Ψ1 , ..., ΨdA −1 ) is an orthonormal basis of W0 . Thus, we conclude that {Ψ1 , ..., ΨdA −1 } is a matrix–valued wavelet set constructed from the A-MMRA.  We now proceed to describe a strategy for constructing a matrix–valued wavelet set associated to a dilation A from a given A-MMRA with a scaling function Φ. According to Theorem 2 the functions Θ0 , . . . , ΘdA −1 defined by (15) are such that T(Θ0 , . . . , ΘdA −1 ) is an orthonormal basis of V1 . Furthermore, since Φ ∈ V0 ⊂ V1 , Lemma 2 implies that there are Zd -periodic matrix–valued functions Hl ∈ L2 (Td , Cn×n ), l = 0, ..., dA − 1, such that b Φ(t) =

dX A −1 l=0

Moreover, Lemma 3 implies that (16)

dX A −1

cl (t) Hl (t)Θ

Hl (t)H∗l (t) = In

a.e. on Rd .

a.e. on Rd ,

l=0

If we denote by J0 the n × ndA matrix of functions defined by J0 (t) = (H0 (t), ..., HdA −1 (t))

22

MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

11

and by vq (t), q = 1, ..., n the vector in the complex vector space CndA defined by the value at t of the q-th row in the matrix J0 (t), the equality (16) implies that the vectors {vq (t) : q = 1, ..., n} are a.e. orthonormal. Note that it is possible to construct a ndA × ndA matrix Q(t) of Zd -periodic functions, a.e. unitary in Rd , such that for q = 1, . . . , n the q-th row is given by the function vector vq (t). The construction of such a matrix can be done by the Gram–Schmidt orthogonalization process. If H1 (t) is symmetric, another method for the completion of a unitary matrix is given in [17]. Finally, if Ψ1 , ..., ΨdA −1 ∈ L2 (Rd , Cn×n ) is defined by T

b b 1 (t), · · · , Ψ b dA −1 (t)) = Q(t)(Θ b 0 (t), Θ b 1 (t), . . . , Θ b dA −1 (t)) (Φ(t), Ψ

T

a.e. on Rd ,

and applying Theorem 4, we conclude that {Ψ1 , ..., ΨdA −1 } is a matrix–valued wavelet set constructed from the given A-MMRA. We have therefore proved the following. Theorem 5. Given an expansive linear map A : Rd → Rd such that A(Zd ) ⊂ Zd and given an A-MMRA, then there exists a set of matrix–valued functions {Ψ1 , ..., ΨdA −1 } in L2 (Rd , Cn×n ) which is a matrix–valued wavelet set constructed from such an A-MMRA. Recalling that a set of matrix–valued functions F ⊂ L2 (Rd , Cn×n ) is a matrix– valued wavelet set constructed from an A-MMRA, {Vj : j ∈ Z}, if and only if T(F) is an orthonormal basis of the subspace W0 defined as the orthogonal complement of V0 in V1 , then the following is a corollary of Theorem 5 and Proposition 2. Corollary 4. Let {Vj : j ∈ Z} be an A-MMRA and let W0 denote the orthogonal complement of V0 in V1 . Then there exists a set of matrix–valued functions F ⊂ L2 (Rd , Cn×n ) such that T(F) is an orthonormal basis of W0 , and any set of matrix– valued functions G in L2 (Rd , Cn×n ) such that T(G) is an orthonormal basis of W0 has exactly dA − 1 matrix–valued functions. From Corollary 4, (14), and Theorem 1, we obtain the following. Corollary 5. Let {Vj : j ∈ Z} be an A-MMRA and let Wj denote the orthogonal complement of Vj in Vj+1 . Then, for every j ∈ {0, 1, 2, ...} there exists a set of matrix–valued functions Fj ⊂ L2 (Rd , Cn×n ) such that T(Fj ) is an orthonormal basis of Wj , and any set of functions Gj in L2 (Rd , Cn×n ) such that T(Gj ) is an orthonormal basis of Wj has exactly (dA − 1)djA matrix–valued functions. Let us continue with the study of the structure of subspaces Vj and Wj . Theorem 6. Let {Vj : j ∈ Z} be an A-MMRA and let F = {f1 , ..., fdA −1 } be a set of matrix–valued functions in L2 (Rd , Cn×n ). If there exists an integer l < 0 such that F ⊂ Vl , then F cannot be a matrix–valued wavelet set. Proof. If F is a matrix–valued wavelet set then T(F) is an orthonormal sequence in Vl . Thus, applying Theorem 1 with the expansive linear map Al , we see that there exist a set of (dA − 1)dlA matrix–valued functions G in L2 (Rd , Cn×n ) such that T(G) is an orthonormal sequence in V0 . Moreover, according to the definition of V0 and Proposition 2, the number (dA − 1)dlA must be less or equal to 1. Since dA ≥ 2, we have a contradiction.  Theorem 7. Let {Vj : j ∈ Z} be an A-MMRA and let F = {f1 , ..., fdA −1 } be a set of matrix–valued functions in L2 (Rd , Cn×n ). If there exists an integer l 6= 0 such that F ⊂ Wl , then F cannot be a matrix–valued wavelet set.

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A. SAN ANTOL´IN AND R. A. ZALIK

Proof. Assume that F is a matrix–valued wavelet set. If l < 0, since Wl is a proper subset of V0 it follows that dA − 1 must be less of equal to 1 and this is impossible. On the other hand, if l > 0, Corollary 5 implies that every set of matrix–valued functions G ⊂ L2 (Rd , Cn×n ) such that T(G) is an orthonormal basis of Wl must have exactly (dA − 1)dlA matrix–valued functions. Since dA < (dA − 1)dlA we get a contradiction in this case as well. 

References [1] S. Bacchelli, M. Cotronei, T. Sauer; Wavelets for multichannel signals Adv. in Appl. Math. 29 (2002), no. 4, 581–598. [2] C. de Boor, R.A. DeVore, A. Ron; On the construction of multivariate (pre)wavelets, Constr. Approx. 9 (1993), 123–166. [3] C. de Boor, R.A. DeVore, A. Ron; The structure of finitely generated shift invariant spaces in L2 (Rd ), J. Funct. Anal. 119 (1994), no. 1, 37–78. [4] Q. Chen, H. Cao, Z. Shi; Design and characterizations of a class of orthogonal multiple vector-valued wavelets with 4-scale, Chaos Solitons Fractals 41 (2009), no. 1, 91–102. [5] Q–J. Chen, Z–X. Cheng, C–L. Wang; Existence and construction of compactly supported biorthogonal multiple vector-valued wavelets, J. Appl. Math. Comput. 22, No. 3, 101-115 (2006). [6] Q. Chen, Z. Shi; Construction and properties of orthogonal matrix-valued wavelets and wavelet packets, Chaos Solitons Fractals 37 (2008), no. 1, 75–86. [7] C. K. Chui; An Introduction to Wavelets, Academic Press, Inc. 1992. [8] L. Cui, B. Zhai, T. Zhang; Existence and design of biorthogonal matrix-valued wavelets, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 2679–2687. [9] I. Daubechies; Ten lectures on wavelets, SIAM, Philadelphia, 1992. [10] K. Gr¨ ochening, W. R. Madych; Multiresolution analysis, Haar bases and self-similar tillings of Rn , IEEE Trans. Inform. Theory, 38(2) (March 1992), 556–568. [11] J. Han, Z. Cheng, Q. Chen; A study of biorthogonal multiple vector–valued wavelets, Chaos Solitons Fractals 40 (2009), no. 4, 1574–1587. [12] J. He, B. Yu; Continuous wavelet transforms on the space L2 (R, H; dx), Appl. Math. Lett. 17 (2004), no. 1, 111–121. [13] E. Hern´ andez, G. Weiss; A first course on Wavelets, CRC Press, Inc., 1996. [14] R.Q. Jia and Z. Shen; Multiresolution and Wavelets, Proc. Edinburgh Math. Soc., 1994, 271–300. [15] W. R. Madych; Some elementary properties of multiresolution analyses of L2 (Rd ), Wavelets - a tutorial in theory and applications, Ch. Chui ed., Wavelet Anal. Appl. 2, Academic Press (1992), 259–294. [16] S. Mallat; Multiresolution approximations and wavelet orthonormal bases for L2 (R), Trans. of Amer. Math. Soc., 315 (1989), 69–87. [17] R. F. Mathis; Completion of a symmetric unitary matrix, SIAM Rev. 11, (1969) 261–263. [18] Y. Meyer; Ondelettes et op´ erateurs. I, Hermann, Paris, 1996 [English Translation: Wavelets and operators, Cambridge University Press, 1992]. [19] A. Ron, Z. Shen; Affine systems in L2 (Rd ) : The analysis of the analysis operator, J. Funct. Anal., 148 (1997), 408–447. [20] W. Rudin; Real and complex analysis, Third edition. McGraw-Hill Book Co., New York, 1987. [21] K. Slavakis, I. Yamada; Biorthogonal unconditional bases of compactly supported matrix valued wavelets, Numer. Funct. Anal. Optim. 22 (2001), no. 1-2, 223–253. [22] X-G.Xia, B. W. Suter; Vector–Valued wavelets and Vector Filter Banks, IEEE Transactions on Signal Processing, vol. 44, no 3, (1996), 508–518. [23] A. T. Walden, A. Serroukh; Wavelet analysis of matrix–valued time–series, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002), no. 2017, 157–179. [24] P. Wojtaszczyk; A mathematical introduction to wavelets, London Math. Soc., Student Texts 37, 1997.

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MATRIX–VALUED WAVELETS AND MULTIRESOLUTION ANALYSIS

13

[25] X. G. Xia; Orthonormal matrix valued wavelets and matrix Karhunen-Love expansion, Wavelets, multiwavelets, and their applications (San Diego, CA, 1997), 159–175, Contemp. Math., 216, Amer. Math. Soc., Providence, RI, 1998. [26] R. A. Zalik; On MRA Riesz wavelets, Proc. Amer. Math. Soc. 135 (2007), no. 3, 787–793. [27] R. A. Zalik;Bases of translates and multiresolution analyses, Appl. Comput. Harmon. Anal. 24 (2008), no. 1, 41–57. [28] P. Zhao, G. Liu, C. Zhao; A matrix-valued wavelet KL-like expansion for wide-sense stationary random processes, IEEE Trans. Signal Process. 52 (2004), no. 4, 914–920. ´ lisis Matema ´ tico, Universidad de Alicante, 03080 Alicante, Departamento de Ana Spain. E-mail address: [email protected] Department of Mathematics and Statistics, Auburn University, Auburn, Al. 36849– 5310 E-mail address: [email protected]

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,26-41,COPYRIGHT 2012 EUDOXUS PRESS,LLC

The Random Motion on the Sphere Generated by the Laplace-Beltrami Operator Dimitra Kouloumpou Department of Mathematics National Technical University of Athens Zografou Campus 157 80, Athens, GREECE. [email protected] and Vassilis G. Papanicolaou Department of Mathematics National Technical University of Athens Zografou Campus 157 80, Athens, GREECE. [email protected] running title: The Random Motion on the Sphere Generated by the Laplace-Beltrami Operator contact author: Vassilis G. Papanicolaou Department of Mathematics National Technical University of Athens Zografou Campus 157 80, Athens, GREECE. Tel: ++30 210 772 1722 Fax: ++30 210 772 1775 email: [email protected] Abstract Using the Laplace-Beltrami operator we construct the Brownian motion process on the n-dimensional sphere, n = 1, 2, 3. Then we evaluate explicitly certain quantities for this process. We start with the transition density and continue with the calculation of some probabilistic quantities regarding the exit times of specific domains possessing certain symmetries.

Key words and phrases: n-dimensional sphere, Laplace-Beltrami operator, Brownian motion, transition densities, exit times.

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1

Introduction

1.1

The n-Sphere

Let n ∈ N = {1, 2, 3, . . .}. The n-dimensional sphere S n with center (c1 , ..., cn+1 ) and radius a > 0 is the set of all points x ∈ Rn+1 satisfying (x1 − c1 )2 + · · · + (xn+1 − cn+1 )2 = a2 The most interesting case in applications is, of course, the case n = 2. For the sake of comparison we will also discuss the cases n = 1 (i.e. the circle) and n = 3. In some cases we will even consider the case of general n.

1.2

Stereographic Projection Coordinates

Consider the n-sphere, n ≥ 2, x21 + · · · + x2n + (xn+1 − a)2 = a2 To each point (x1 , ..., xn , xn+1 ) of this sphere, other than its “north pole” N = (0, ..., 0, 2a) we associate the coordinates ξ1 =

2ax1 2axn , . . . , ξn = 2a − xn+1 2a − xn+1

Given the coordinates (ξ1 , ..., ξn ) of a point on the sphere with Cartesian coordinates (x1 , ..., xn , xn+1 ), we have
2a ξ12 + · · · + ξn2 4a2 ξn 4a2 ξ1 , . . . , xn = 2 , xn+1 = 2 . x1 = 2 ξ1 + · · · + ξn2 + 4a2 ξ1 + · · · + ξn2 + 4a2 ξ1 + · · · + ξn2 + 4a2

1.3

Spherical Coordinates

The points of the n-sphere x21 + · · · + x2n + x2n+1 = a2 may also be described in spherical coordinates (θ1 , ..., θn−1 , ϕ) as follows: • For n = 1, x1 = a cos ϕ, x2 = a sin ϕ, where 0 ≤ ϕ < 2π. • For n = 2, (θ1 = θ) x1 = a cos θ sin ϕ, x2 = a sin θ sin ϕ, x3 = a cos ϕ, where 0 ≤ θ < 2π and 0 ≤ ϕ ≤ π. • For n = 3, x1 = a cos θ1 sin θ2 sin ϕ, x2 = a sin θ1 sin θ2 sin ϕ, x3 = a cos θ2 sin ϕ, x4 = a cos ϕ, where 0 ≤ θ1 < 2π, 0 ≤ θ2 ≤ π, and 0 ≤ ϕ ≤ π. • In general for n ≥ 4 x1 = a cos θ1 sin θ2 sin θ3 ... sin θn−1 sin ϕ, x2 = a sin θ1 sin θ2 sin θ3 ... sin θn−1 sin ϕ, xk = a cos θk−1 sin θk ... sin θn−1 sin ϕ, for k = 3, 4, ..., n and xn+1 = a cos ϕ, where 0 ≤ θ1 < 2π, 0 ≤ θi ≤ π, for i = 2, 3, ..., n − 1, and 0 ≤ ϕ ≤ π.

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1.4

The Laplace-Beltrami Operator

In spherical coordinates: The Laplace-Beltrami operator of a smooth function f on S 1 is 1 ∂2f ∆1 f = 2 · a ∂ϕ2 The Laplace-Beltrami operator of a smooth function f on S 2 is   1 fθθ ∆2 f = 2 + fϕ cos ϕ + fϕϕ sin ϕ a sin ϕ sin ϕ In the case where f is independent of θ we have ∆2 f =

1 (fϕϕ + fϕ cot ϕ) a2

The Laplace-Beltrami operator of a smooth function f on S 3 is      1 ∂f ∂f 1 ∂2f ∂ ∂ 1 2 ∆3 f = 2 2 · · sin θ2 + sin ϕ + sin θ2 ∂θ2 ∂θ2 ∂ϕ ∂ϕ a sin ϕ sin2 θ2 ∂θ12 and if f is independent of θ1 and θ2 , ∆3 f =

1 (fϕϕ + 2fϕ cot ϕ) a2

In stereographic projection coordinates: The Laplace-Beltrami operator of a smooth function f on S n , n ≥ 2 is #
2 " n n X ∂2f X ξ12 + · · · + ξn2 + 4a2 2(n − 2) ∂f ξi ∆n f = − 2 16a4 ∂ξi2 (ξ1 + · · · + ξn2 + 4a2 ) i=1 ∂ξi i=1 In particular, for n = 2 we get ∆2 f =

1.5

ξ12 + ξ22 + 4a2 16a4


2 

∂2f ∂2f + 2 2 ∂ξ1 ∂ξ2



Brownian motion on S n (starting at x ∈ S n )

The Brownian motion on S n is a diffusion (Markov) process Xt , t ≥ 0, on S n whose transition density is a function P (t, x, y) on (0, ∞) × S n × S n satisfying ∂P 1 = ∆n P, ∂t 2 P (t, x, y) → δx (y)

as t → 0+

where ∆n is the Laplace-Beltrami operator of S n acting on the x-variables and δx (y) is the delta mass at x, i.e. P (t, x, y) is the heat kernel of S n . The heat kernel exists, it is unique, positive, and smooth in (t, x, y) [4].

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1.5.1

Further Properties of the Heat Kernel P (t, x, y)

It is well known that p(t, x, y) satisfies the following properties [4] 1. Symmetry: P (t, x, y) = P (t, y, x) 2. The semigroup identity: For any s ∈ (0, t), Z P (t, x, y) = P (s, x, z)P (t − s, z, y)dµ(z) Sn

where dµ is the n-th dimensional surface area. 3. For all t > 0 and x ∈ S n Z P (t, x, y)dµ(y) = 1 Sn

4. As t → ∞, P (t, x, y) approaches the uniform density on S n , i.e. lim P (t, x, y) =

t→∞

1 An

where An is nth dimensional surface area of S n with radius a. It is well known that [8] n+1 2π 2 an for n odd An = n−1 , ( 2 )! n

2n ( n2 − 1)!π 2 an An = , (n − 1)!

for n even.

Finally, the symmetry of S n implies that P (t, x, y) depends only on t and d(x, y), the distance between x and y. Thus in spherical coordinates it depends on t and the angle ϕ between x and y. Hence P (t, x, y) = p(t, ϕ), where p(t, ϕ) satisfies   ∂p 1 1 ∂p ∂2p = ∆n p = 2 (n − 1) cot ϕ · + ∂t 2 2a ∂ϕ ∂ϕ2 and lim aAn−1 p(t, ϕ) · sinn−1 ϕ = δ(ϕ).

t→0+

Here δ(·) is the standard Dirac delta function on R.

2

Explicit Form of the Heat Kernel

Reminder (Poisson Summation Formula). Let f (x) be a function in the Schwartz space S(R), where S(R) consists of the set of all infinitely differentiable functions f on R so that f and all its derivatives f (l) are rapidly decreasing, in the sense that k sup |x| f (l) (x) < ∞ for every k, l ≥ 0. x∈R

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Then X

f (x + 2πn) =

n∈Z

1 X F (n) exp(inx), 2π n∈Z

where F (ξ) is the Fourier transform of f (x), i.e. Z

+∞

f (x) exp(−iξx)dx,

F (ξ) =

ξ ∈ R.

−∞

For example, if f (x) = exp(−Ax2 + Bx), then

r F (ξ) =

2.1

π exp A



B ∈ C,

A > 0, (iξ − B)2 4A



The Case of S 1

Proposition 2.1 The transition density function of the Brownian motion Xt , t ≥ 0 on S 1 with radius a is the function   1 X n2 t p(t, ϕ) = exp − 2 + inϕ . 2πa 2a n∈Z

Equivalently     1 X n2 t 1 p(t, ϕ) = exp − 2 cos(nϕ) − πa 2a 2πa n∈N

and

Proof. If

  2 1 X a 2 p(t, ϕ) = √ exp − (ϕ − 2πn) . 2t 2πt n∈Z     1 X n2 t 1 p(t, ϕ) = exp − 2 cos(nϕ) − , πa 2a 2πa n∈N

then

  ∂p(t, ϕ) 1 X 2 n2 t =− n cos(nϕ) exp − ∂t 2πa3 2a2

(2.1)

  n2 t ∂ 2 p(t, ϕ) 1 X 2 n cos(nϕ) exp − = − . ∂ϕ2 πa 2a2

(2.2)

n∈N

and

n∈N

Therefore

∂p(t, ϕ) 1 ∂ 2 p(t, ϕ) = 2 . ∂t 2a ∂ϕ2

We will now show that lim ap(t, ϕ) = δ(ϕ).

t→0+

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If ϕ ∈ (0, 2π), then lim ap(t, ϕ) = 0.

t→0+

Next we observe that     Z Z 2π n2 t 1 2π X exp − 2 cos(nϕ) dϕ − 1. ap(t, ϕ)dϕ = π 0 2a 0

(2.3)

(2.4)

n∈N

For t > 0 let us consider the functions fn : [0, 2π] → R, with

n ∈ N,

  n2 t fn (ϕ) = cos(nϕ) exp − 2 . 2a

Notice that fn (ϕ) are integrable functions on [0, 2π]. Furthermore +∞ X

fn (ϕ)

n=1

converges uniformly on [0, 2π] because   n2 t |fn (ϕ)| ≤ exp − 2 2a and the series

∞ X

n2 t exp − 2 2a n=1 



converges. Therefore (2.4) gives   Z 2π Z 2π 1X n2 t ap(t, ϕ)dϕ = −1 + exp − 2 cos(nϕ)dϕ, π 2a 0 0 n∈N

thus Z

2π

ap(t, ϕ)dϕ = 1, for every t > 0.

(2.5)

0

Therefore from (2.4) and (2.5) lim ap(t, ϕ) = δ(ϕ)

t→0+

and this complete the proof.

2.2



The Case of S 2

Let Xt , t ≥ 0 be the Brownian motion on a 2-dimensional sphere S 2 of radius a. The transition density function p(t, ϕ) of Xt is the unique solution of  2  1 ∂ p(t, ϕ) ∂p ∂p = 2 sin ϕ + cos ϕ (2.6) ∂t 2a sin ϕ ∂ϕ2 ∂ϕ 6

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and lim 2πa2 sin(ϕ)p(t, ϕ) = δ(ϕ).

(2.7)

t→0+

The solution to the diffusion equation   1 ∂K(t, ϕ) ∂ 2 K(t, ϕ) ∂K(t, ϕ) = cos ϕ + sin ϕ ∂t sin ϕ ∂ϕ ∂ϕ2

(2.8)

with initial condition lim 2π sin(ϕ)K(t, ϕ) = δ(ϕ)

(2.9)

t→0+

is given by the function (see [3]) K(t, ϕ) =

 √  1 X (2n + 1) exp −n(n + 1) 2t Pn0 (cos ϕ). 4π

(2.10)

n∈N

Here Pn0 (·) is the associated Legendre polynomials of order zero, i.e. Pn0 (x) =

 1 dn  2 · (x − 1)n . n n 2 n! dx

(2.11)

This fact implies the following Proposition 2.2 The transition density function of the Brownian motion Xt , t ≥ 0, on S 2 with radius a it is given by the function √   n(n + 1) t 1 X (2n + 1) exp − p(t, ϕ) = Pn0 (cos ϕ). (2.12) 4πa2 a n∈N

2.3

The Case of S 3

Proposition 2.3 Let Xt , t ≥ 0 be the Brownian motion on a 3-dimensional sphere S 3 of radius a. The transition density function p(t, ϕ) of Xt is given by
  X exp 2at 2 (ϕ + 2nπ)2 a2 p(t, ϕ) = (ϕ + 2nπ) exp − , 2t (2πt)3/2 sin ϕ n∈Z

where Z is the set of all integers. Equivalently   X i t(n2 − 1) p(t, ϕ) = − 2 3 n exp − + iϕn 4π a sin ϕ 2a2 n∈Z

and p(t, ϕ) =

1 2π 2 a3 sin ϕ

X n∈N

t(n2 − 1) n sin(nϕ) exp − 2a2 

 .

Furthermore p(t, ϕ) is analytic about ϕ = 0 and ϕ = π. In fact   1 X 2 t(n2 − 1) p(t, 0) = lim+ p(t, ϕ) = n exp − 2π 2 a3 2a2 ϕ→0 n∈N

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and   1 X 2 t(n2 − 1) n p(t, π) = lim p(t, ϕ) = n (−1) exp − . 2π 2 a3 2a2 ϕ→π − n∈N

Reminder. The ϑ3 function of Jacobi is ∞ X

ϑ3 (z, r) = 1 + 2


exp iπrn2 cos(2nz),

n=0

where r ∈ C with Im {r} > 0. It follows that     ∂ t ϕ ti 1 exp ϑ3 , . p(t, ϕ) = − 2 3 4π a sin ϕ 2a2 ∂ϕ 2 2a2 π Sketch of Proof. First we will prove that p(t, ϕ) satisfies the differential equation ∂p 1 = ∆3 p. ∂t 2 After that, we will show that lim 4πa3 sin2 (ϕ)p(t, ϕ) = δ(ϕ).

t→0+

For arbitrarily small  > 0, let Z  I = 4πa3 sin2 (ϕ)p(t, ϕ)dϕ. 0

We have 4πa3 exp 2at 2 lim+ I = lim+ t→0 t→0 (2tπ)3/2


Z



  ϕ2 a2 dϕ ϕ sin(ϕ) exp − 2t 0   ! X Z  (ϕ + 2nπ)2 a2 + (ϕ + 2nπ) sin(ϕ) exp − dϕ , 2t 0 ∗ n∈Z

∗

where Z = Z − {0}. However   2 2 2  X Z  X Z  n π a (ϕ + 2nπ)2 a2 dϕ ≤ dϕ (2|n|+1)π exp − (ϕ + 2nπ) sin(ϕ) exp − ∗ 0 2t 2t 0 ∗ n∈Z

n∈Z

and X Z n∈Z∗

0



n2 π 2 a 2 (2|n| + 1)π exp − 2t 

 dϕ = 

n2 π 2 a2 (2|n| + 1)π exp − 2t ∗

X



 ,

n∈Z

which converges to 0 as t → 0+ , by Lebesgue’s Dominated Convergence Theorem. Therefore
Z    4πa3 exp 2at 2 ϕ2 a2 ϕ sin ϕ exp − dϕ. lim+ I = lim+ 2t t→0 t→0 (2tπ)3/2 0 8

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By using the Laplace method for integrals [1]       Z  Z  Z ∞ ϕ2 a2 ϕ2 a2 ϕ2 a2 ϕ sin(ϕ) exp − dϕ ∼ ϕ2 exp − dϕ ∼ ϕ2 exp − dϕ, 2t 2t 2t 0 0 0 as t → 0+ . Here A ∼ B means that

A B

→ 1. Hence
  Z ∞ 4πa3 exp 2at 2 ϕ2 a2 2 lim I = lim+ ϕ exp − dϕ, 2t t→0+ t→0 (2tπ)3/2 0

or, for

ϕa u= √ t  Z ∞ 2  2 t u u √ lim+ I = lim+ 2 exp exp − du. 2 2a 2 t→0 t→0 2π 0

i.e. lim I = 1.

(2.13)

t→0+

Furthermore, for every t > 0, we have Z π I= 4πa3 sin2 (ϕ)p(t, ϕ)dϕ,

(2.14)

0

hence, 4πa3 exp 2at 2 I= (2tπ)3/2 The series X


Z 0

π

X

(ϕ + 2nπ)2 a2 (ϕ + 2nπ) sin(ϕ) exp − 2t 

 dϕ.

n∈Z

  (ϕ + 2nπ)2 a2 (ϕ + 2nπ) sin(ϕ) exp − 2t

n∈Z

converges uniformly on [0, π] for every t > 0, because   2 2 2  2 2 (ϕ + 2nπ) sin(ϕ) exp − (ϕ + 2nπ) a ≤ 2|n|π exp − n π a 2t 2t and the series X

Mn ,

n∈Z

where

 2 2 2 n π a Mn = 2|n|π exp − 2t

converges. Therefore (2.14), implies that
  Z 4πa3 exp 2at 2 X π (ϕ + 2nπ)2 a2 I= (ϕ + 2nπ) sin(ϕ) exp − dϕ. 2t (2tπ)3/2 n∈Z 0 Hence I=


  Z a exp 2at 2 X π (ϕ + 2nπ)2 a2 √ [exp(iϕ) + exp(−iϕ)] exp − dϕ. 2t 2tπ n∈Z 0 (2.15) 9

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Let u = ϕ + 2nπ, then (2.15) gives a I=√ 2tπ for every t > 0. In particular Z lim t→0+

! √ √ 2tπ 2tπ =1 + 2a 2a

π

4πa3 sin2 (ϕ)p(t, ϕ)dϕ = 1.

(2.16)

0

From (2.13) and (2.16) we have that that lim 4πa3 sin2 (ϕ)p(t, ϕ)dϕ = δ(ϕ)

t→0+

and this complete the proof.

3



Stochastic Differential Equation (SDE) of Xt in Local Coordinates

In spherical coordinates: The Brownian motion on S 1 satisfies the SDE 1 dBt . a The Brownian motion on S 2 satisfies the SDE      1 cos ϕ 0 dB1 (t) a sin ϕ dXt = 0, 2 dt + . 1 dB2 (t) 0 2a sin ϕ a dXt =

The Brownian motion on S 3 satisfies the SDE  1   a sin θ2 sin ϕ cos θ2 cos ϕ 0 dt+ dXt = 0, 2 , 2a sin θ2 sin2 ϕ a2 sin ϕ 0

0 1 a sin ϕ

0

  0 dB1 (t) 0   dB2 (t)  . 1 dB3 (t) a

In stereographic projection coordinates: The Brownian motion on S 2 satisfies the SDE   ξ12 + ξ22 + 4a2 dB1 (t) dXt = . dB2 (t) 4a2 The Brownian motion on S 3 satisfies the SDE

 dB1 (t)  ξ12 + ξ22 + ξ32 + 4a2 ξ12 + ξ22 + ξ32 + 4a2  dB2 (t)  . dXt = − (ξ1 , ξ2 , ξ3 ) dt + 16a4 4a2 dB3 (t) The Brownian motion on S n , n ≥ 2 satisfies the SDE 

ξ12 + · · · + ξn2 + 4a2 ξ12 + . . . + ξn2 + 4a2   (ξ1 , · · · , ξn ) dt+ dXt = (2−n)  16a4 4a2 

dB1 (t) dB2 (t) .. . dBn (t)

10

35

   . 

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

4

Expectations of exit times

Let Xt be the Brownian motion in S n and D ⊂ S n . The random variable T = inf{ t ≥ 0| Xt ∈ / D} is called the (first) exit time D. Reminder. If u(x) = E x [T ], (here the superscript x indicates that X0 = x) then u(x) satisfies 1 ∆n u = −1 2 u|∂D = 0 Let ϕ1 , ϕ2 ∈ [0, 2π), ϕ1 < ϕ2 . Consider the set D = (ϕ1 , ϕ2 ) . If Xt is the Brownian motion on S 1 starting at the point ϕ ∈ D, then E ϕ [T ] = a2 (ϕ − ϕ1 ) (ϕ2 − ϕ) Let ϕ0 ∈ (0, π) be fixed. We consider the set D in S n , n ≥ 2, such that D = { (θ1 , . . . , θn−1 , ϕ)| ϕ ∈ [0, ϕ0 )} . If Xt is the Brownian motion on S n starting at the point A = (θ1 , . . . , θn−1 , ϕ) ∈ D then A

2

Z

ϕ0

E [T ] = u(ϕ) = 2a

ϕ

Rx 0

(sin ω)n−1 dω dx (sin x)n−1

Notice that u(ϕ) is an elementary function. For n = 2 we obtain   1 + cos ϕ A 2 E [T ] = 2a ln . 1 + cos ϕ0 For n = 3 we obtain E A [T ] = a2 (ϕ cot ϕ − ϕ0 cot ϕ0 ) . Let ϕ1 , ϕ2 ∈ (0, π), ϕ1 < ϕ2 . Consider the set D in S n , n ≥ 2, D = { (θ1 , . . . , θn−1 , ϕ)| ϕ ∈ (ϕ1 , ϕ2 )} . If Xt is the Brownian motion on S n starting at the point A = (θ1 , . . . , θn−1 , ϕ) ∈ D

11

36

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

then  Z E A [T ] = 2a2 

ϕ1

Rx

ϕ

0

n−1

(sin ω) dω dx + (sin x)n−1

Rx

(sin ω)n−1 dω dx (sin x)n−1 ϕ1 R ϕ2 1 dx ϕ1 (sin x)n−1

R ϕ2

0

Z

ϕ

· ϕ1

 1 dx . (sin x)n−1

For n = 2 we obtain 4a2

         cos (ϕ1 /2) sin (ϕ/2) cos (ϕ1 /2) sin (ϕ2 /2)  E [T ] = ln ln − ln ln . 2 /2) cos (ϕ2 /2) sin (ϕ1 /2) cos (ϕ/2) sin (ϕ1 /2) ln tan(ϕ tan(ϕ1 /2) A



For n = 3 we obtain E A [T ] =

a2 [(ϕ − ϕ1 ) cot ϕ cot ϕ1 + (ϕ1 − ϕ2 ) cot ϕ1 cot ϕ2 + (ϕ2 − ϕ) cot ϕ2 cot ϕ] . cot ϕ1 − cot ϕ2

Notice that the formulas for n = 2 and n = 3 are quite different.

5

Hitting Probabilities

Let Xt be the Brownian motion in S n , D ⊂ S n , and T its exit time. Reminder. Let Γ ⊂ D and u(x) = P x {XT ∈ Γ}, then u(x) satisfies ∆n u = 0 u|Γ = 1,

u|∂D\Γ = 0

Consider the subset D = (ϕ1 , ϕ2 ) of S 1 , 0 < ϕ1 < ϕ2 < 2π. If Γ1 = {ϕ1 }, then ϕ2 − ϕ ϕ2 − ϕ1

P ϕ {XT ∈ Γ1 } =

Let ϕ1 , ϕ2 ∈ (0, π), ϕ1 < ϕ2 . Consider the set D in S n , n ≥ 2, D = { (θ1 , . . . , θn−1 , ϕ)| ϕ ∈ (ϕ1 , ϕ2 )} and the point A = (θ1 , . . . , θn−1 , ϕ) ∈ D. If Γ1 = {(θ1 , . . . , θn−1 , ϕ1 )}, then R ϕ2 A

P {XT ∈ Γ1 } =

1 dx ϕ (sin x)n−1 R ϕ2 . 1 dx ϕ1 (sin x)n−1

For n = 2 we obtain 

tan(

ϕ2 2 ϕ 2



tan(

ϕ2 2 ϕ1 2

ln P A {XT ∈ Γ1 } = ln

12

37

) tan( )

tan(

) )

 .

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

For n = 3 we obtain P A {XT ∈ Γ1 } =

cot ϕ − cot ϕ2 sin ϕ1 sin(ϕ2 − ϕ) = . cot ϕ1 − cot ϕ2 sin ϕ sin(ϕ2 − ϕ1 )

Let D be domain on S 2 whose stereographic coordinate description is D = {(ξ1 , ξ2 )| b < ξ2 < c}, i.e. D is the domain bounded by two circles passing through the north pole. If A = (ξ1 , ξ2 ) ∈ D and Γ1 = {(ξ1 , b)| ξ1 ∈ R}, then P A {XT ∈ Γ1 } =

6

c − ξ2 . c−b

(5.1)

The Moment Generating Function of T

Reminder. Assume that λ > −λ1 /2, where λ1 is the first Dirichlet eigenvalue of D ⊂ S n . If u(x) = E x [e−λT ], then u(x) satisfies 1 ∆n u = λu 2 u|∂D = 1 Suppose D ⊂ S 1 is the domain D = (ϕ1 , ϕ2 ),

0 ≤ ϕ1 < ϕ2 < 2π.

Then, for ϕ ∈ (ϕ1 , ϕ2 )  √   √  sinh a 2λ(ϕ2 − ϕ) + sinh a 2λ(ϕ − ϕ1 )  √  E ϕ [e−λT ] = sinh a 2λ(ϕ2 − ϕ1 ) provided λ>−

π2 − ϕ1 )2

2a2 (ϕ2

Let Xt be the Brownian motion on S 2 starting at the point A = (θ, ϕ) ∈ D, where D is the domain D = { (θ, ϕ)| θ ∈ [0, 2π), Then E A [exp(−λT )] =

13

38

and ϕ ∈ [0, ϕ0 )} . Pν (cos ϕ) , Pν (cos ϕ0 )

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

where ν is such that ν(ν + 1) = −2a2 λ and Pν (·) is the Legendre function Z ν p 1 π Pν (z) = P−ν−1 (z) = z + z 2 − 1 cos φ dφ, π 0 √
ν where the multiple-valued function z + z 2 − 1 cos φ is to be determined in such a way that for φ = π/2 it is equal to (the principal value of) z ν (which is, in particular, real for positive z and real ν). Let Xt be the Brownian motion on S n starting at the point A ∈ D, where D = { (θ1 , . . . , θn−1 , ϕ)| θ1 ∈ [0, 2π), θi ∈ [0, π] Then A

E [exp(−λT )] =

1− n 2

Pνµ (cos ϕ)

1− n 2

Pνµ (cos ϕ0 )

(sin ϕ) (sin ϕ0 )

for i = 2, . . . , n − 1

and ϕ ∈ [0, ϕ0 )} .

,

(6.1)

where

 1 p 1 (n − 1)2 − 8a2 λ − 1 and µ = (n − 2). 2 2 The function Pνµ (·) is the associated Legendre function ν=

Pνµ (z)

1 = Γ(−ν)Γ(ν + 1)



1+z 1−z

µ/2 X ∞

Γ(n − ν)Γ(n + ν + 1) Γ(n + 1 − µ)n! n=0



1−z z

n .

Here Γ(·) denotes the Gamma function.

7

The Reflection Principle

We will discuss the reflection principle on S 2 . Everything extends easily to S n . Notation. For every point A = (x1 , x2 , x3 ) ∈ S 2 we denote by Aˆ the symmetric of A with respect to the x1 x2 -plane. In other words Aˆ = (x1 , x2 , −x3 ) ∈ S 2 Theorem 7.1 Let Xt , t ≥ 0, be the Brownian motion on S 2 starting at the point A = (θ, ϕ) (in spherical coordinates). We assume that A ∈ D, where D is the lower hemisphere, i.e. D = { (θ, ϕ)| θ ∈ [0, 2π)

and

ϕ ∈ ( π/2, π] }

If T = inf { t ≥ 0| Xt ∈ / D} , then P A {T < t} = 2P A {Xt ∈ / D} . Sketch of Proof. P A {T < t} = P A {T < t, Xt ∈ / D} + P A {T < t, Xt ∈ D} .

14

39

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

However, if Xt ∈ / D, then, of course, T < t. Thus P A {T < t, Xt ∈ / D} = P A {Xt ∈ / D} . On the other hand, if we set X˜t =



Xt , Xˆt ,

if t ≤ T if t > T

then, by the strong Markov property of Xt n o P A {T < t, Xt ∈ D} = P A T < t, X˜t ∈ D , but X˜t ∈ D if and only if Xt ∈ / D. Hence, n o P A T < t, X˜t ∈ D = P A {T < t, Xt ∈ / D} = P A {Xt ∈ / D} and P A {T < t, Xt ∈ D} = P A {Xt ∈ / D} . Therefore P A {T < t} = 2P A {Xt ∈ / D}.

7.1



Applications of the Reflection Principle

The reflection principle can help to calculate the distribution functions of certain exit times. Let Xt be the Brownian motion on S 2 starting at the south pole S, where S = (0, π) in spherical coordinates. If D is the lower hemisphere and T its exit time, then √   ∞ X (2n)!(2n + 3) (2n + 1)2 t · P S {T < t} = 1 − (−1)n exp − a 22n+1 n! n=0 The case of S 1 : Let Xt be the Brownian motion on S 1 starting at ϕ ∈ D = (π, 2π). If T is the exit time of D, then   4 X 1 n2 t ϕ P {T < t} = 1 + exp − 2 sin(nϕ) π n 2a n odd

3

The case of S : Let Xt be the Brownian motion on S 3 starting at the south pole S, where S = (0, 0, π) in spherical coordinates. If D is the lower hemisphere, namely  D = (θ1 , θ2 , ϕ) ∈ S 3 θ1 ∈ [0, 2π), θ2 ∈ [0, π], ϕ ∈ (π/2, π ] and T the exit time of D, then   ∞ (4n2 − 1)t 16 X n 2 (−1) n exp − P {T < t} = 1 + . π n=1 2a2 S

Acknowledgment. This work was partially supported by a Π.E.B.E. Grant of the National Technical University of Athens. 15

40

RANDOM MOTION ON THE SPHERE...

D. Kouloumpou and V.G. Papanicolaou

References [1] Bender M. and Orszag S., Advanced Mathematical Methods for Scientists and Engineers McGraw-Hill Book Company, USA (1978) [2] Camporesi R., Harmonic Analysis and Propagators on Homogeneous Spaces, Phisycs Reports 196 (7) 1–134 (1990). [3] Chung M.K., Heat Kernel Smoothing On Unit Sphere in Proceedings of IEEE International Symposium on Biomedical Imaging (ISBI), 992–995 (2006). [4] Dodziuk J., Maximum Principle for Parabolic Inequalities and the Heat Flow on Open Manifolds, Indiana Oniv. Math. J. 32 no.5 115–142 (1983). [5] Dynkin E.B., Markov Processes vol 2 Springer, Berlin (1965). [6] Grigor’yan A. and Saloff-Coste L., Hitting Probabilities for Brownian Motion on Riemannian Manifolds, J. Math. Pures et Appl. 81 115–142 (2002) Company, Malabar, Florida, (1987). [7] Hsu E.P., A Brief Introduction to Brownian Motion On A Riemannian Manifold, Lecture notes. [8] John F., Partial Differential Equations Springer, USA (1982) [9] Karatzas I. and Shreve S.E., Brownian Motion and Stochastic Calculus Springer, USA (1991). [10] Klebaner F.C., Introduction to Stochastic Calculus with Applications Imerial College Press, Melbourne (2004). [11] Oksendal B., Stochastic Differential Equations. Springer-Verlag (1995).

16

41

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,42-53,COPYRIGHT 2012 EUDOXUS PRESS,LLC

Quantitative Estimates in the Overconvergence of Some Multivariate Singular Integrals∗ George A. Anastassiou and Sorin G. Gal Department of Mathematical Sciences The University of Memphis Memphis, TN 38152, U.S.A. [email protected] and Department of Mathematics and Computer Science University of Oradea Str. Universitatii No. 1 410087 Oradea, ROMANIA [email protected] Abstract In this paper we obtain quantitative estimates in the overconvergence phenomenon in polystrips in Cm , the weighted and non-weighted cases, for some multivariate singular integrals of Picard, Poisson-Cauchy and Gauss-Weierstrass.

AMS 2000 Mathematics Subject Classification : 32E30, 41A35, 41A25. Key words and phrases : Multivariate Picard, Poisson-Cauchy and GaussWeierstrass singular integrals of complex variables, overconvergence in polystrips in Cm , weighted approximation.

1

Introduction

Let ξ1 , ξ2 , . . . , ξm > 0 and f ; Rm → R be continuous on Rm , m ∈ N. If f is 2π- periodic or non periodic and bounded on Rm or of some exponential or polynomial growth on Rm , the following integrals are well defined : Pξ1,...,ξm (f ) (x1 , . . . , xm ) =

1 Πm j=1 (2ξj )

Z

Z

∞

∞

... −∞

−∞

−|tj |/ξj f (x1 + t1 , . . . , xm + tm ) Πm dt1 . . . dtm , j=1 e

∗ This

paper was written during the 2009 Spring Semester when the second author was a Visiting Professor at the Department of Mathematical Sciences, The University of Memphis, TN, U.S.A.

1

42

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

Qξ1,...,ξm (f ) (x1 , . . . xm ) Z ∞ ∞ f (x1 + t1 , . . . , xm + tm ) j=1 ξj Qm 2 ... dt1 . . . dtm , 2 m π −∞ −∞ j=1 (tj + ξj ) µ ¶m Y Z ∞ Z ∞ m 2 ξj3 ... Rξ1,..., ξm (f )(x1,...,xm ) = π −∞ −∞ j=1

Qm =

Z

f (x1 + t1 , . . . , xm + tm ) Qm 2 dt1 . . . dtm , 2 2 j=1 (tj + ξj ) and

=

π

1 Q m m/2

1/2 j=1 ξj

Z

∞

Wξ1,..., ξm (f )(x1 , . . . , xm ) Z ∞ m Y 2 ... e−tj /ξj dt1 . . . dtm . f (x1 + t1 , . . . , xm + tm )

−∞

−∞

j=1

Here Pξ1 ,...,ξm (f )(x1 , . . . xm ) is called of Picard type, Qξ1,... ξm (f )(x1 , . . . xm ), Rξ1 ,...,ξm (f )(x1 , . . . xm ) are called Poisson-Cauchy type and the singular integral Wξ1 ,...,ξm (f ), is called of Gauss-Weierstrass type. The approximation of f (x1 , ..., xm ) by the above singular integrals in the case of real variables as ξj → 0 , j = 1, ..., m, was studied in [1]. A quite natural problem would be the study of the overconvergence phenomenon for these singular integrals in polystrips, that is the approximation of the continuous function f (z1 , . . . zm ) by the complex singular integrals obtained by replacing xj ∈ R by zj ∈ C, j = 1, . . . , m in the above formulae of definition. This case for m = 1 was made in [2]. The aim of the present article is to extend the results from [2] for general m ∈ N.

2

Main Result

The first main result follows. Theorem 1. Let d1 , . . . , dm > 0 and suppose that f : ×m j=1 Sdj → C is bounded and uniformly continuous in the multivariate strip ×m j=1 Sdj , where Sdj = {z = x + iy ∈ C; x ∈ R, |y| ≤ dj }. (i) Denoting Z ∞ Z ∞ 1 ... f (z1 + t1 , . . . , zm + tm ) Pξ1 ,...,ξm (f )(z1 , . . . , zm ) = Qm −∞ j=1 (2ξj ) −∞ ·

m Y

e−|tj |/ξj dt1 , . . . dtm

j=1

for all ξj > 0 and zj ∈ Sdj , j = 1, . . . , m, we have |Pξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ (m + 1)ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

2

43

,

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

where for δj ≥ 0, j = 1, . . . , m, we define ω1 (f ; δ1 , . . . , δm )×m

j=1 Sdj

= sup {|f (u1 , . . . , um ) − f (v1 , . . . , vm )| : |uj − vj | ≤ δj , ª with uj , vj ∈ Sdj , for j = 1, . . . , m . (ii) Denoting µ ¶m Y Z ∞ Z ∞ m 2 3 ξj ... Rξ1 ,...,ξm (f )(z1 , . . . , zm ) = π −∞ −∞ j=1 f (z1 + t1 , . . . , zm + tm ) Qm 2 dt1 . . . dtm , 2 2 j=1 (tj + ξj ) for ξj > 0 and zj ∈ Sdj , j = 1, . . . , m, we have |Rξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ¶ µ 2m m ω1 (f ; ξ1 , . . . , ξm )×m Sd . ≤ 2 + j=1 j π (iii) Denoting

=

π

Z

1 Q m m/2

1/2 j=1 ξj

∞

−∞

Wξ1 ,...,ξm (f )(z1 , . . . , zm ) Z ∞ m Y 2 ... f (z1 + t1 , . . . , zm + tm ) e−tj /ξj dt1 . . . dtm , −∞

j=1

for ξj > 0 and zj ∈ Sdj , j = 1, . . . , m, we have |Wξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ¶ ³ µ p p ´ m ω1 f ; ξ1 , . . . , ξm m . ≤ 2m + √ π ×j=1 Sdj (iv) Denoting Qm Qξ1 ,...,ξm (f )(z1 , . . . , zm ) =

j=1 ξj πm

Z

Z

∞

∞

... −∞

−∞

f (z1 + t1 , . . . , zm + tm ) Qm 2 dt1 . . . dtm , 2 j=1 (tj + ξj ) for ξj > 0 and zj ∈ Sdj , j = 1, . . . , m, and supposing in addition, that f is of Lipschitz class (α1 , . . . , αm ) ∈ (0, 1)m in ×m j=1 Sdj , that is there exists a constant M > 0 such that   m X |f (u1 , . . . , um ) − f (v1 , . . . , vm )| ≤ M  |uj − vj |αj  , j=1

3

44

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

for all uj , vj ∈ Sdj , j = 1, . . . , m, we obtain   m X α |Qξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ CM  ξj j  , j=1

where

2 max π j=1,m

C=

½Z

¾ v αj dv . v2 + 1

∞

0

Proof. (i) If zj ∈ Sdj , j = 1, . . . , m, then clearly for all t ∈ R we have zj + t ∈ Sdj and since f is bounded on ×m (denote the j=1 Sdj bound by M(f ) ) it clearly follows |Pξ1 ,...,ξm (f )(z1 , . . . , zm )| ≤ M(f ) for all (z1 , . . . , zm ) ∈ ×m j=1 Sdj . Therefore Pξ1 ,...,ξm (f )(z1 , . . . , zm ) exists for all m (z1 , . . . , zm ) ∈ ×m j=1 Sdj . Also, the uniform continuity of f on ×j=1 Sdj implies that lim ω1 (f ; ξ1 , . . . , ξm )×m Sd = 0. ξ1 ,...,ξm →0

For all (z1 , . . . , zm ) ∈

1 j=1 (2ξj )

≤ Qm

j=1

×m j=1 Sdj

we have

|Pξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ... |f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )| −∞

−∞

· 1 j=1 (2ξj )

Z

≤ Qm

m Y

e−|tj |/ξj dt1 . . . dtm

j=1

Z

∞

∞

··· −∞

·

−∞

m Y

Z

m

2

j=1 (2ξj )

·

m Y j=1

Z

∞

ω1 (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

e−|tj |/ξj dt1 . . . dtm

j=1

= Qm

j

∞

··· 0

0

¶ µ ξm ξ1 ω1 f ; t1 , . . . , tm ξ1 ξm ×m

j=1 Sdj

Z

1 e−tj /ξj dt1 . . . dtm ≤ Qm

j=1 ξj

ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

Z

∞

∞

··· 0

0



 1 +

m m X tj  Y −tj /ξj e dt1 , . . . , dtm ξ j=1 j=1 j

" = ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

Z

1 Qm

j=1 ξj

4

45

Z

∞

··· 0

0

m ∞ Y j=1

e−tj /ξj dt1 . . . dtm

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

Z

1

+ Qm

j=1 ξj

∞

··· 0

   m m Y X ∗ t j   e−tj /ξj  dt1 . . . dtm  ∗ ξ j ∗ j=1 j =1 

Z

∞

0

= (m + 1)ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

,

proving the claim. (ii) We obtain µ ¶m Y m 2 ξj3 |Rξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤ π j=1 Z

Z

∞

∞

(|f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )|) Qm 2 dt1 . . . dtm 2 2 −∞ −∞ j=1 (tj + ξj ) µ ¶m Y Z ∞ Z ∞ ω1 (f ; |t1 |, . . . , |tm |) m m ×j=1 Sdj 2 Qm 2 ξj3 ··· dt1 . . . dtm ≤ 2 2 π −∞ −∞ j=1 (tj + ξj ) j=1 ´ ³   tm t1 , . . . , ξ ω f ; ξ µ ¶m Y Z Z m 1 1 m ∞ ∞ ξ1 ξm ×m 2 j=1 Sdj Qm 2 ξj3  ... dt1 . . . dtm = 2m  2 )2 π (t + ξ 0 0 j j=1 j j=1 ! Ã Qm 22m j=1 ξj3 ≤ ω1 (f ; ξ1 , . . . , ξm )×m Sd j=1 j πm ´ ³ Z ∞ Z ∞ 1 + Pm tj j=1 ξj Qm 2 · ··· 2 2 dt1 . . . dtm (t 0 0 j=1 j + ξj ) ! "Z Ã Qm Z ∞ ∞ 22m j=1 ξj3 d t . . . d tm Qm 1 2 = ω1 (f ; ξ1 , . . . , ξm )×m Sd · · · 2 2 m j=1 j π 0 0 j=1 (tj + ξj )  Z ∞ m Z ∞ X 1 tj ∗  Qm + ··· 2 2 dt1 . . . dtm ∗ 2 + ξ ) ξ (t j j 0 0 j ∗ j=1 j =1 ·

···

= ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj



! ∗ ∗ t d t j j · 2 + πm ξj ∗ (tj ∗ + ξj2∗ )2 0 j ∗ =1  ! ÃZ m  ∞ Y tj  ·  2 2 2  (t + ξ ) 0  j j j=1 j6=j ∗    Ã ! m Qm 3 m   2m  X Y 2 ξ 1 π   j=1 j = ω1 (f ; ξ1 , . . . , ξm )×m Sd 2m +  j=1 j  2ξj3∗ j=1 4ξj3  πm  j ∗ =1  Ã

m

! m (ÃZ Qm 22m j=1 ξj3 X

∞

j6=j ∗

5

46

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

= ω1 (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

¸ · 2m , 2m + π

proving the claim. (iii) We further have

≤

π m/2

|Wξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| Z ∞ Z ∞ ... |f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )| 1/2

1 Qm

j=1 ξj

−∞

−∞

·

m Y

2

e−tj /ξj dt1 . . . dtm

j=1

≤

π

Z

1 Qm m/2

−∞

· Z

=

π

2 Qm m/2

m Y

Z

0

∞

0

·

m Y

ω1 (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

2

j=1 ∞

−∞

e−tj /ξj dt1 . . . dtm

...

1/2 j=1 ξj

∞

...

1/2 j=1 ξj

m

Z

∞

¶ µ p t1 p tm ω1 f ; ξ1 √ , . . . , ξm √ ξ1 ξm × m

j=1 Sdj

2

e−tj /ξj dt1 . . . dtm

j=1

³ ≤ ω1

p ´ p f ; ξ1 , . . . , ξm

×m j=1 Sdj m Y

·

2m Qm p πξj j=1

m

= 2 ω1

p ´ p f ; ξ1 , . . . , ξm ·

m Y

0

" ×m j=1 Sdj

1 Qm p πξj j=1

Z

Z

∞

∞

··· 0

0

2

e−uj /ξj du1 . . . dum

j=1

1 + Qm p πξj j=1

∞

... 0

 m X u j 1 + p  ξj j=1 

Z

∞

e−uj2 /ξj du1 . . . dum

j=1

³

Z

  m m Y X ∗ 2 u j −u /ξ  p  e j j du1 . . . dum  ··· ξj ∗ j=1 0 0 j ∗ =1 ³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm m

Z

∞

Z



∞

×j=1 Sdj



 Z ∞ m m Z ∞ √ 2 Y X ∗ 1 u 1 + Qm p pj e−(uj / ξj ) du1 . . . dum  ··· ∗ πξ ξ 0 0 j j j=1 j=1 j ∗ =1 6

47

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm  1 +

"µZ m X

×m j=1 Sdj

uj ∗ −(uj ∗ /√ξj ∗ )2 duj∗ p p e π m/2 j ∗ =1 ξj ∗ ξj∗ 0  ! ÃZ m ∞ √ 2 Y  −(uj / ξj ) duj p · e  ξ 0 j j=1 1

∞

!

j6=j ∗

³ p p ´ = 2m ω1 f ; ξ1 , . . . , ξm  µZ m X  

  1 +

= 2m ω1

1 π m/2 ³

j ∗ =1

∞

2

uj ∗ e−uj∗ duj ∗

³

¶ Y m µZ

0

p ´ p f ; ξ1 , . . . , ξm

j=1 j6=j ∗

 ×m j=1 Sdj

1 +

1 π m/2 ·

p ´ p f ; ξ1 , . . . , ξm

×m j=1 Sdj ∞

 ¶ 2  e−uj duj 

0

" µ √ ¶m−1 # 1 π  2 2 ∗ j =1 m X

m m−1 1 + m/2 · m π 2 = 2 ω1 2 ×m S π j=1 dj ¸ · ³ p p ´ m m 2 +√ , = ω1 f ; ξ1 , . . . , ξm m π ×j=1 Sdj m

1

¸

proving the claim. (iv) We observe that Qm

j=1 ξj πm

|Qξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ≤

! |f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )| Qm 2 dt1 . . . dtm · ··· 2 −∞ −∞ j=1 (tj + ξj ) ! Qm Z ∞ Z ∞ Ã Pm αj j=1 ξj j=1 |tj | Qm 2 ··· dt1 . . . dtm ≤M 2 πm −∞ −∞ j=1 (tj + ξj ) !Z Ã Qm αj ∗ ! Z ∞ Ã Pm ∞ j=1 (2ξj ) j ∗ =1 tj ∗ Qm 2 ... dt1 . . . dtm =M 2 πm 0 0 j=1 (tj + ξj )    ÃZ ! Ã ! ¶ µ αj ∗ Z m m ∞ ∞ X  Y 2ξj tj ∗ d tj  2ξj ∗  ∗ =M    2 2 2 2 dtj π (t + ξ ) π (t ∗ + ξj ∗ ) 0 0 j j j ∗ j=1 j =1 Z

∞

Z

∞

Ã

j6=j ∗

=M

m X j ∗ =1

"

2ξj ∗ π

Z

∞

0

7

48

α

∗

tj ∗j dtj ∗ (t2j ∗ + ξj2∗ )

#

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

=M

m X j ∗ =1

where

"

α

2ξj ∗j π

∗

µZ

∞

0

  ¶# m X uαj∗ αj ∗ du ≤ CM  ξj ∗  , u2 + 1 ∗ j =1

2 C = · ∗ max π j =1,...,m

½Z

∞

0

uαj∗ du u2 + 1

¾ < ∞,

finishing the proof of the theorem. ¤ In what follows for Pn,ξ1 ,...,ξm (f ; (z1 , . . . , zm ) and Wξ1 ,...,ξm (f )(z1 , . . . , zm ) we will consider the weighted approximation on ×m j=1 Sdj , which seems to be m S is unbounded in C , m ∈ N. For this purpose, more natural because ×m j=1 dj first we need some general notations. Let w : ×m S j=1 dj → R+ be a continuous weighted functions in ×m S , with the properties that w(z1 , . . . , zm ) > 0 for j=1 dj any (z1 , . . . , zm ) ∈ ×m S and lim w(z , . . . , z 1 m ) = 0. j=1 dj |zj |→∞, j=1,...,m

Define the space

¡ ¢ Cw × m j=1 Sdj

© ª m = f : ×m j=1 Sdj → C; f is continuous in ×j=1 Sdj and kf kw < ∞ , where © ª kf kw := sup w(z1 , . . . , zm )|f (z1 , . . . , zm )|; (z1 , . . . , zm ) ∈ ×m j=1 Sdj . ¡ ¢ Also, for f ∈ Cw ×m j=1 Sdj define the weighted modulus of continuity ω1,w (f ; δ1 , . . . , δm )×m

j=1 Sdj

= sup {w(z1 , . . . , zm ) |f (z1 + h1 , . . . , zm + hm ) − f (z1 , . . . , zm )| ; ª (z1 , . . . , zm ) ∈ ×m j=1 Sdj , hj ∈ R with |hj | ≤ δj , j = 1, . . . , m Remark. The last modulus of continuity has the properties: a) it is increasing as a function of each δj , j = 1, . . . , m, b) ω1,w (f ; 0, . . . , 0)×m Sd = 0, j=1

j

c)

 ω1,w (f ; λ1 δ1 , . . . , λm δm )×m

j=1 Sdj

≤ 1 +

m X

 λj  ω1,w (f ; δ1 , . . . , δm )×m

j=1 Sdj

j=1

,

for all λj ≥ 0, j = 1, . . . , m. We present Theorem 2. Let dj > 0, j = 1, . . . , m, and suppose that f : ×m j=1 Sdj → C is Qm m continuous in ×j=1 Sdj . Let the Freud-type weight w(z1 , . . . , zm ) = j=1 e−qj |zj | ¡ m ¢ with qj > 0 , fixed, j = 1, . . . , m; and f ∈ Cw ×j=1 Sdj . Then (i) kPξ1 ,...,ξm (f ) − f kw ≤ (m + 1)ωn,w (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

8

49

,

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

for all 0 < ξj < (ii)

1 qj ,

j = 1, . . . , m, and

¶ µ ³ p p ´ m m ω1,w f ; ξ1 , . . . , ξm m , − f kw ≤ 2 + √ π ×j=1 Sdj

kWξ∗ (f )

for all 0 < ξj < 1, j = 1, . . . , m. Proof. The continuity of f in ×m j=1 Sdj immediately implies the continuity of Pξ1 ,...,ξm (f ) and Wξ1 ,...,ξm (f )(z1 , . . . , zm ). (i) In addition we have |w(z1 , . . . , zm )Pξ1 ,...,ξm (f )(z1 , . . . , zm )| ¯ Z ∞ Z ∞ ¯ 1 ¯ ··· w(z1 + t1 , . . . , zm + tm )f (z1 + t1 , . . . , tm ) = ¯ Qm ¯ j=1 (2ξj ) −∞ −∞ ¯ ¯ m Y ¯ w(z1 , . . . , zm ) −|tj |/ξj e dt1 . . . dtm ¯¯ · w(z1 + t1 , . . . , zm + tm ) j=1 ¯   ³ ´ Z ∞ m m Y Y 1 |tj | qj − ξ1 j e dtj ≤  Cξj ,qj  kf kw , ≤ kf kw 2ξ j −∞ j=1 j=1 where Cξj ,qj

1 = 2ξj

Z

∞

e

³ ´ |tj | qj − ξ1 j

dtj < ∞,

−∞

for all j = 1, . . . , m. Passing to supremum over all (z1 , . . . , zm ) ∈ ×m S , it follows that j=1 ¡ ¢ dj kPξ1 ,...,ξm (f )kw < ∞, that is Pξ1 ,...,ξm (f ) ∈ Cw ×m S , for 0 < ξj < q1j , j = j=1 dj 1, . . . , m. Next for all zj ∈ Sdj , j = 1, . . . , m, we derive w(z1 , . . . , zm )|Pξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ¯Z Z ∞ w(z1 , . . . , zm ) ¯¯ ∞ ··· (f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )) = Qm ¯ −∞ −∞ j=1 (2ξj ) ¯ ¯ m Y ¯ 1 · e−|tj |/ξj dt1 . . . dtm ¯¯ ≤ Qm ¯ j=1 (2ξj ) j=1 Z

Z

∞

·

∞

··· −∞

w(z1 , . . . , zm ) |f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )| −∞

·

m Y

e−|tj |/ξj dt1 . . . dtm

j=1

9

50

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

1 j=1 (2ξj )

Z

≤ Qm

Z

∞

∞

··· −∞ m Y

·

ω1,w (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

−∞

e−|tj |/ξj dt1 . . . dtm

j=1

Z

1

= Qm

j=1 ξj

Z

∞

µ

∞

···

ω1,w

0

0 m Y

·

tm t1 f ; ξ1 , . . . , ξm ξ1 ξm

¶ ×m j=1 Sdj

e−tj /ξj dt1 . . . dtm ,

j=1

  Z ∞ m ω1,w (f ; ξ1 , . . . , ξm )×m Sd Z ∞ X tj  j=1 j 1 + Qm ··· ≤ ξ ξ 0 0 j=1 j j=1 j ·

m Y

e−tj /ξj dt1 . . . dtm = (m + 1)ω1,w (f ; ξ1 , . . . , ξm )×m

j=1 Sdj

j=1

,

proving the claim. (ii) Next we observe |w(z1 , . . . , zm )Wξ1 ,...,ξm (f )(z1 , . . . , zm )| ¯ Z ∞ Z ∞ ¯ 1 ¯ · · · w(z1 + t1 , . . . , zm + tm )f (z1 + t1 , . . . , zm + tm ) =¯ ¯ π m/2 Qm ξ 1/2 −∞ −∞ j j=1 ¯ ¯ m Y ¯ 2 w(z1 , . . . , zm ) −tj /ξj · e dt1 . . . dtm ¯¯ w(z1 + t1 , . . . , zm + tm ) j=1 ¯ Z ∞ 1 p e|tj |(qj −|tj |/ξj ) dtj πξ j −∞ j=1 ! à Z ∞ m Y 1 m tj (qj −tj /ξj ) p dtj . = 2 kf kw e πξj 0 j=1 ≤ kf kw

m Y

But we can write Z ∞ Z tj (qj −tj /ξj ) e dtj = 0

Z

qj +1

e

tj (qj −tj /ξj )

∞

dtj +

etj (qj −tj /ξj ) dtj .

qj +1

0

For 0 < ξj ≤ 1 and tj ≥ qj + 1 we get tj (qj − tj /ξj ) ≤ −tj etj (qj −tj /ξj ) ≤ e−tj , which implies Z ∞ Z ∞ tj (qj −tj /ξj ) e dtj ≤ e−tj dtj = e−(qj +1) , qj +1

qj +1

10

51

and

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

for j = 1, . . . , m. In conclusion, from the above considerations we get |w(z1 , . . . , zm )Wξ1 ,...,ξm (f )(z1 , . . . , zm )| ≤ Cm kf kw , for some Cm > 0. Passing to sup over (z1 , . . . , zm ) ∈ ×m j=1 Sdj , it follows that Wξ1 ,...,ξm (f ) ∈ ¡ ¢ Cw × m S , for all 0 < ξ ≤ 1, j = 1, . . . , m. j j=1 dj For the estimate, for all (z1 , . . . , zm ) ∈ ×m j=1 Sdj we find w(z1 , . . . , zm ) |Wξ1 ,...,ξm (f )(z1 , . . . , zm ) − f (z1 , . . . , zm )| ¯Z Z w(z1 , . . . , zm ) ¯¯ ∞ ∞ (f (z1 + t1 , . . . , zm + tm ) − f (z1 , . . . , zm )) = ¯ 1/2 ¯ −∞ −∞ π m/2 Πm j=1 ξj ¯ ¯ m Y ¯ −t2j /ξj · e dt1 . . . dtm ¯¯ ¯ j=1 Z ∞ Z ∞ 1 ≤ Qm p ··· w(z1 , . . . , zm )|f (z1 +t1 , . . . , zm +tm )−f (z1 , . . . , zm )| πξj −∞ −∞ j=1 · 1 ≤ Qm p πξj j=1

m Y

2

e−tj /ξj dt1 . . . dtm

j=1

Z

Z

∞

∞

··· −∞

·

ω1,w (f ; |t1 |, . . . , |tm |)×m

j=1 Sdj

−∞

m Y

,

2

e−tj /ξj dt1 . . . dtm

j=1

2

Z

m

= Qm p πξj j=1

Z

∞

∞

··· 0

0

·

¶ µ p p t1 tm ω1,w f ; ξ1 √ , . . . , ξm √ ξ1 ξm × m

m Y

j=1 Sdj

2

e−tj /ξj dt1 . . . dtm

j=1

µ ³ p p ´ ≤ ω1,w f ; ξ1 , . . . , ξ1 

¶Ã ×m j=1 Sdj



2m Qm p πξj j=1

!Z

Z

∞

∞

··· 0

0

m m Y X t pj  e−tj2 /ξj dt1 . . . dtm ξ j j=1 j=1 ¶ µ ³ p p ´ m ω1,w f ; ξ1 , . . . , ξm m = 2m + √ π ×j=1 Sdj

1 +

proving the claim and finishing the proof of theorem. 11

52

¤

ANASTASSIOU, GAL: OVERCONVERGENCE OF SINGULAR INTEGRALS

References [1] G. A. Anastassiou and S. G. Gal, Convergence of generalized singular integrals to the unit, multivariate case, in : Applied Mathematics Reviews (G. A. Anastassiou ed.) , vol. 1, World Scientific, Singapore, New Jersey, London, Hong Kong, 2000, pp. 1-8. [2] G. A. Anastassiou and S. G. Gal, Quantitative estimates in the overconvergence of some singular integrals, Communications in Applied Analysis, under press.

12

53

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,54-61,COPYRIGHT 2012 EUDOXUS PRESS,LLC

A note on strong di¤erential superordinations using S¼al¼agean and Ruscheweyh operators Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In the present paper we establish several strong di¤erential superordinations regardind the new operator SRm de…ned by convolution product of the extended S¼ al¼ agean operator and Ruscheweyh derivative, SRm : An ! An ; SRm f (z; ) = (S m Rm ) f (z; ) ; z 2 U; 2 U ; where Rm f (z; ) denote the extended Ruscheweyh derivative, S m f (z; ) is the extended S¼ al¼ agean operator and An = ff 2 H(U U ); f (z; ) = z + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g is the class of normalized analytic functions.

Keywords: strong di¤erential superordination, convex function, best subordinant, extended di¤erential operator, convolution product. 2000 Mathematical Subject Classi…cation: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane U = fz 2 C : jzj < 1g, U = fz 2 C : jzj 1g the closed unit disc of the complex plane and H(U U ) the class of analytic functions in U U . Let An = ff 2 H(U U ); f (z; ) = z + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g; where ak ( ) are holomorphic functions in U for k 2; and H [a; n; ] = ff 2 H(U U ); f (z; ) = a + an ( ) z n + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g; for a 2 C, n 2 N; ak ( ) are holomorphic functions in U for k n: We also extend the well known di¤erential operators to the new class of analytic functions An introduced in [5]. De…nition 1.1 [1] For f 2 An , n; m 2 N; the operator S m is de…ned by S m : An ! An , S 0 f (z; ) 1

S f (z; ) S m+1 f (z; ) Remark 1.2 [1] If f 2 An , f (z; ) = z + 2 U.

= f (z; ) ; = zfz0 (z; ); :::; 0

= z (S m f (z; ))z , P1

j=n+1

z 2 U;

2 U:

aj ( ) z j , then S m f (z; ) = z +

P1

j=n+1

j m aj ( ) z j , z 2 U;

De…nition 1.3 [1] For f 2 An , n; m 2 N; the operator Rm is de…ned by Rm : An ! An , R0 f (z; ) 1

R f (z; ) (m + 1) Rm+1 f (z; )

= f (z; ) ; = zfz0 (z; ) ; :::; 0

= z (Rm f (z; ))z + mRm f (z; ) , 1

54

z 2 U;

2 U:

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

Remark 1.4 [1] If f 2 An , f (z; ) = z + z 2 U; 2 U :

P1

j=n+1

aj ( ) z j , then Rm f (z; ) = z +

P1

j=n+1

m Cm+j

1 aj

( ) zj ,

As a dual notion of strong di¤erential subordination G.I. Oros has introduced and developed the notion of strong di¤erential superordinations in [4]. De…nition 1.5 [4] Let f (z; ), H (z; ) analytic in U U : The function f (z; ) is said to be strongly superordinate to H (z; ) if there exists a function w analytic in U , with w (0) = 0 and jw (z)j < 1; such that H (z; ) = f (w (z) ; ) ; for all 2 U . In such a case we write H (z; ) f (z; ) ; z 2 U; 2 U : Remark 1.6 [4] (i) Since f (z; ) is analytic in U U , for all 2 U ; and univalent in U; for all 2 U , De…nition 1.5 is equivalent to H (0; ) = f (0; ) ; for all 2 U ; and H U U f U U : (ii) If H (z; ) H (z) and f (z; ) f (z) ; the strong superordination becomes the usual notion of superordination. De…nition 1.7 [3] We denote by Q the set of functions that are analytic and injective on U U nE (f; ), where E (f; ) = fy 2 @U : lim f (z; ) = 1g; and are such that fz0 (y; ) 6= 0 for y 2 @U U nE (f; ). The z!y

subclass of Q for which f (0; ) = a is denoted by Q (a). We have need the following lemmas to study the strong di¤erential superordinations. Lemma 1.8 [3] Let h (z; ) be a convex function with h(0; ) = a and let 2 C be a complex number with p(z; ) + 1 zp0z (z; ); Re 0. If p 2 H [a; n; ] \ Q ; p(z; ) + 1 zp0z (z; ) is univalent in U U and h(z; ) Rz 1 z 2 U; 2 U ;then q(z; ) p(z; ); z 2 U; 2 U ; where q(z; ) = h (t; ) t n dt; z 2 U; 2 U : The nz n 0 function q is convex and is the best subordinant. Lemma 1.9 [3] Let q (z; ) be a convex function in U U and let h(z; ) = q(z; ) + 1 zqz0 (z; ); z 2 U; 2 U ; where Re 0. If p 2 H [a; n; ] \ Q , p(z; ) + 1 zp0z (z; ) is univalent in U U and q(z; ) + 1 zqz0 (z; ) Rz p(z; ) + 1 zp0z (z; ) ; z 2 U; 2 U ; then q(z; ) p(z; ); z 2 U; 2 U ; where q(z; ) = h (t; ) t n 1 dt; 0 nz n

z 2 U,

2

2 U : The function q is the best subordinant.

Main results

De…nition 2.1 [2] Let m 2 N [ f0g. Denote by SRm the operator given by the Hadamard product (the convolution product) of the extended S¼al¼agean operator S m and the extended Ruscheweyh operator Rm , SRm : A n ! An , SRm f (z; ) = (S m Rm ) f (z; ) : Remark 2.2 [2] If f 2 An , f (z; ) = z+ z 2 U; 2 U :

P1

j=n+1

aj ( ) z j ; then SRm f (z; ) = z+

P1

j=n+1

m Cm+j

1j

m 2 aj

( ) zj ;

Theorem 2.3 Let h (z; ) be a convex function in U U with h (0; ) = 1. Let m 2 N, f (z; ) 2 An ; Rz c 0 t f (t; ) dt, z 2 U; 2 U , Re c > 2; and suppose that (SRm f (z; ))z is F (z; ) = Ic (f ) (z; ) = zc+2 c+1 0 0 univalent in U U , (SRm F (z; ))z 2 H [1; n; ] \ Q and 0

z 2 U;

2 U;

0

z 2 U;

2 U;

h (z; )

(SRm f (z; ))z ,

q (z; )

(SRm F (z; ))z ,

then where q(z; ) =

c+2 nz

c+2 n

Rz 0

h(t; )t

c+2 n

1

dt: The function q is convex and it is the best subordinant.

2

55

(1)

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

Rz Proof. We have z c+1 F (z; ) = (c + 2) 0 tc f (t; ) dt and di¤erentiating it, with respect to z, we obtain 0 (c + 1) F (z; ) + zFz0 (z; ) = (c + 2) f (z; ) and (c + 1) SRm F (z; ) + z (SRm F (z; ))z = (c + 2) SRm f (z; ) ; z 2 U; 2 U : Di¤erentiating the last relation with respect to z we have 0

(SRm F (z; ))z +

1 00 0 z (SRm F (z; ))z2 = (SRm f (z; ))z , z 2 U; c+2

2 U:

(2)

Using (2), the strong di¤erential superordination (1) becomes 0

(SRm F (z; ))z +

h (z; )

1 00 z (SRm F (z; ))z2 : c+2

(3)

0

(4)

Denote p (z; ) = (SRm F (z; ))z ; z 2 U; Replacing (4) in (3) we obtain h (z; ) p (z; ) + Using Lemma 1.8 for = c + 2; we have q (z; ) where q(z; ) =

c+2 c+2 nz n

Rz 0

p (z; ) ; z 2 U; h(t; )t

c+2 n

1

1 0 c+2 zpz

(z; ),

2 U ; i.e. q (z; )

2 U: z 2 U;

2 U: 0

(SRm F (z; ))z , z 2 U;

2 U;

dt: The function q is convex and it is the best subordinant.

Corollary 2.4 Let h (z; ) = +(21+z )z ; where 2 [0; 1). Let m 2 N, f (z; ) 2 An ; F (z; ) = Ic (f ) (z; ) = Rz c 0 c+2 U, 2 U ; Re c > 2; and suppose that (SRm f (z; ))z is univalent in U z c+1 0 t f (t; ) dt, z 2 U; 0 m (SR F (z; ))z 2 H [1; n; ] \ Q and 0

(SRm f (z; ))z , z 2 U;

h (z; )

2 U;

(5)

then q (z; ) where q is given by q(z; ) = 2

+

0

(SRm F (z; ))z ,

2(c+2)(

the best subordinant.

)

c+2 nz n

Rz

t

0

c+2 n

1

t+1

z 2 U;

dt; z 2 U;

2 U; 2 U : The function q is convex and it is 0

Proof. Following the same steps as in the proof of Theorem 2.3 and considering p(z; ) = (SRm F (z; ))z , 1 the strong di¤erential superordination (5) becomes h(z; ) = +(21+z )z p (z; ) + c+2 zp0z (z; ) ; z 2 U; 2 U: Rz c+2 By using Lemma 1.8 for = c + 2, we have q(z; ) p(z; ), i.e. q(z; ) = c+2 h(t; )t n 1 dt = c+2 0 nz n c+2 R z +(2 0 )t c+2 1 ) Rz t n 1 c+2 t n (SRm F (z; ))z ; z 2 U; 2 U : dt = 2 + 2(c+2)( c+2 c+2 1+t t+1 dt 0 0 nz

nz

n

n

The function q is convex and it is the best subordinant.

1 zqz0 (z; ) ; where z 2 U; Theorem 2.5 Let q (z; ) be a convex function in U U and let h (z; ) = q (z; ) + c+2 Rz c c+2 2 U ; Re c > 2: Let m 2 N, f (z; ) 2 An ; F (z; ) = Ic (f ) (z; ) = zc+1 0 t f (t; ) dt, z 2 U; 2 U ; and 0 0 suppose that (SRm f (z; ))z is univalent in U U , (SRm F (z; ))z 2 H [1; n; ] \ Q and 0

h (z; )

(SRm f (z; ))z ,

q (z; )

(SRm F (z; ))z ,

z 2 U;

2 U;

z 2 U;

2 U;

then where q(z; ) =

c+2 nz

c+2 n

Rz 0

h(t; )t

c+2 n

1

0

dt: The function q is the best subordinant.

3

56

(6)

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

Proof. We obtain that z c+1 F (z; ) = (c + 2)

Z

z

tc f (t; ) dt:

(7)

0

Di¤erentiating (7), with respect to z, we have (c + 1) F (z; ) + zFz0 (z; ) = (c + 2) f (z; ) and 0

(c + 1) SRm F (z; ) + z (SRm F (z; ))z = (c + 2) SRm f (z; ) ;

2 U:

z 2 U;

(8)

Di¤erentiating (8) with respect to z we have 1 00 0 z (SRm F (z; ))z2 = (SRm f (z; ))z , c+2

0

(SRm F (z; ))z +

2 U:

z 2 U;

(9)

Using (9), the strong di¤erential superordination (6) becomes 1 zq 0 (z; ) c+2 z

h (z; ) = q (z; ) +

0

(SRm F (z; ))z +

1 00 z (SRm F (z; ))z2 : c+2

(10)

2 U:

(11)

Denote 0

p (z; ) = (SRm F (z; ))z ; Replacing (11) in (10) we obtain h (z; ) = q (z; ) + Using Lemma 1.9 for = c + 2; we have q (z; ) where q(z; ) =

c+2 c+2 n

nz

Rz 0

p (z; ) ; z 2 U; h(t; )t

c+2 n

1

1 0 c+2 zqz

z 2 U; (z; )

1 0 c+2 zpz

p (z; ) +

(z; ), z 2 U;

0

(SRm F (z; ))z , z 2 U;

2 U ; i.e. q (z; )

2 U:

2 U;

dt: The function q is the best subordinant.

Theorem 2.6 Let h (z; ) be a convex function, h(0; ) = 1: Let m 2 N; f (z; ) 2 An and suppose that 0

(SRm f (z; ))z is univalent and

SRm f (z; ) z

2 H [1; n; ] \Q . If 0

(SRm f (z; ))z ;

h(z; ) then

SRm f (z; ) ; z

q(z; ) where q(z; ) =

1 1

nz n

Rz 0

1

h(t; )t n

1

z 2 U; z 2 U;

2 U;

(12)

2 U;

dt: The function q is convex and it is the best subordinant.

P1 m m 2 j m P1 z+ Cm+j 1 j aj ( )z j=n+1 m Proof. Consider p (z; ) = SR zf (z; ) = = 1 + j=n+1 Cm+j z Evidently p 2 H [1; n; ]. 0 Di¤erentiating with respect to z, we obtain p (z; ) + zp0z (z; ) = (SRm f (z; ))z : Then (12) becomes h(z; ) p(z; ) + zp0z (z; ); z 2 U; 2 U : By using Lemma 1.8 for = 1, we have q(z; ) where q(z; ) =

1 1

nz n

p(z; ); Rz 0

z 2 U;

1

h(t; )t n

Corollary 2.7 Let h(z; ) =

1

2 U;

z 2 U;

m 2 aj

( ) zj

1

:

2 U;

dt: The function q is convex and it is the best subordinant.

+(2 )z 1+z m

be a convex function in U 0 ))z

f (z; ) 2 An and suppose that (SR f (z;

0

then q(z; ) +

is univalent and

(SRm f (z; ))z ;

h(z; )

where q is given by q(z; ) = 2 best subordinant.

SRm f (z; ) ; z

i.e. q(z; )

1j

2( 1

nz n

U , where 0

SRm f (z; ) z

z 2 U;

< 1. Let m 2 N [ f0g;

2 H [1; n; ] \ Q . If

2 U;

(13)

SRm f (z; ) ; z 2 U; 2 U ; z 1 ) R z tn 1 dt; z 2 U; 2 U : The function q is convex and it is the 0 1+t 4

57

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

m

Proof. Following the same steps as in the proof of Theorem 2.6 and considering p(z; ) = SR zf (z; ) , the strong di¤erential superordination (13) becomes h(z; ) = +(21+z )z p(z; ) + zp0z (z; ); z 2 U; 2 U : Rz 1 By using Lemma 1.8 for = 1, we have q(z; ) p(z; ), i.e. q(z; ) = 1 1 0 h (t; ) t n 1 dt = nz n R z 1 1 +(2 R z n1 1 )t SRm f (z; ) 1 tn dt = 2 + 2( 1 ) 0 t1+t dt ; z 2 U; 2 U : 1 1+t z 0 nz n nz n The function q is convex and it is the best subordinant. Theorem 2.8 Let q (z; ) be convex in U U and let h be de…ned by h (z; ) = q (z; )+zqz0 (z; ) : If m 2 N[f0g, m 0 f (z; ) 2 An , suppose that (SRm f (z; ))z is univalent, SR zf (z; ) 2 H [1; n; ] \ Q and satis…es the strong di¤ erential superordination 0

h(z; ) = q (z; ) + zqz0 (z; ) then

SRm f (z; ) ; z

q(z; ) 1

where q(z; ) =

1

nz n

Rz 0

1

1

h(t; )t n

(SRm f (z; ))z ;

z 2 U;

z 2 U;

2 U;

(14)

2 U;

dt: The function q is the best subordinant. P1

m m 2 j P1 Cm+j 1 j aj ( )z j=n+1 m m 2 j 1 : Evidently = = 1+ j=n+1 Cm+j Proof. Let p (z; ) = 1 j aj ( ) z z p 2 H [1; n; ]. 0 Di¤erentiating with respect to z, we obtain p(z; ) + zp0z (z; ) = (SRm f (z; ))z ; z 2 U; 2 U ; and (14) p(z; ) + zp0z (z; ) ; z 2 U; 2 U : becomes q(z; ) + zqz0 (z; ) Using Lemma 1.9 for = 1, we have Z z 1 1 SRm f (z; ) h(t; )t n 1 dt , z 2 U; 2 U ; q(z; ) p(z; ); z 2 U; 2 U ; i.e. q(z; ) = 1 z nz n 0

z+

SRm f (z; ) z

and q is the best subordinant. Theorem 2.9 Let h (z; ) be a convex function, h(0; ) = 1: Let m 2 N [ f0g; f (z; ) 2 An and suppose that zSRm+1 f (z; ) SRm f (z; )

0

z

is univalent and

SRm+1 f (z; ) SRm f (z; )

zSRm+1 f (z; ) SRm f (z; )

h(z; ) then

1 1

nz n

Rz 0

1

h(t; )t n

Proof. Consider p (z; ) = dently p 2 H [1; n; ].

We have p0z (z; ) =

1

where q(z; ) =

1 1 nz n

Rz 0

SRm+1 f (z; ) SRm f (z; ) 0

(SRm+1 f (z; ))z SRm f (z; )

=

p (z; )

p(z; ); z 2 U; 1

h(t; )t n

;

z

z 2 U;

z 2 U;

2 U;

(15)

2 U;

dt: The function q is convex and it is the best subordinant. P1 m+1 m+1 2 Cm+j j aj ( Pj=n+1 1

z+

z+

m Cm+j j=n+1

(SRm f (z; ))0z SRm f (z; )

Then (15) becomes h(z; ) p(z; ) + zp0z (z; ); z 2 U; By using Lemma 1.8 for = 1, we have q(z; )

0

SRm+1 f (z; ) ; SRm f (z; )

q(z; ) where q(z; ) =

2 H [1; n; ] \ Q . If

1

2 U ; i.e. q(z; )

m 2 1 j aj (

)z j )z j

=

1+

P1 m+1 m+1 2 Cm+j j aj ( Pj=n+1 1

1+

m Cm+j j=n+1

and p (z; ) + zp0z (z; ) =

m 2 1 j aj (

2 U;

dt: The function q is convex and it is the best subordinant.

5

58

1

)z j

1

zSRm+1 f (z; ) SRm f (z; )

2 U: SRm+1 f (z; ) ; z 2 U; SRm f (z; )

)z j

0 z

:

: Evi-

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

Corollary 2.10 Let h(z; ) = f (z; ) 2 An and suppose that

+(2 )z be a convex function in U U , 1+z 0 m+1 zSRm+1 f (z; ) f (z; ) is univalent, SR m SR f (z; ) SRm f (z; ) z

then

2(

+

;

2 H [1; n; ] \ Q : If 2 U;

z 2 U;

z

< 1. Let m 2 N [ f0g;

(16)

SRm+1 f (z; ) ; z 2 U; 2 U ; SRm f (z; ) 1 ) R z tn 1 dt; z 2 U; 2 U : The function q is convex and it is the 0 1+t

q(z; ) where q is given by q(z; ) = 2 best subordinant.

0

zSRm+1 f (z; ) SRm f (z; )

h(z; )

where 0

1

nz n

m

Proof. Following the same steps as in the proof of Theorem 2.9 and considering p(z; ) = SR zf (z; ) , the strong di¤erential superordination (16) becomes h(z; ) = +(21+z )z p(z; ) + zp0z (z; ); z 2 U; 2 U : Rz 1 By using Lemma 1.8 for = 1, we have q(z; ) p(z; ), i.e. q(z; ) = 1 1 0 h (t; ) t n 1 dt = nz n R z 1 1 +(2 R z n1 1 )t SRm+1 f (z; ) 1 tn dt = 2 + 2( 1 ) 0 t1+t dt 2 U: 1 1+t SRm f (z; ) ; z 2 U; 0 nz n nz n The function q is convex and it is the best subordinant. U and let h be de…ned by h (z; ) = q (z; ) + zqz0 (z; ) : If m 2

Theorem 2.11 Let q (z; ) be convex in U

zSRm+1 f (z; ) SRm f (z; )

N [ f0g, f (z; ) 2 An , suppose that strong di¤ erential superordination

0

z

SRm+1 f (z; ) ; SRm f (z; )

q(z; ) where q(z; ) =

1 1 nz n

Rz 0

1

1

h(t; )t n

Proof. Let p (z; ) =

SRm+1 f (z; ) SRm f (z)

SRm+1 f (z; ) SRm f (z; )

2 U;

z 2 U;

z

z 2 U;

=

P1 m+1 m+1 2 Cm+j j aj ( Pj=n+1 1

z+

z+

m Cm+j j=n+1

m 2 1 j aj (

)z j

(17)

2 U;

zqz0 (z;

becomes q(z; ) + ) p(z; ) + Using Lemma 1.9 for = 1, we have 2 U;

zp0z

(z; ) ; z 2 U;

i.e. q(z; ) =

1 1

nz n

1+

=

)z j

Di¤erentiating with respect to z, we obtain p(z; ) + zp0z (z; ) =

p(z; ); z 2 U;

;

2 H [1; n] \ Q and satis…es the

dt: The function q is the best subordinant.

p 2 H [1; n; ].

q(z; )

0

zSRm+1 f (z; ) SRm f (z; )

h(z; ) = q (z; ) + zqz0 (z; ) then

is univalent,

P1 m+1 m+1 2 j aj ( Cm+j Pj=n+1 1

1+

m Cm+j j=n+1

zSRm+1 f (z; ) SRm f (z; )

0

z

m 2 1 j aj (

; z 2 U;

)z j

1

)z j

1

: Evidently

2 U ; and (17)

2 U:

Z

z

1

h(t; )t n

1

SRm+1 f (z; ) , z 2 U; SRm f (z; )

dt

0

2 U;

and q is the best subordinant. Theorem 2.12 Let h (z; ) be a convex function, h(0; ) = 1: Let m 2 N [ f0g; f (z; ) 2 An and suppose that 0 1 m+1 f (z; ) is univalent and (SRm f (z; ))z 2 H [1; n; ] \ Q . If z SR h(z; )

1 SRm+1 f (z; ) ; z

q(z; )

(SRm f (z; ))z ;

z 2 U;

2 U;

then where q(z; ) =

m+1 nz

m+1 n

Rz 0

h(t; )t

m+1 n

1

0

z 2 U;

2 U;

dt: The function q is convex and it is the best subordinant. 6

59

(18)

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

P1 0 m m+1 2 Proof. With notation p (z; ) = (SRm f (z; ))z = 1 + j=n+1 Cm+j aj ( ) z j 1 and p (0; ) = 1, we 1j P1 1 1 obtain for f (z) = z + j=n+1 aj ( ) z j ; p (z; ) + m+1 zp0z (z; ) = z SRm+1 f (z; ) : Evidently p 2 H [1; n; ]. 1 zp0z (z; ); z 2 U; 2 U : Then (18) becomes h(z; ) p(z; ) + m+1 By using Lemma 1.8 for = m + 1, we have q(z; ) where q(z; ) =

m+1 nz

m+1 n

Rz 0

p(z; ); z 2 U; h(t; )t

Corollary 2.13 Let h(z; ) = f (z; ) 2 An and suppose that

m+1 n

1

0

(SRm f (z; ))z ; z 2 U;

2 U ; i.e. q(z; )

2 U;

dt: The function q is convex and it is the best subordinant.

+(2 )z be a 1+z 1 m+1 SR f (z; z

< 1. Let m 2 N [ f0g; convex function in U U , where 0 0 ) is univalent and (SRm f (z; ))z 2 H [1; n; ] \ Q . If

1 SRm+1 f (z; ) ; z

h(z; )

z 2 U;

2 U;

(19)

then q(z; ) where q is given by q(z; ) = 2

+

2(

0

(SRm f (z; ))z ; )(m+1) m+1 nz n

is the best subordinant.

Rz

t

m+1 n

1+t

0

1

z 2 U; dt; z 2 U;

2 U; 2 U : The function q is convex and it 0

Proof. Following the same steps as in the proof of Theorem 2.12 and considering p(z; ) = (SRm f (z; ))z , 1 the strong di¤erential superordination (19) becomes h(z; ) = +(21+z )z p(z; ) + m+1 zp0z (z; ); z 2 U; 2 U: Rz m+1 By using Lemma 1.8 for = m + 1, we have q(z; ) p(z; ), i.e. q(z; ) = m+1 h (t; ) t n 1 dt = m+1 0 nz n m+1 1 R z m+1 1 +(2 0 )t )(m+1) R z t n m+1 dt = 2 + 2( m+1 (SRm f (z; ))z ; z 2 U; 2 U : t n m+1 1+t 1+t dt 0 0 nz

nz

n

n

The function q is convex and it is the best subordinant.

1 Theorem 2.14 Let q (z; ) be convex in U U and let h be de…ned by h (z; ) = q (z; ) + m+1 zqz0 (z; ) : If 0 1 m+1 m m 2 N [ f0g, f (z; ) 2 An , suppose that z SR f (z; ) is univalent, (SR f (z; ))z 2 H [1; n; ] \ Q and satis…es the strong di¤ erential superordination

h(z; ) = q (z; ) +

1 zq 0 (z; ) m+1 z

1 SRm+1 f (z; ) ; z

z 2 U;

2 U;

(20)

then q(z; ) where q(z; ) =

m+1 nz

m+1 n

Rz 0

h(t; )t

m+1 n

1

0

(SRm f (z; )) ;

z 2 U;

2 U;

dt: The function q is the best subordinant.

P1 0 m m+1 2 Proof. Let p (z; ) = (SRm f (z; ))z = 1 + j=n+1 Cm+j aj ( ) z j 1 . 1j 1 0 Di¤erentiating with respect to z, we obtain p(z; ) + m+1 zpz (z; ) = z1 SRm+1 f (z; ) ; z 2 U; 2 U ; and 1 1 (20) becomes q(z; ) + m+1 zqz0 (z; ) p(z; ) + m+1 zp0z (z; ) ; z 2 U; 2 U : Using Lemma 1.9 for = m + 1, we have Z m+1 m+1 z 0 h(t; )t n 1 dt q(z; ) p(z; ); z 2 U; 2 U ; i.e. q(z; ) = (SRm f (z; )) , z 2 U; 2 U ; m+1 n nz 0 and q is the best subordinant.

7

60

LUPAS: STRONG DIFFERENTIAL SUPERORDINATIONS...

References [1] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, On special strong di¤ erential subordinations using S¼al¼agean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, 2012 (to appear). [2] A. Alb Lupa¸s, A note on strong di¤ erential subordinations using S¼al¼agean and Ruscheweyh operators, Libertas Mathematica, submitted 2010. [3] A. Alb Lupa¸s, On special strong di¤ erential superordinations using S¼al¼agean and Ruscheweyh operators, submitted 2011. [4] G.I. Oros, Strong di¤ erential superordination, Acta Universitatis Apulensis, Nr. 19, 2009, 101-106. [5] G.I. Oros, On a new strong di¤ erential subordination, (to appear).

8

61

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,62-68,COPYRIGHT 2012 EUDOXUS PRESS,LLC

Certain strong di¤erential superordinations using a generalized S¼al¼agean operator and Ruscheweyh operator Alina Alb Lupa¸s Department of Mathematics and Computer Science University of Oradea str. Universitatii nr. 1, 410087 Oradea, Romania [email protected] Abstract In the present paper we establish several strong di¤erential superordinations regardind the new operator DRm de…ned by convolution product of the extended generalized S¼ al¼ agean operator and Ruscheweyh derivative, DRm : A ! A ; DRm f (z; ) = (Dm Rm ) f (z; ) ; z 2 U; 2 U ; where Rm f (z; ) dem note the extended Ruscheweyh derivative, D f (z; ) is the extended generalized S¼ al¼ agean operator and An = ff 2 H(U U ); f (z; ) = z + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g; with A1 = A ; is the class of normalized analytic functions.

Keywords: strong di¤erential superordination, convex function, best subordinant, extended di¤erential operator, convolution product. 2000 Mathematical Subject Classi…cation: 30C45, 30A20, 34A40.

1

Introduction

Denote by U the unit disc of the complex plane U = fz 2 C : jzj < 1g, U = fz 2 C : jzj 1g the closed unit disc of the complex plane and H(U U ) the class of analytic functions in U U . Let An = ff 2 H(U U ); f (z; ) = z + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g; with A1 = A ; where ak ( ) are holomorphic functions in U for k 2; and H [a; n; ] = ff 2 H(U U ); f (z; ) = a + an ( ) z n + an+1 ( ) z n+1 + : : : ; z 2 U; 2 U g; for a 2 C, n 2 N; ak ( ) are holomorphic functions in U for k n: We also extend the well known di¤erential operators to the new class of analytic functions An introduced in [10]. 0 and m 2 N, the operator Dm is de…ned by Dm : A ! A ,

De…nition 1.1 [5] For f 2 A , D0 f (z; ) 1

= f (z; ) ;

D f (z; )

=

Dm+1 f (z; )

=

) f (z; ) + zfz0 (z; ) = D f (z; ) ; :::;

(1

0

) Dm f (z; ) + z (Dm f (z; ))z = D (Dm f (z; )) , z 2 U; 2 U : P1 P1 m Remark 1.2 [5] If f 2 A and f (z) = z + j=2 aj ( ) z j , then Dm f (z; ) = z + j=2 [1 + (j 1) ] aj ( ) z j , z 2 U; 2 U . (1

De…nition 1.3 [4] For f 2 A , m 2 N; the operator Rm is de…ned by Rm : A ! A , R0 f (z; ) 1

R f (z; ) (m + 1) Rm+1 f (z; )

= f (z; ) ; = zfz0 (z; ) ; :::; 0

= z (Rm f (z; ))z + mRm f (z; ) , 1

62

z 2 U;

2 U:

LUPAS: Using Salagean and Ruscheweyh operators

Remark 1.4 [4] If f 2 A , f (z; ) = z + 2 U:

P1

j=2

aj ( ) z j , then Rm f (z; ) = z +

P1

j=2

m Cm+j

1 aj

( ) z j , z 2 U;

As a dual notion of strong di¤erential subordination G.I. Oros has introduced and developed the notion of strong di¤erential superordinations in [9]. De…nition 1.5 [9] Let f (z; ), H (z; ) analytic in U U : The function f (z; ) is said to be strongly superordinate to H (z; ) if there exists a function w analytic in U , with w (0) = 0 and jw (z)j < 1; such that H (z; ) = f (w (z) ; ) ; for all 2 U . In such a case we write H (z; ) f (z; ) ; z 2 U; 2 U : Remark 1.6 [9] (i) Since f (z; ) is analytic in U U , for all 2 U ; and univalent in U; for all 2 U , De…nition 1.5 is equivalent to H (0; ) = f (0; ) ; for all 2 U ; and H U U f U U : (ii) If H (z; ) H (z) and f (z; ) f (z) ; the strong superordination becomes the usual notion of superordination. De…nition 1.7 [6] We denote by Q the set of functions that are analytic and injective on U U nE (f; ), where E (f; ) = fy 2 @U : lim f (z; ) = 1g; and are such that fz0 (y; ) 6= 0 for y 2 @U U nE (f; ). The z!y

subclass of Q for which f (0; ) = a is denoted by Q (a). We have need the following lemmas to study the strong di¤erential superordinations. Lemma 1.8 [6] Let h (z; ) be a convex function with h(0; ) = a and let 2 C be a complex number with Re 0. If p 2 H [a; n; ] \ Q ; p(z; ) + 1 zp0z (z; ) is univalent in U U and h(z; ) p(z; ) + 1 zp0z (z; ); Rz 1 z 2 U; 2 U ; then q(z; ) p(z; ); z 2 U; 2 U ; where q(z; ) = h (t; ) t n dt; z 2 U; 2 U : The nz n 0 function q is convex and is the best subordinant. Lemma 1.9 [6] Let q (z; ) be a convex function in U U and let h(z; ) = q(z; ) + 1 zqz0 (z; ); z 2 U; 2 U ; where Re 0. If p 2 H [a; n; ] \ Q , p(z; ) + 1 zp0z (z; ) is univalent in U U and q(z; ) + 1 zqz0 (z; ) Rz p(z; ) + 1 zp0z (z; ) ; z 2 U; 2 U ; then q(z; ) p(z; ); z 2 U; 2 U ; where q(z; ) = h (t; ) t n 1 dt; 0 nz n

z 2 U,

2

2 U : The function q is the best subordinant.

Main results

De…nition 2.1 [2] Let 0 and m 2 N[f0g. Denote by DRm the operator given by the Hadamard product (the convolution product) of the extended generalized S¼al¼agean operator Dm and the extended Ruscheweyh operator Rm , DRm : A ! A , DRm f (z; ) = (Dm Rm ) f (z; ) : P1 Remark 2.2 [2] If f 2 A , f (z; ) = z + j=2 aj ( ) z j ; then P 1 m m DRm f (z; ) = z + j=2 Cm+j 1) ] a2j ( ) z j ; z 2 U; 2 U : 1 [1 + (j Remark 2.3 For = 1 we obtain the Hadamard product SRm ([1], [3], [7], [8]) of the extended S¼al¼agean operator S m and extended Ruscheweyh operator Rm .

0; f (z; ) 2 A ; Theorem 2.4 Let h (z; ) be a convex function in U U with h (0; ) = 1. Let m 2 N, Rz c 0 m F (z; ) = Ic (f ) (z; ) = zc+2 t f (t; ) dt, z 2 U; 2 U , Re c > 2; and suppose that (DR f (z; ))z is c+1 0 0 m univalent in U U , (DR F (z; ))z 2 H [1; 1; ] \ Q and 0

z 2 U;

2 U;

0

z 2 U;

2 U;

h (z; )

(DRm f (z; ))z ,

q (z; )

(DRm F (z; ))z ,

then where q(z; ) =

c+2 z c+2

Rz 0

h(t; )tc+1 dt: The function q is convex and it is the best subordinant. 2

63

(1)

LUPAS: Using Salagean and Ruscheweyh operators

Rz Proof. We have z c+1 F (z; ) = (c + 2) 0 tc f (t; ) dt and di¤erentiating it, with respect to z, we obtain 0 (c + 1) F (z; )+zFz0 (z; ) = (c + 2) f (z; ) and (c + 1) DRm F (z; )+z (DRm F (z; ))z = (c + 2) DRm f (z; ) ; z 2 U; 2 U : Di¤erentiating the last relation with respect to z we have 0

(DRm F (z; ))z +

1 00 0 z (DRm F (z; ))z2 = (DRm f (z; ))z , z 2 U; c+2

2 U:

(2)

Using (2), the strong di¤erential superordination (1) becomes 0

(DRm F (z; ))z +

h (z; )

1 00 z (DRm F (z; ))z2 : c+2

(3)

0

(4)

Denote p (z; ) = (DRm F (z; ))z ; z 2 U; 1 0 c+2 zpz

2 U:

(z; ), z 2 U;

Replacing (4) in (3) we obtain h (z; ) p (z; ) + Using Lemma 1.8 for n = 1 and = c + 2; we have

2 U: 0

q (z; ) p (z; ) ; z 2 U; 2 U ; i.e. q (z; ) (DRm F (z; ))z , z 2 U; 2 U ; Rz h(t; )tc+1 dt: The function q is convex and it is the best subordinant. where q(z; ) = zc+2 c+2 0

Corollary 2.5 Let h (z; ) = +(21+z )z ; where 2 [0; 1). Let m 2 N, 0; f (z; ) 2 A ; F (z; ) = Rz c 0 m t f (t; ) dt, z 2 U; 2 U ; Re c > 2; and suppose that (DR f (z; ))z is univalent in Ic (f ) (z; ) = zc+2 c+1 0 0 m U U , (DR F (z; ))z 2 H [1; 1; ] \ Q and h (z; )

0

(DRm f (z; ))z , z 2 U;

2 U;

(5)

then 0

(DRm F (z; ))z , z 2 U; 2 U ; ) R z tc+1 + 2(c+2)( dt; z 2 U; 2 U : The function q is convex and it is the z c+2 0 t+1

q (z; ) where q is given by q(z; ) = 2 best subordinant.

0

Proof. Following the same steps as in the proof of Theorem 2.4 and considering p(z; ) = (DRm F (z; ))z , 1 the strong di¤erential superordination (5) becomes h(z; ) = +(21+z )z p (z; )+ c+2 zp0z (z; ) ; z 2 U; 2 U : Rz By using Lemma 1.8 for n = 1 and = c + 2, we have q(z; ) p(z; ), i.e. q(z; ) = zc+2 h(t; )tc+1 dt = c+2 0 R R c+1 z z 0 +(2 )t c+1 ) c+2 t t dt = 2 + 2(c+2)( dt (DRm F (z; ))z ; z 2 U; 2 U : The function q is z c+2 0 1+t z c+2 0 t+1 convex and it is the best subordinant. 1 Theorem 2.6 Let q (z; ) be a convex function in U U and let h (z; ) = q (z; ) + c+2 zqz0 (z; ) ; where z 2 U; 2 U ; Re c > 2: Rz c Let m 2 N, 0; f (z; ) 2 A ; F (z; ) = Ic (f ) (z; ) = zc+2 t f (t; ) dt, z 2 U; 2 U ; and suppose c+1 0 0 0 that (DRm f (z; ))z is univalent in U U , (DRm F (z; ))z 2 H [1; 1; ] \ Q and 0

h (z; )

(DRm f (z; ))z ,

q (z; )

(DRm F (z; ))z ,

z 2 U;

2 U;

z 2 U;

2 U;

(6)

then where q(z; ) =

c+2 z c+2

Rz 0

c+1

h(t; )t

0

dt: The function q is the best subordinant. 0

Proof. Following the same steps as in the proof of Theorem 2.4 and considering p (z; ) = (DRm F (z; ))z ; 1 z 2 U; 2 U ; the strong di¤erential superordination (6) becomes h (z; ) = q (z; ) + c+2 zqz0 (z; ) 1 0 p (z; ) + c+2 zpz (z; ), z 2 U; 2 U : Using Lemma 1.9 for n = 1 and = c + 2; we have 0

q (z; ) p (z; ) ; z 2 U; 2 U ; i.e. q (z; ) (DRm F (z; ))z , z 2 U; Rz where q(z; ) = zc+2 h(t; )tc+1 dt: The function q is the best subordinant. c+2 0 3

64

2 U;

LUPAS: Using Salagean and Ruscheweyh operators

Theorem 2.7 Let h (z; ) be a convex function, h(0; ) = 1: Let m

(DR f (z;

0 ))z

is univalent and

DRm f (z; ) z

0; m 2 N; f (z; ) 2 A and suppose that

2 H [1; 1; ] \ Q . If 0

(DRm f (z; ))z ;

h(z; )

2 U;

z 2 U;

(7)

then

DRm f (z; ) ; z 2 U; 2 U ; z R z where q(z; ) = z1 0 h(t; )dt: The function q is convex and it is the best subordinant. q(z; )

P z+ 1 C m

DRm f (z; )

[1+(j 1) ]m a2 ( )z j

m+j 1 j j=2 Proof. Consider p (z; ) = = z z P1 m m 1 + j=2 Cm+j 1) ] a2j ( ) z j 1 : Evidently p 2 H [1; 1; ]. 1 [1 + (j 0 0 We have p (z; ) + zpz (z; ) = (DRm f (z; ))z , z 2 U; 2 U . 0 Then (7) becomes h(z; ) p(z; ) + zpz (z; ); z 2 U; 2 U : By using Lemma 1.8 for n = 1 and = 1, we have

=

DRm f (z; ) q(z; ) p(z; ); z 2 U; 2 U ; i.e. q(z; ) ; z 2 U; z Rz where q(z; ) = z1 0 h(t; )dt: The function q is convex and it is the best subordinant.

Corollary 2.8 Let h(z; ) =

+(2 )z 1+z m

be a convex function in U 0 ))z

f (z; ) 2 A and suppose that (DR f (z; then q(z; ) +

0 ))z

(DRm f (z;

h(z; )

where q is given by q(z; ) = 2 best subordinant.

is univalent and

2( z

;

U , where 0

DRm f (z; ) z

z 2 U;

2 U;

< 1. Let

0; m 2 N;

2 H [1; 1; ] \ Q . If 2 U;

(8)

DRm f (z; ) ; z 2 U; 2 U ; z ) ln (1 + z) ; z 2 U; 2 U : The function q is convex and it is the DRm f (z; ) , z

Proof. Following the same steps as in the proof of Theorem 2.7 and considering p(z; ) = p(z; ) + zp0z (z; ); z 2 U; strong di¤erential superordination (8) becomes h(z; ) = +(21+z )z Rz Rz using Lemma 1.8 for n = 1 and = 1, we have q(z; ) p(z; ), i.e. q(z; ) = z1 0 h (t; ) dt = z1 0

=2

+ 2(

)

z

ln (1 + z)

m

DR f (z; ) ; z

z 2 U;

the 2 U : By

+(2 )t dt 1+t

2 U : The function q is convex and it is the best subordinant.

0, Theorem 2.9 Let q (z; ) be convex in U U and let h be de…ned by h (z; ) = q (z; ) + zqz0 (z; ) : If DRm f (z; ) 0 m 2 N, f (z; ) 2 A , suppose that (DRm f (z; ))z is univalent, 2 H [1; 1; ] \ Q and satis…es the z strong di¤ erential superordination 0

h(z; ) = q (z; ) + zqz0 (z; )

(DRm f (z; ))z ;

z 2 U;

2 U;

(9)

then

DRm f (z; ) ; z 2 U; z R z where q(z; ) = z1 0 h(t; )dt: The function q is the best subordinant. q(z; )

DRm f (z; )

P z+ 1 C m

[1+(j 1) ]m a2 ( )z j

2 U;

m+j 1 j j=2 Proof. Let p (z; ) = = = z z P1 m 2 m j 1 1 + j=2 Cm+j 1 [1 + (j 1) ] aj ( ) z : Evidently p 2 H [1; 1; ]. 0 Di¤erentiating, we obtain p(z; ) + zp0z (z; ) = (DRm f (z; ))z ; z 2 U; 2 U ; and (9) becomes q(z; ) + zqz0 (z; ) p(z; ) + zp0z (z; ) ; z 2 U; 2 U : Using Lemma 1.9 for n = 1 and = 1, we have Z 1 z DRm f (z; ) q(z; ) p(z; ); z 2 U; 2 U ; i.e. q(z; ) = h(t; )dt ; z 2 U; 2 U ; z 0 z

and q is the best subordinant.

4

65

LUPAS: Using Salagean and Ruscheweyh operators

Theorem 2.10 Let h (z; ) be a convex function, h(0; ) = 1: Let 0

zDRm+1 f (z; ) DRm f (z; )

DRm+1 f (z; ) DRm f (z; )

is univalent and

z

then

1 z

Rz 0

P 1+ 1 C m+1 [1+(j 1) ]m+1 a2j ( )z j Pj=2 m+j m 2 m j 1+ 1 j=2 Cm+j 1 [1+(j 1) ] aj ( )z

(DR

m+1

q(z; ) 1 z

DRm+1 f (z; ) DRm f (z; )

Rz 0

=

1 1

z 2 U;

2 U;

(10)

2 U;

z 2 U;

P z+ 1 C m+1 [1+(j 1) ]m+1 a2j ( )z j Pj=2 m+j m 2 m j z+ 1 j=2 Cm+j 1 [1+(j 1) ] aj ( )z

=

: Evidently p 2 H [1; 1; ]. 0

f (z; )) z DRm f (z; )

Then (10) becomes h(z; ) we have

where q(z; ) =

;

z

h (t; ) dt: The function q is convex and it is the best subordinant.

Proof. Consider p (z; ) =

We have p0z (z; ) =

0

DRm+1 f (z; ) ; DRm f (z; )

q(z; ) where q(z; ) =

2 H [1; 1; ] \ Q . If

zDRm+1 f (z; ) DRm f (z; )

h(z; )

0; m 2 N; f (z; ) 2 A and suppose that

0 (DRm f (z; ))0z zDRm+1 f (z; ) 0 : DRm f (z; ) : Then p (z; ) + zpz (z; ) = DRm f (z; ) z 0 zpz (z; ); z 2 U; 2 U : By using Lemma 1.8 for n = 1 and

p (z; ) p(z; ) +

p(z; ); z 2 U;

2 U;

DRm+1 f (z; ) ; z 2 U; DRm f (z; )

i.e. q(z; )

2 U;

h (t; ) dt: The function q is convex and it is the best subordinant. +(2 )z be a convex function in U U , 1+z 0 m+1 zDR f (z; ) DRm+1 f (z; ) is univalent, DRm f (z; ) DRm f (z; ) z

Corollary 2.11 Let h(z; ) = f (z; ) 2 A and suppose that

then

+

2(

) z

;

where 0

z 2 U;

ln (1 + z) ; z 2 U;

< 1. Let

2 U;

(11)

2 U;

2 U : The function q is convex and it is the

Proof. Following the same steps as in the proof of Theorem 2.10 and considering p(z; ) = using Lemma 1.8 for n = 1 and =2

+

) z

ln (1 + z)

= 1, we have q(z; ) DRm+1 f (z; ) DRm f (z; ) ;

z 2 U;

m+1

zDR f (z; ) DRm f (z; )

q(z; ) where q(z; ) =

1 z

Rz 0

2 U : By

2 U : The function q is convex and it is the best subordinant.

0

z

is univalent,

DRm+1 f (z; ) DRm f (z; )

zDRm+1 f (z; ) DRm f (z; )

h(z; ) = q (z; ) + zqz0 (z; ) then

the

+(2 )t dt 1+t

U and let h be de…ned by h (z; ) = q (z; ) + zqz0 (z; ) : If

Theorem 2.12 Let q (z; ) be convex in U m 2 N, f (z; ) 2 A , suppose that the strong di¤ erential superordination

+(2 )z 1+z

DRm+1 f (z; ) DRm f (z; ) ,

p(z; ) + zp0z (z; ); z 2 U; Rz Rz p(z; ), i.e. q(z; ) = z1 0 h (t; ) dt = z1 0

strong di¤erential superordination (11) becomes h(z; ) =

0, m 2 N;

2 H [1; 1; ] \ Q : If

z 2 U;

z

DRm+1 f (z; ) ; DRm f (z; )

q(z; ) where q is given by q(z; ) = 2 best subordinant.

0

zDRm+1 f (z; ) DRm f (z; )

h(z; )

2(

= 1,

DRm+1 f (z; ) ; DRm f (z; )

z 2 U;

h (t; ) dt: The function q is the best subordinant. 5

66

0 z

;

2 H [1; 1; ] \ Q and satis…es

z 2 U;

2 U;

0;

2 U;

(12)

LUPAS: Using Salagean and Ruscheweyh operators

DRm+1 f (z; ) DRm f (z; )

Proof. Let p (z; ) =

=

Evidently p 2 H [1; 1; ].

P m+1 [1+(j 1) ]m+1 a2j ( )z j z+ 1 Cm+j Pj=2 1 m 2 m j z+ j=2 Cm+j 1 [1+(j 1) ] aj ( )z

becomes q(z; q(z; )

)

p(z;

p(z; ); z 2 U;

) + zp0z

(z; ) ; z 2 U; 1 z

2 U ; i.e. q(z; ) =

P m+1 1+ 1 Cm+j [1+(j 1) ]m+1 a2j ( )z j Pj=2 1 m 2 m j 1+ j=2 Cm+j 1 [1+(j 1) ] aj ( )z

zDRm+1 f (z; ) DRm f (z; )

Di¤erentiating with respect to z, we obtain p(z; ) + zp0z (z; ) = ) + zqz0 (z;

=

0

z

; z 2 U;

2 U : Using Lemma 1.9 for n = 1 and

Z

1 1

:

2 U ; and (12) = 1, we have

DRm+1 f (z; ) ; z 2 U; 2 U ; DRm f (z; )

z

h (t; ) dt

0

and q is the best subordinant. Theorem 2.13 Let h (z; ) be a convex function, h(0; ) = 1: Let m+1 m+1 f (m +1)z DR

m(1 ) m (m +1)z DR f

(z; )

0

(z; ) is univalent and (DR f (z; ))z 2 H [1; 1; ] \ Q . If

m+1 DRm+1 f (z; ) (m + 1) z

h(z; )

0; m 2 N; f (z; ) 2 A and suppose that

m

m (1 ) DRm f (z; ) ; (m + 1) z

z 2 U;

2 U;

(13)

then 0

(DRm f (z; ))z ;

q(z; ) where q(z; ) =

m +1 z

m +1

Rz 0

h (t; ) t

(m

1) +1

z 2 U;

2 U;

dt: The function q is convex and it is the best subordinant. 0

Proof. With notation p (z; ) = (DRm f (z; ))z = 1 + P1 p (0; ) = 1, we obtain for f (z; ) = z + j=2 aj ( ) z j ;

P1

j=2

m Cm+j

1

m

1) ] ja2j ( ) z j

[1 + (j

1

and

0

m+1 p (z; ) + zp0z (z; ) = m+1 f (z; ) m 1 + 1 (DRm f (z; ))z m(1z ) DRm f (z; ) and z DR m(1 ) m+1 m+1 0 m f (z; ) (m p (z; ) + m +1 zpz (z; ) = (m +1)z DR +1)z DR f (z; ) : Evidently p 2 H [1; 1; ].

Then (13) becomes h(z; ) = m + 1 , we have q(z; )

where q(z; ) =

m +1 z

m +1

Rz 0

p(z; ); z 2 U; h (t; ) t

Corollary 2.14 Let h(z; ) = f (z; ) 2 A and suppose that H [1; 1; ] \ Q : If h(z; )

p (z; ) +

(m

1) +1

0 m +1 zpz

(z; ) ; z 2 U;

2 U : By using Lemma 1.8 for n = 1 and 0

(DRm f (z; ))z ; z 2 U;

2 U ; i.e. q(z; )

2 U;

dt: The function q is convex and it is the best subordinant.

+(2 )z be a convex 1+z m+1 m+1 DR f (z; (m +1)z

function in U

U , where 0

0, m 2 N;

(z; ) is univalent, (DRm f (z; ))z 2

m (1 ) DRm f (z; ) ; (m + 1) z

m+1 DRm+1 f (z; ) (m + 1) z

< 1. Let

0

m(1 ) m (m +1)z DR f

)

z 2 U;

2 U;

(14)

then 0

(DRm f (z; ))z ;

q(z; ) where q is given by q(z; ) = 2 it is the best subordinant.

+

2(

)(m z

+1) R z

m +1

t

m +1

1+t

0

z 2 U; 1

2 U;

dt; z 2 U;

2 U : The function q is convex and 0

Proof. Following the same steps as in the proof of Theorem 2.13 and considering p(z; ) = (DRm f (z; ))z , the strong di¤erential superordination (14) becomes h(z; ) = +(21+z )z p(z; )+zp0z (z; ); z 2 U; 2 U : By Rz (m 1) +1 using Lemma 1.8 for n = 1 and = m +1 , we have q(z; ) p(z; ), i.e. q(z; ) = mm+1 h (t; ) t dt +1 0 z m +1 1 R R (m 1) +1 z 0 +(2 )t )(m +1) z t dt = 2 + 2( m dt (DRm f (z; ))z ; z 2 U; 2 U : = mm+1 t +1 +1 1+t 1+t 0 0 z

z

The function q is convex and it is the best subordinant.

6

67

LUPAS: Using Salagean and Ruscheweyh operators

Theorem 2.15 Let q (z; ) be convex in U

U and let h be de…ned by h (z; ) = q (z; ) +

0; m; n 2 N. If f (z; ) 2 An ; suppose that m

(DR f (z;

0 ))z

m+1 m+1 f (m +1)z DR

(z; )

m(1 ) m (m +1)z DR f

0 m +1 zqz

(z; ),

(z; ) is univalent and

2 H [1; 1; ] \ Q and satis…es the strong di¤ erential superordination zq 0 (z; ) m +1 z m (1 ) DRm f (z; ) ; z 2 U; 2 U ; (m + 1) z

h(z; ) = q (z; ) + m+1 DRm+1 f (z; ) (m + 1) z

(15)

then q(z; ) where q(z; ) =

m +1 z

m +1

Rz 0

h (t; ) t

(m

1) +1

0

(DRm f (z; ))z ;

z 2 U;

2 U;

dt: The function q is the best subordinant.

P1 m 0 m 1) ] ja2j ( ) z j 1 . Proof. Let p (z; ) = (DRm f (z; )) = 1 + j=2 Cm+j 1 [1 + (j Di¤erentiating, we obtain 0 m+1 p (z; ) + zp0z (z; ) = m+1 f (z; ) m 1 + 1 (DRm f (z; ))z m(1z ) DRn f (z; ) and z DR m(1 ) m+1 p (z; ) + m +1 zp0z (z; ) = (mm+1 f (z; ) (m +1)z DRm f (z; ) ; z 2 U; 2 U ; and (15) becomes +1)z DR q(z; ) +

0 m +1 zqz (z;

we have q(z; )

)

2 U : Using Lemma 1.9 for n = 1 and = m + 1 , Rz (m 1) +1 0 h (t; ) t dt (DRm f (z; ))z , z 2 U; 2 U ; i.e. q(z; ) = mm+1 +1 0

p(z; ) +

p(z; ); z 2 U;

0 m +1 zpz

(z; ) ; z 2 U; z

2 U ; and q is the best subordinant.

References [1] A. Alb Lupa¸s, Certain strong di¤ erential subordinations using S¼al¼agean and Ruscheweyh operators, Advances in Applied Mathematical Analysis, Volume 6, Number 1 (2011), 27–34. [2] A. Alb Lupa¸s, A note on strong di¤ erential subordinations using a generalized S¼al¼agean operator and Ruscheweyh operator, Bulletin of Mathematical Analysis and Applications, submitted 2010. [3] A. Alb Lupa¸s, A note on strong di¤ erential subordinations using S¼al¼agean and Ruscheweyh operators, Libertas Mathematica, submitted 2010. [4] A. Alb Lupa¸s, G.I. Oros, Gh. Oros, On special strong di¤ erential subordinations using S¼al¼agean and Ruscheweyh operators, Journal of Computational Analysis and Applications, Vol. 14, 2012 (to appear). [5] A. Alb Lupa¸s, On special strong di¤ erential subordinations using a generalized S¼al¼agean operator and Ruscheweyh derivative, submitted 2011. [6] A. Alb Lupa¸s, On special strong di¤ erential superordinations using S¼al¼agean and Ruscheweyh operators, submitted 2011. [7] A. Alb Lupa¸s, Certain strong di¤ erential superordinations using S¼al¼agean and Ruscheweyh operators, submitted 2011. [8] A. Alb Lupa¸s, A note on strong di¤ erential superordinations using S¼al¼agean and Ruscheweyh operators, submitted 2011. [9] G.I. Oros, Strong di¤ erential superordination, Acta Universitatis Apulensis, Nr. 19, 2009, 101-106. [10] G.I. Oros, On a new strong di¤ erential subordination, (to appear).

7

68

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,69-80,COPYRIGHT 2012 EUDOXUS PRESS,LLC

A NEW FIXED POINT THEOREM FOR m MAPINGS ON m COMPLETE METRIC SPACES Luljeta Kikina and Kristaq Kikina Department of Mathematics and Computer Science Faculty of Natural Sciences, University of Gjirokastra Albania [email protected], [email protected] Abstract: A new fixed point theorem for m ∈ N mappings, satisfying implicit relations, was proved. This result generalizes and unifies several of well-known fixed point theorems for one, two and three mappings and extends then to an arbitrary number m of mappings. Keywords: Cauchy sequence, complete metric space, fixed point, implicit relation. Mathematics Subject Classification: 47H10, 54H25 1. INTRODUCTION In [1], [2], [8], [11] etc, are proved fixed point theorems on metric spaces for mappings satisfying implicit relations. In this paper, we will prove a related fixed point theorem for m mappings on m metric spaces, m − 1 of mappings must be continuous. This result generalizes and unifies the theorems of Rhoades [10], Banach [3], Kannan [7], Bianchini [4], Reich [9], Fisher [5], Jain et al [6], etc. and in the same time, it extends them for an arbitrary number m of mappings. These generalizations and unifications have been done using a wide class of implicit relations. In [10], [5] and [6], the following theorems are proved. Theorem 1.1 (Rhoades [10]) Let ( X , d ) be a complete metric space and T : X → X a self map of X. If for some c ∈[0,1) we have

d (Tx, Ty ) ≤ c max{d ( x, y ), d ( x, Tx), d ( y, Ty )} for all x, y ∈ X , then T has a unique fixed point α in X. Theorem 2.2 Fisher [5]) Let ( X , d ) and (Y , ρ ) are complete metric spaces and S : X → Y , R : Y → X be two maps, at least one of them being continuous. If for some

c ∈ (0,1) the following inequalities are satisfied:

d ( RSx, RSx ') ≤ c max{d ( x, x ') , d ( x, RSx ) , d ( x ', RSx ') , ρ ( Sx, Sx ')}

ρ ( SRy, SRy ') ≤ c max{ρ ( y, y ') , ρ ( y, SRy ) , ρ ( y ', SRy ') , d ( Ry, Ry ')}

69

1

KIKINA: FIXED POINT THEOREM

for all x, x'∈X; y, y'∈Y , then RS has a unique fixed point α ∈ X and SR has a unique fixed point β ∈ Y . Moreover, Sα = β and. Rβ = α Theorem 1.3 ( Jain et al [6] ) Let ( X , d ), (Y , ρ ) and ( Z , σ ) be complete metric spaces. If T is a continuous mapping of X into Y , S is a continuous mapping of Y into Z and R is a mapping of Z into X satisfying the inequalities

d ( RSTx, RSTx′) ≤ c max{d ( x, x′), d ( x, RSTx), d ( x′, RSTx′) ρ (Tx, Tx′), σ (STx, STx′)} ρ (TRSy, TRSy′) ≤ c max{ρ ( y, y′), ρ ( y, TRSy), ρ ( y′, TRSy′)σ (Sy, Sy′), d ( RSy, RSy′)}

σ (STRz, STRz′) ≤ c max{σ ( z, z′), σ ( z, STRz), σ ( z′, STRz′), d ( Rz, Rz′), ρ (TRz, TRz′)}

for all x, x′ ∈ X ; y, y′ ∈ Y and z , z ′ ∈ Z where 0 ≤ c < 1 . Then RST has a unique fixed point u in X , TRS has a unique fixed point v in Y and STR has a unique fixed point w in Z . Further, Tu = v, Sv = w and Rw = u .

2. MAIN RESULTS Before stating the main theorem we define new classes implicit functions, whose role will be crucial. Let R+ = [0, +∞) . We denote by Φ k , for k ≥ 4 , the set of all functions with k variables

ϕ : R+k → R satisfying the properties: (a). ϕ is upper semi-continuous in each coordinate variable t1 , t2 ,..., tk (b). If ϕ (u , v, v, u , v1 , v2 ,..., vk − 4 ) ≤ 0 or ϕ (u , v, u , v, v1 , v2 ,..., vk − 4 ) ≤ 0 or ϕ (u , u , v, v, v1 , v2 ,..., vk − 4 ) ≤ 0 , for all u, v, v1 , v2 ,..., vk −4 ≥ 0 , then there exists a real constant 0 ≤ c < 1 such that u ≤ c max{v, v1 , v2 ,..., vk − 4 } . Every such function will be called a Φ k -function with constant c. Note: If k1 < k2 , then Φ k1 ⊂ Φ k2 . In the case k = 4 , we have

ϕ (u , v, v, u , v1 , v2 ,..., vk − 4 ) = ϕ (u , v, v, u ) and max{v, v1 , v2 ,..., vk − 4 } = v Example 2.1 The function ϕ (t1 , t2 ,..., tk ) = t1p − c max{t2p , t3p ,..., tkp } , where 0 ≤ c < 1 and p > 0 , is Φ k -function with constant c. Proof: (a) is clear since ϕ is continuous. Suppose that u , v, v1 , v2 ,..., vk − 4 ≥ 0 and then

ϕ (u, v, v, u, v1 , v2 ,..., vk − 4 ) = u p − c max{v p , v p , u p , v1p , v2p ,..., vkp− 4 } = = u p − c max{u p , v p , v1p , v2p ,..., vkp− 4 } ≤ 0 If u p ≥ max{v p , v1p , v2p ,..., vkp− 4 }, then u p ≤ c max{u p , v p , v1p , v2p ,..., vkp− 4 } = cu p < u p , a contradiction. Therefore, u p ≤ c max{v p , v1p , v2p ,..., vkp− 4 } and so u ≤ c1 max{v, v1 , v2 ,..., vk − 4 } where c1 = c < 1 . Similarly, if ϕ (u , v, u , v, v1 , v2 ,..., vk − 4 ) ≤ 0 or p

70

2

KIKINA: FIXED POINT THEOREM

ϕ (u, u, v, v, v1 , v2 ,..., vk − 4 ) ≤ 0 , then u ≤ c1 max{v, v1 , v2 ,..., vk −4 } .

The proof of (b) is completed. 1 Example 2.2 The function ϕ (t1 , t2 ,..., tk ) = t1 − (a2t2p + a3t p3 +... + ak tkp ) p , where p > 0 k

k

i =2

i=2

and 0 ≤ ai , ∑ ai < 1, i = 2,3,..., k , is Φ k -function with constant c = ∑ ai

Proof: (a) is clear since ϕ is continuous. Suppose that u , v, v1 , v2 ,..., vk −4 ≥ 0 and then 1

ϕ (u, v, v, u, v1 , v2 ,..., vk − 4 ) = u − (a2 v p + a3v p + a4u p + a5v1p ... + ak vkp− 4 ) ≤ 0 p

If u p ≥ max{v p , v1p , v2p ,..., vkp− 4 }, then 1

1

u ≤ (a2v p + a3v p + a4u p + a5v1p + ... + ak vkp−4 ) p ≤ (a2u p + a3u p + a4u p + a5u p ... + ak u p ) p = 1

1

= [(a2 + a3 + a4 + ... + ak )u p ] p = (a2 + a3 + a4 + ... + ak ) p u = cu < u , 1

a contradiction, where c = (a2 + a3 + a4 + ... + ak ) p < 1 . Therefore, 1

u ≤ [(a2 + a3 + a4 + ... + ak ) max{v p , v1p , v2p ,..., vkp− 4 }] p = c max{v, v1 , v2 ,..., vk − 4 } . Similarly,

if ϕ (u , v, u , v, v1 , v2 ,..., vk −4 ) ≤ 0 or ϕ (u , u , v, v, v1 , v2 ,..., vk −4 ) ≤ 0 , then

u ≤ c1 max{v, v1 , v2 ,..., vk −4 } . The proof of (b) is completed. We denote by Fk the set of all continuous functions with k variables f : R+k → R satisfying the properties: (a’). f is non decreasing in respect with each variable. (b’). f (t , t ,..., t ) ≤ t , t ∈ R+ Every such function will be called a Fk -function. Denote I k = {1, 2,..., k} . Some examples of Fk -function are as follows:

1. f (t1 , t2 ,..., tk ) = max{t1 , t2 ,..., tk } 1

2. f (t1 , t2 ,..., tk ) = [max{t1t2 , t2t3 ,..., tk −1tk , tk t1}] 2 1

3. f (t1 , t2 ,..., tk ) = [max{t1p , t2p ,..., tkp }] p , p > 0 a t + a t + ... + ak tk 4. f (t1 , t2 ,..., tk ) = 1 1 2 2 , where p > 0 and ai ≥ 0 a1 + a2 + ... + ak The following relationship between Fk −1 -functions and Φ k -functions holds:

Lemma 2.3 If f ∈ Fk −1 and 0 ≤ c < 1 , then the function ϕ(t1,t2,...,tk ) = t1 −cf (t2,t3,...,tk ) is Φ k -function with constant c Proof. (a) is clear since ϕ is continuous. Suppose that u , v, v1 , v2 ,..., vk − 4 ≥ 0 and then ϕ (u, v, v, u, v1 , v2 ,..., vk −4 ) = u − cf (v, v, u , v1 ,..., vk −4 ) ≤ o (*) We have u ≤ max{v, v1 , v2 ,..., vk −4 } since in contrary, ( if u > max{v, v1 , v2 ,..., vk − 4 } ), by using the properties of f we get: f (v, v, u , v1 ,..., vk − 4 ) ≤ f (u , u ,..., u ) ≤ u and by (*) it follows u ≤ cu < u , a contradiction. Therefore, after replacing the coordinates of the point

71

3

KIKINA: FIXED POINT THEOREM

(v, v, u , v1 , v2 ,..., vk − 4 ) by max{v, v1 , v2 ,..., vk −4 } and using the properties of f we get

u ≤ c max{v, v1 , v2 ,..., vk − 4 } .

Similarly,

ϕ (u, v, u , v, v1 , v2 ,..., vk − 4 ) ≤ 0

if

or

ϕ (u, u, v, v, v1 , v2 ,..., vk − 4 ) ≤ 0 , then u ≤ c max{v, v1 , v2 ,..., vk − 4 } . The proof of (b) is

completed.

The above lemma gives us the possibility to establish other functions of type Φ k :

Example 2.4 ϕ (t1 , t2 ,..., tk ) = t1 − c[max{t2t3 , t3t4 ,..., tk −1tk }] , where 0 ≤ c < 1 . 1

Example 2.5 ϕ ( t1 , t2 ,..., t k ) = t1 − c

2

t2 + t3 + ... + tk , where 0 ≤ c < 1 etc. k −1

Now, we prove the following theorem for m mappings on m metric spaces. Theorem 2.6 Let ( X i , di ) be m complete metric spaces and Ti m mappings such that Ti : X i → X i +1 for i = 1, 2,..., m − 1, Tm : X m → X 1 and from which ( m − 1 ) are continuous. If satisfying the inequalities:

( ( (

) ( ) ) ( )

 d1 TmTm −1...T2T1 x1 , TmTm −1...T2T1 x1' , d1 x1 , x1' , d1 ( x1 , TmTm −1...T1 x1 ) ,    ' ' ' '  ≤0 ϕ1 d1 x1 , TmTm −1...T1 x1 , d 2 T1 x1 , T1 x1 , d 3 T2T1 x1 , T2T1 x1 ,...,    d T T ...T x , T T ...T x '   m m −1 m − 2 1 1 m −1 m − 2 1 1 

) (

)

(1)

for all x1 , x1' ∈ X 1

( ( (

) ( ) ) ( ) (

 d 2 T1Tm ...T3T2 x2 , T1Tm ...T3T2 x2' , d 2 x2 , x2' , d 2 ( x2 , T1Tm ...T2 x2 ) ,  ϕ 2  d 2 x2' , T1Tm ...T2 x2' , d3 T2 x2 , T2 x2' , d 4 T3T2 x2 , T3T2 x2' ,...,   d T T ...T x , T T ...T x ' , d T T ...T x , T T ...T x '  m m −1 m − 2 2 2 m −1 m − 2 2 2 1 m m −1 2 2 m m −1 2 2

) (

)

)

  ≤0   

(2)

for all x2 , x2' ∈ X 2 , in general

(

) (

) )

' '  di Ti −1Ti −2 ...TT 1 mTm−1...Ti xi , Ti −1Ti − 2 ...TT 1 mTm−1...Ti xi , di xi , xi ,   d ( x , T T ...TT T ...T x ) , d x' , T T ...TT T ...T x' , i i i −1 i − 2 1 m m−1 i i i i i −1 i −2 1 m m−1 i i ϕi   d T x , T x' ,..., d T T ...T x , T T ...T x' , d T T ...T x , T T ...T x' , m m−1 m−2 i i m−1 m−2 i i 1 m m−1 i i m m−1 i i  i +1 i i i i  d TT T ...T x , TT T ...T x' ,..., d T T ...TT T ...T x , T T ...TT T ...T x' i −1 i −2 i −3 1 m m−1 i i i − 2 i −3 1 m m−1 i i  2 1 m m−1 i i 1 m m−1 i i

(

(

)

(

(

)

) (

(

)

)

    (i)    

for all xi , xi' ∈ X i for i = 3,..., m − 1 , and

72

4

KIKINA: FIXED POINT THEOREM

( (

) ( ) ) ( )

' '  dm Tm−1Tm−2...TT  1 m xm ,Tm−1Tm−2...TT 1 m xm , dm xm , xm , dm ( xm ,Tm−1Tm−2 ...TT 1 m xm ) ,   ' ' ' '   ϕm dm xm,Tm−1Tm−2...TT 1 m xm , d1 Tm xm ,Tm xm , d2 TT 1 m xm ,TT 1 m xm ,...,   '  d T T ...TT x ,T T ...TT x   m−1 m−2 m−3 1 m m m−2 m−3 1 m m 

) (

(

)

(m)

for all xm , xm' ∈ X m , where ϕi ∈ Φ m +3 for i = 1, 2,..., m . Then the maps

TmTm −1...T1 , T1TmTm −1...T2 , …, Ti −1Ti − 2 ...T1TmTm −1...Ti ,…, Tm −1Tm − 2 ...T1Tm have unique fixed point α1 ∈ X 1 , α 2 ∈ X 2 , …, α i ∈ X i , …, α m ∈ X m , respectively. Further, Tiα i = α i +1 for i = 1,..., m − 1 and Tmα m = α1

Proof. Let x 0(1 ) be an arbitrary point in X 1 . We define the sequences

{ x } , { x } , ..., { x } , ..., { x } i n (1 ) n

(2) n

(i) n

(m ) n

X 1 , X 2 , ..., X i , ..., X

m

respectively, as

follows:

xn(1) = (TmTm −1 ...T1 ) n x0(1) , xn(2) = T1 xn(1)−1 ,..., xn( i ) = Ti −1 xn( i −1) ,..., xn( m ) = Tm −1 xn( m −1) , n ∈ N

{ }

We prove that xn(i ) are Cauchy sequences for i=1, 2… m. Denote

d n( i ) = d i ( x n( i ) , x n( i+)1 ), i = 1, 2 , ..., m . We will assume that xn( i ) ≠ xn( i+)1 for all n. Otherwise, if xn(1) = xn(1)+1 for some n, then

xn( i+)1 = xn( i+)2 for i=2, 3,…, m and we could put α i = xn( i+)1 . Applying the inequality (2) for x2 = xn(2)−1 and x2′ = xn(2) , we have:

( ( (

)

(

)

(

)

 d 2 T1Tm ...T3T2 x n( 2−)1 , T1Tm ...T2 x n( 2 ) , d 2 x n( 2−)1 , x n( 2 ) , d 2 x n( 2−)1 , T1Tm ...T2 x n( 2−)1 ,  ϕ 2  d 2 x n( 2 ) , T1Tm ...T2 x n( 2 ) , d 3 T2 x n( 2−)1 , T2 x n( 2 ) , d 4 T3T2 x n( 2−)1 , T3T2 x n( 2 ) , ...,   d T T ...T x ( 2 ) , T T ...T x ( 2 ) , d T T ...T x ( 2 ) , T T ...T x ( 2 ) 1 m m −1 2 n −1 m m −1 2 n  m m −1 m − 2 2 n −1 m −1 m − 2 2 n

)

(

)

) (

(

(

)

)

)

  =   

(4) (m) (1) = ϕ 2 d n( 2 ) , d n( 2−1) , d n( 2−1) , d n( 2 ) , d n( 3) −1 , d n −1 , ..., d n −1 , d n −1 ≤ 0

{

}

(2) (2) (3) (4) ( m) (1) and from (b), we have dn ≤ c max dn−1 , dn−1 , dn−1 ,..., dn−1 , dn−1 or

{

dn(2) ≤ c max dn(1)−1 , dn(2)−1 , dn(3)−1 , dn(4)−1 ,..., dn( m−1)

}

(2’)

We have denoted by c = max{c1 , c2 ,..., cm } , where ci is the constant of Φ k -function ϕi . Applying inequality (i) for xi = xn( i−)1 and xi′ = xn( i ) , we obtain:

73

5

KIKINA: FIXED POINT THEOREM

( (

) (

) )

 d i Ti −1Ti − 2 ...T1TmTm −1...Ti xn( i−)1 , Ti −1Ti − 2 ...T1TmTm −1...Ti xn( i ) , di xn( i−)1 , xn( i ) ,   d x ( i ) , T T ...T T T ...T x ( i ) , d x ( i ) , T T ...T T T ...T x ( i ) , d T x ( i ) , T x ( i ) i +1 i n −1 i n  i n −1 i −1 i − 2 1 m m −1 i n −1 i n i −1 i − 2 1 m m −1 i n ϕi  d i + 2 Ti +1Ti xn( i−)1 , Ti +1Ti xn( i ) ,..., d m Tm −1Tm − 2 ...Ti xn( i−)1 , Tm −1Tm − 2 ...Ti xn( i ) ,   d T T ...T x (i ) , T T ...T x ( i ) , d T T T ...T x ( i ) , T T T ...T x ( i ) ,..., 2 1 m m −1 i n −1 1 m m −1 i n  1 m m −1 i n −1 m m −1 i n ( i ) ( i )  d T T ...T T T ...T x , T T ...T T T ...T x  i −1 i − 2 i −3 1 m m −1 i n −1 i − 2 i −3 1 m m −1 i n

(

(

) (

)

( ) (

(

)

(

)

)

)

  ,  =     

= ϕi (d n(i ) , d n(i−)1 , d n(i−)1 , d n(i ) , d n(i−+11) , d n(i−+12) ,..., d n( m−1) , d n(1)−1 , d n(2) ,..., d n(i −1) ) ≤ 0 and from (b), we have

d ni ≤ c max(d n(i−)1 , d n(i−+11) , d n(i−+12) ,..., d n( m−1) , d n(1)−1 , d n(2) , d n(3) ,..., d n(i −1) )

(*)

By (*), for i=3 we have: (m) (1) (2) d n3 ≤ c max(d n(3)−1 , d n(4) −1 ,..., d n −1 , d n −1 , d n )

By this inequality and (2’) it follows:

{

(3) (4) (m) d n(3) ≤ c max d n(1)−1 , d n(2) −1 , d n −1 , d n −1 ,..., d n −1

}

(3’)

In similar way, for i=4,5,…,m-1 and by the inequalities (2’),(3’),...,((i-1)’) we get:

{

(3) (4) (m) d n( i ) ≤ c max d n(1)−1 , d n(2) −1 , d n −1 , d n −1 ,..., d n −1

} i = 2, 3,..., m − 1

(i)

Applying inequality (m) for xm = xn( m−1) and xm′ = xn( m ) , we have:

( (

)

(

m m m m m m  dm Tm−1Tm−2...TT 1 m xn−1, Tm−1Tm−2 ...TT 1 m xn , dm ( xn−1, xn ), dm xn−1, Tm−1Tm−2 ...TT 1 m xn−1  m m m m m ϕm  dm xnm ,Tm−1Tm−2...TT 1 m xn , d1 Tm xn−1, Tm xn , d2 TT 1 m xn−1, TT 1 m xn ,...,  m m  1 m xn−1, Tm−2Tm−3...TT 1 m xn  dm−1 Tm−2Tm−3...TT

) (

(

)

) (

)

) , 

=   

= ϕ m ( d n( m ) , d n( m−1) , d n( m−1) , d n( m ) , d n(1)−1 , d n(2) , d n(3) ,..., d n( m −1) ) and from (b), we have:

d nm ≤ c max( d n( m−1) , d n(1)−1 , d n(2) , d n(3) ,..., d n( m −1) ) By this inequality and by (2’),(3’),…, ((m-1)’) it follows: (3) (m) d n( m ) ≤ c max(d n(1)−1 , d n(2) −1 , d n −1 ,..., d n −1 )

(m’)

Applying inequality (1) for x1 = xn(1)−1 and x1′ = xn(1) , we have:

74

6

KIKINA: FIXED POINT THEOREM

( ( (

) ( ) ( ) ( )

 d1 TmTm−1...T2T1xn(1)−1, TmTm−1...T2T1xn(1) , d1 xn(1)−1, xn(1) , d1 xn(1)−1, TmTm−1...T1xn(1)−1  ϕ1  d1 xn(1) , TmTm−1...T1xn(1) , d2 T1xn(1)−1, T1xn(1) , d3 T2T1xn(1)−1, T2T1xn(1) ,...,   d T T ...T x(1) , T T ...T x(1)  m m−1 m−2 1 n m−1 m−2 1 n

) (

)

) , 

=   

= ϕ1 (dn(1) , dn(1)−1, dn(1)−1, dn(1) , dn(2) , dn(3) ,..., dn(m) ) ≤ 0 and from (b) we have:

d n(1) ≤ c max{d n(1)−1 , d n(2) , d n(3) ,..., d n( m ) } By this inequality and by (2’),(3’),…,(m’) it follows: (3) (m) d n(1) ≤ c max{d n(1)−1 , d n(2) −1 , d n −1 ,..., d n −1 }

(1’)

It now follows from (1’),(2’),…,(m’) that for large enough m (3) (m) d n( i ) = di ( xn( i ) , xn(i+)1 ) ≤ c max{d n(1)−1 , d n(2) −1 , d n −1 ,..., d n −1 } (3) (m) ≤ c 2 max{d n(1)− 2 , d n(2) − 2 , d n − 2 ,..., d n − 2 } (3) (m) ≤ c 3 max{d n(1)−3 , d n(2) − 3 , d n −3 ,..., d n − 3 }

... ≤ c n −1 max{d1(1) , d1(2) , d1(3) ,..., d1( m ) } = c n −1l where

l = max{d1(1) , d1(2) , d1(3) ,..., d1( m ) } Since 0 ≤ c < 1 , it follows that {xn( i ) } are Cauchy sequences with the limits

α i in X i for i = 1, 2,..., m. Now suppose that Ti for i = 1, 2,..., m − 1 are continuous, we have lim xn(2) = lim T1 xn(1)−1 ⇒ T1α1 = α 2 and lim xn(i++11) = lim Ti xn(i ) ⇒ Tiα i = α i +1 n →∞

n →∞

n →∞

n →∞

for i = 2,3,..., m − 1 Later we will show that Tmα m = α1 . To prove that α1 is a fixed point of TmTm −1...T1 Using the inequality (1) for x1 = α1 and x1' = xn(1)−1 , we obtain:

( (

) (

) )

(

)

 d1 TmTm−1...T2T1α1 , xn(1) , d1 α1 , xn(1)−1 , d1 (α1 , TmTm−1...T1α1 ) , d1 xn(1)−1 , xn(1) , ϕ1   d T α , T x(1) , d T T α , T T x(1) ,..., d T T ...T α , T T ...T x(1) 3 2 1 1 2 1 n −1 1 1 m −1 m − 2 1 n −1 m m −1 m − 2  2 1 1 1 n−1

)

(

(

)

 ≤0  

Letting n tend to infinity and using (a) and the continuality of Ti for i=1,2,…,m-1, we have

75

7

KIKINA: FIXED POINT THEOREM

 d1 (TmTm −1...T2T1α1 , α1 ) , d1 (α1 , α1 ) , d1 (α1 , TmTm −1...T1α1 ) , d1 (α1 , α1 ) ,  =  d (T α , T α ) , d (T T α , T T α ) ,..., d (T T ...T α , T T ...T α )  3 2 1 1 2 1 1 1 1 m −1 m − 2 1 1  m m −1 m − 2  2 1 1 1 1

ϕ1 

= ϕ1 ( d1 (TmTm −1...T2T1α1 , α1 ) , 0, d1 (α1 , TmTm −1...T1α1 ) , 0, 0,..., 0 ) ≤ 0 and from (b), we have: d1 (TmTm −1...T2T1α1 , α1 ) ≤ c max{0, 0,..., 0} = 0 Thus TmTm −1...T1α1 = α1 and so α1 is a fixed point of TmTm −1...T1 . We now have T1TmTm −1...T2α 2 = T1TmTm −1...T2T1α1 = T1α1 = α 2 In general

Ti −1Ti − 2 ...T1TmTm −1...Tiα i = Ti −1 (Ti − 2 ...T1TmTm −1...TT i i −1α i −1 ) = Ti −1α i −1 = α i , i = 2,3,..., m Hence α i are fixed points of

Ti −1Ti − 2 ...T1TmTm −1...Ti , i = 2,3,..., m We now prove the uniqueness of the fixed point α i . Let us prove for α1 . Suppose that TmTm −1...T1 has a second fixed point α1' ≠ α1 .Using the inequality (1) for

x1 = α1 and x1' = α1' we have:

( (

) )

(

 d1 TmTm−1...T2T1α1,TmTm−1...T2T1α1' , d1 (α1,α1' ), d1 (α1,TmTm−1...T1α1 ) , d1 α1' , TmTm−1...T1α1' ϕ1   d Tα ,Tα ' , d T Tα ,T Tα ' ,..., d T T ...Tα , T T ...Tα ' 1 1 m−1 m−2 1 1 m m−1 m−2  2 1 1 1 1 3 21 1 21 1

) (

(

(

) (

)

) (

)

)

) ,   

=

 d1 (α1 , α1' ), d1 (α1 , α1' ), 0, 0, d 2 T1α1 , T1α1' , d 3 T2T1α1 , T2T1α1' ,...,   = ϕ1   d T T ...T α , T T ...T α '   m m −1 m − 2 1 1 m −1 m − 2 1 1 

(

)

And from (b), we have:

( (

d 2 T1α1 , T1α1' , d 3 T2T1α1 , T2T1α1' ,...,    d1 (α1 , α ) ≤ c max   ' d m Tm −1Tm − 2 ...T1α1 , Tm −1Tm − 2 ...T1α1  ' 1

)

(1’’)

In similar way, applying the inequality (2) for x2 = T1α1 and x2' = T1α1' , by the property (b) of ϕ2 and taking in consideration (1’’) we obtain:

( (

) (

)

d3 T2T1α1 , T2T1α1' , d 4 T3T2T1α1 , T3T2T1α1' ,...,    d 2 T1α1 , T α ≤ c max   ' d m Tm −1Tm − 2 ...T2T1α1 , Tm −1Tm − 2 ...T2T1α1 , 

(

' 1 1

)

)

(2’’)

Similarly, applying the inequality (i ) for xi = Ti −1Ti − 2 ...T2T1α1 and xi' = Ti −1Ti − 2 ...T2T1α1' and using these inequalities (1''), (2 ''),..., ((i − 1) ''), we have

76

8

KIKINA: FIXED POINT THEOREM

(

)

di Ti −1Ti − 2 ...T1α1 , Ti −1Ti − 2 ...T1α1' ≤

(

)

(

' '  di +1 TT i i −1 ...T1α1 , TT i i −1 ...T1α1 , d i + 2 Ti +1Ti ...T1α1 , Ti +1Ti ...T1α1 ≤ c max   ,..., d T T ...T α , T T ...T α ' m m −1 m − 2 1 1 m −1 m − 2 i 1 

(

By (i’’) for i=m-1 we get:

(

)

)

) 

(

(i’’)

 

d m −1 Tm − 2Tm −3 ...T1α1 , Tm − 2Tm −3 ...T1α1' ≤ cd m Tm −1Tm − 2 ...T1α1 , Tm −1Tm − 2 ...T1α1'

)

((m-1)’’)

Applying the inequality (m), for xm = Tm −1 , Tm − 2 ,..., T1α1 , xm' = Tm −1 , Tm − 2 ,..., T1α1' and using the property (b) of ϕ m and these inequalities ((m − 1) ''), ((m − 2) ''),..., (1''), we now have

(

)

(

d m Tm −1Tm − 2 ...T1α1 , Tm −1Tm − 2 ...T1α1' ≤ cd m Tm −1Tm − 2 ...T1α1 , Tm −1Tm − 2 ...T1α1' and so

(

)

d m Tm −1Tm − 2 ...T1α1 , Tm −1Tm − 2 ...T1α1' = 0

)

(m’’)

(
)

Returning back and using (
), ((m − 1) ''), ((m − 2) ''),..., (1'') we get:

(

)

d1 α1 , α1' = 0 ⇔ α1 = α1' And so, α1 = α1' , then the uniqueness of α1 is proved. Similarly, it can be proved that α i is the unique fixed point of Ti −1Ti − 2 ...T1TmTm −1...Ti , for i=2, 3… m. We finally prove that also we have Tmα m = α1 . To do this, note that Tmα m = Tm (Tm −1Tm − 2 ...T1Tmα m ) = TmTm −1Tm − 2 ...T1 (Tmα m ) and so, Tmα m is a fixed point of TmTm −1Tm − 2 ...T1 . Since α1 is the unique fixed point, it follows that Tmα m = α1 . This completes the proof of the theorem.

3. COROLLARIES The next corollary follows from Theorem 2.6 in the case m = 1 and T1 = T

Corollary 3.1. Let ( X , d ) be a complete metric space and T : X → X a self map of X. If for some c ∈[0,1) we have

ϕ (d (Tx, Ty ), d ( x, y ), d ( x, Tx), d ( y, Ty )) ≤ 0 for all x, y ∈ X and ϕ ∈ Φ 4 , then T has a unique fixed point α in X. This corollary is a generalization of Rhoades theorem [10] The next corollary follows from corollaries 3.1 in the case

ϕ (t1 , t2 , t3 , t4 ) = t1 − cf (t2 , t3 , t4 ) , where f ∈ F3

77

9

KIKINA: FIXED POINT THEOREM

Corollary 3.2. Let ( X , d ) be a complete metric space and T : X → X a self map of X. If for some c ∈[0,1) we have

d (Tx, Ty ) ≤ cf {d ( x, y ), d ( x, Tx), d ( y, Ty )} for all x, y ∈ X and f ∈ F 3 , then T has a unique fixed point α in X. For different expressions of ϕ , in the Corollary 3.1, and of f , in the Corollary 3.2, we get different theorems. For example: For ϕ ( t1 , t2 , t3 , t4 ) = t1 − c max {t2 , t3 , t4 } we have the Theorem 1.1 (Rhoades theorem [10]). For ϕ ( t1 , t2 , t3 , t4 ) = t1 − ct2 we have the Banach theorem [3] For ϕ ( t1 , t 2 , t3 , t 4 ) = t1 − c

t3 + t 4 we have the Kannan theorem [7] 2

For ϕ ( t1 , t2 , t3 , t4 ) = t1 − c max {t3 , t4 } we have the Bianchini theorem [4]

at2 + bt3 + ct4 where a, b, c are nonnegative numbers such that, a+b+c a + b + c < 1 , we have the Reich theorem [9].

For ϕ ( t1 , t2 , t3 , t4 ) = t1 −

The next corollary follows from Theorem 2.6 in the case m = 2 , T1 = S and T2 = R

Corollary

3.3

Let

( X , d ) and (Y , ρ)

are

complete

metric

spaces

and

S : X → Y , R : Y → X are two maps, at least one of them being continuous. If the following

inequalities are satisfied:

ϕ1 ( d ( RSx, RSx ') , d ( x, x ' ) , d ( x, RSx ) , d ( x ', RSx ' ) , ρ ( Sx, Sx ' )) ≤ 0 ϕ 2 ( ρ ( SRy, SRy ' ) , ρ ( y, y ' ) , ρ ( y, SRy ) , ρ ( y ', SRy ' ) , d ( Ry, Ry ' )) ≤ 0 for all x, x'∈X; y, y'∈Y and ϕ1,ϕ2 ∈Φ5 , then RS has a unique fixed point α ∈ X and SR has a unique fixed point β ∈ Y . Moreover, Sα = β and. Rβ = α This corollary is a generalization of Fisher theorem [5]. The next corollary follows from corollaries 3.3 in the case

ϕi (t1 , t2 , t3 , t4 , t5 ) = t1 − ci fi (t2 , t3 , t4 , t5 ) , where fi ∈ F4 for i=1,2 Corollary

3.4

Let

( X , d ) and (Y , ρ)

are

complete

metric

spaces

and

S : X → Y , R : Y → X be two maps, at least one of them being continuous. If the following

inequalities are satisfied:

d ( RSx, RSx ') ≤ c1 f1 (d ( x, x ') , d ( x, RSx ) , d ( x ', RSx ') , ρ ( Sx, Sx '))

78

10

KIKINA: FIXED POINT THEOREM

ρ ( SRy, SRy ') ≤ c2 f 2 ( ρ ( y, y ') , ρ ( y, SRy ) , ρ ( y ', SRy ') , d ( Ry, Ry ')) for all x, x'∈X; y, y'∈Y and f1, f2 ∈F4 , then RS has a unique fixed point α ∈ X and SR has a unique fixed point β ∈ Y . Moreover, Sα = β and. Rβ = α For the different expressions of ϕ1 and ϕ 2 in the Corollary 3.3 and of f1 and f 2 in the Corollary 3.4, we get different theorems. For example: For ϕ1 = ϕ2 = ϕ , where ϕ ( t1 , t2 , t3 , t4 , t5 ) = t1 − c max {t2 , t3 , t4 , t5 } , we have the Theorem 1.2 (Fisher theorem [5]). For ϕ1 ( t1 , t2 , t3 , t 4 , t5 ) = t1 −

ϕ2 ( t1 , t2 , t3 , t 4 , t5 ) = t1 −

a1t2 + a2t3 + a3t4 + a4t5 and a1 + a2 + a3 + a4

b1t2 + b2t3 + b3t4 + b4t5 , we have an extension of Reich Theorem [9] b1 + b2 + b3 + b4

from one metric space to two metric spaces.

Corollary 3.5. Let ( X , d ) and (Y , ρ ) are complete metric spaces and S : X → Y , R : Y → X be two maps, at least one of them being continuous. If the following

inequalities are satisfied:

d ( RSx, RSx ') ≤ a1d ( x, x′) + a2 d ( x, RSx ) + a3d ( x ', RSx ') + a4 ρ ( Sx, Sx ')

ρ ( SRy, SRy ') ≤ b1 ρ ( y, y′) + b2 ρ ( y, SRy ) + b3 ρ ( y ', SRy ') + b4 d ( Ry, Ry ') for all x, x'∈X; y, y'∈Y , where a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 are nonnegative numbers such that

0 ≤ a1 + a2 + a3 + a4 < 1 , 0 ≤ b1 + b2 + b3 + b4 < 1 , then RS has a unique fixed point α ∈ X and SR has a unique fixed point β ∈ Y . Moreover, Sα = β and Rβ = α .

Remark 3.6 In case m = 2 , in similar way we obtained the above corollary, we can obtained analogues corollaries which extend the Banach, Kannan, Bianchin theorems etc. from one metric space to two metric spaces. The next corollary follows from Theorem 2.6 in the case m = 3 , T1 = T , T2 = S and

T3 = R :

Corollary 3.7 Let ( X , d ), (Y , ρ ) and ( Z , σ ) be complete metric spaces and T : X → Y , S:Y → Z and R : Z → X be three maps, at least one of them being continuous. If the

following inequalities are satisfied

ϕ1 ( d ( RSTx, RSTx′), d ( x, x′), d ( x, RSTx), d ( x′, RSTx′), ρ (Tx, Tx′), σ ( STx, STx′)}) ≤ 0 ϕ2 ( ρ (TRSy, TRSy′), ρ ( y, y′), ρ ( y, TRSy ), ρ ( y′, TRSy′), σ ( Sy, Sy′), d ( RSy, RSy′) ) ≤ 0

ϕ3 (σ ( STRz , STRz ′), σ ( z , z ′), σ ( z , STRz ), σ ( z ′, STRz ′), d ( Rz , Rz ′), ρ (TRz , TRz′) ) ≤ 0

79

11

KIKINA: FIXED POINT THEOREM

for all x, x′ ∈ X ; y, y′ ∈ Y ; z , z ′ ∈ Z and ϕ1,ϕ2 ,ϕ3 ∈Φ6 , then RST has a unique fixed point u in X , TRS has a unique fixed point v in Y and STR has a unique fixed point w in Z . Further, Tu = v, Sv = w and Rw = u . This corollary is a generalization of Jain et all theorem [6]. In the special case for ϕ1 =ϕ2 =ϕ3 =ϕ where ϕ ( t1 , t2 , t3 , t4 , t5 , t6 ) = t1 − c max {t2 , t3 , t4 , t5 , t6 } , we have the theorem 1.2 (Jain et all theorem [6]). Remark 3.8 From the corollary 3.6 we can obtain other propositions determined by the form of implicit relations ϕ1 , ϕ 2 and ϕ 3 . For example the corollary which extended the theorem of Reich [6], from one metric space to three metric spaces (in similar way as above we extended to two metric spaces), etc. At last, we would like to emphasize the fact that all the above corollaries does not hold only for 1, 2 and 3 metric spaces, but also for an arbitrary number of metric spaces.

REFERENCES [1] A. Aliouche and B. Fisher. Fixed point theorems for mappings satisfying implicit relation on two complete and compact metric spaces, Applied Mathematics and Mechanics, 7 ( 9), (2006), 1217-1222. [2] A. Aliouche and B. Fisher, A related fixed point theorem for two pairs of mappings on two complete metric spaces. Hacettepe Journal of Mathematics and statistics. V.34 (2005), 39-45. [3] S. Banach. Theorie des operations lineaires Manograie, Mathematyezne (Warsaw, Poland), 1932. [4] R. M. T. Bianchini, Su un problema di S.Reich riguardante la teoria dei punti fissi, Boll. Un. Mat. Ital.5 ( 1972) 103-108. [5] B. Fisher. Related fixed points on two metric spaces. Math. Sem. Notes, Kobe univ., 10, 17 -26, 1982. [6] R. K. Jain, H. K. Sahu, and B. Fisher. Related fixed points theorems for three metric spaces, Novi Sad. I. Math, Vol 26,No.1, 11-17, 1996. [7] R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969), 405408. [8] V. Popa, On some fixed point theorems for mappings satisfying a new type of implicit relation, Mathematica Moravica, (7), pp. 61-66, 2003. [9] S. Reich, Some remarks concerning contraction mappings, Canad. Math. Bull. 14(1971) 121-124. [10] B. E. Rhoades, A comparison of various definitions of contractive mappings. Trans. Amer. Math. soc. 226, (1977), 256-290. [11] M. Telci, Fixed points on two complete and compact metric spaces, Applied Mathematics and Mechanics, 22 (5) , (2001), 564-568.

80

12

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,81-91,COPYRIGHT 2012 EUDOXUS PRESS,LLC

Direct and converse theorems for generalized Bernstein polynomials based on the q integers I·brahim Büyükyaz¬c¬ Department of Mathematics, Gazi University, Çubuk - Ankara, Turkey e-mail: [email protected] Abstract In this paper, we discuss the convergence properties of a new generalization of q Bernstein polynomials. The uniform convergence is established, and estimates for the rate of convergence are provided. We also prove that if the rate of convergence for these operators is C=[n]qn , then f 2 Lip( ; [0; 1]), for some positive constant C and exponent 0 < < 1. Keywords: q

Bernstein polynomials, rate of convergence, inverse theorem.

2000 Mathematics Subject Classi…cation: 41A10, 41A25, 41A36.

1

Introduction and de…nitions

In some branches of mathematics (for example, approximation theory, probability theory, number theory, the solution of integral and di¤erential equations), Bernstein polynomials have important applications. The reasons for these applications are the simple structure and important properties of the Bernstein polynomials (for example, shape preserving, reproducing linear functions, degree reducing on the set of polynomials, interpolating f at both end points of [0; 1]). The nth Bernstein polynomial for a given function f on [0; 1] is de…ned as Bn (f ; x) =

n X

f

k=0

where

k n

Pk;n (x) =

Pk;n (x) n k x (1 k

(1) x)n

k

;

0

x

1;

n = 1; 2; 3; : : :

If f 2 C[0; 1], then the sequence fBn (f ; x)g converges uniformly to f (x) on [0; 1]. In [5], J. D. Cao introduced the following generalized Bernstein polynomials (which we call the Cao polynomials): for f 2 C[0; 1], the Cao polynomials of f are Cn;sn (f ; x) =

n sn 1 1 XX k+i f( )Pk;n (x); 0 sn n + sn 1

x

1;

n = 1; 2; 3; : : :

(2)

k=0 i=0

sn where fsn g is a sequence of natural numbers with the properties lim = 0 and sn 1. n!1 n When sn = 1, we recover the Bernstein polynomials. Before introducing the new operators based on q integers, we recall the following de…nitions of the q calculus (quantum calculus). Given a value of q > 0 and any nonnegative integer n, we de…ne the q integer [n]q as ( (1 q n )=(1 q) if q 6= 1 [n]q = n if q = 1 1 81

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

and the q

factorial [n]q ! as ( [n]q [n [n]q ! = 1

For integers n and k, with 0 follows (see [9]):

k

1]q

[1]q

n, the q n k

= q

if n 1 : if n = 0

binomial coe¢ cients are then de…ned as

[n]q ! : [k]q ![n k]q !

Due to the intensive development of the q calculus, generalizations of Bernstein polynomials based on the q integers have emerged. In [16], G. M. Phillips proposed the q Bernstein polynomials (which we also call the Phillips polynomials): for each positive integer n, and f 2 C[0; 1], the q Bernstein polynomial of f is Bn;q (f ; x) =

n X k=0

[k]q f [n]q

n k

k

x q

nY k 1

(1

q s x);

0

x

1;

n = 1; 2; 3; : : :

(3)

s=0

where an empty product is taken to equal 1. Note that the Phillips polynomials reduce to the Bernstein polynomials when we choose q = 1 in (3). Explicit expressions for Bn;q (tr ; x) for r = 0; 1; 2 can be obtained by direct calculation, giving Bn;q (1; x) = 1 Bn;q (t; x) = x

(4)

Bn;q (t2 ; x) = x2 +

x(1 x) : [n]q

In recent years, the q Bernstein polynomials and some generalizations have been studied by several researchers because of their potential applications in approximation theory and numerical analysis, and some features of the Bernstein polynomials are inherited by the q Bernstein polynomials, especially for 0 < q < 1 (see [4], [10]-[20]). In this paper, as Phillips has done for the Bernstein polynomials, we consider similar modi…cation of the Cao polynomials: q Bn;

(f ; x) = n

n [ n] 1 [k]q + i 1 X X )Pk;n;q (x); f( [ n] [n]q + n 1 k=0

where

(5)

i=0

Pk;n;q (x) =

n k

xk q

nY k 1

(1

q s x); 0

x

1; n = 1; 2; 3; : : :

s=0

where n is a positive real number and [ n ], as usual, denotes the greatest integer less than n . There should be no confusion between the two notations [ n ] and [n]q . When n = 1; we recover the Phillips polynomials; when q = 1 and n = sn we recover the Cao polynomials. Note that throughout the paper, we always assume that fqn g is a sequence of real numbers such that 0 < qn < 1 for all n and lim qn = 1, and where f n g is a sequence of n!1

positive real numbers with the properties lim [n]n = 0 and [ n ] 1: qn n!1 The outline of this paper is as follows: in Section 2, we give the convergence properties of the polynomials (5). Section 3 deals with the rate of convergence, using di¤erent methods. Finally, we prove an inverse theorem for the polynomials (5). Now, we give the following basic de…nitions:

2 82

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

De…nition 1.1. The q

derivative of a function f with respect to x is Dq (f (x)) =

f (qx) qx

f (x) ; x

and this is also known as the Jackson derivative. The q derivative ‘product rule’is Dq (f (x)g(x)) = Dq (f (x)) g(qx) + f (x) Dq (g(x)). Note also that Dq (xn ) = [n]q xn 1 . De…nition 1.2. The q integral of a function f is Z b 1 X f (x)dq x = (1 q)b q m f (bq m ) (1 a

q)a

m=0

Note that as q ! 1, the q derivative and the q and the Riemann integral, respectively. De…nition 1.3. For f 2 C[0; 1] and

1 X

q m f (aq m ):

m=0

integral approach the usual derivative

> 0, the usual modulus of continuity is de…ned by

!(f ; ) =

sup jf (x)

f (y)j :

jx yj x;y2[0;1]

It is known that lim

!0 !(f ;

De…nition 1.4. For > 0 and functional is de…ned as K(f ; ) =

) = 0 and, for any C 2 [0; 1] inf

g2C 2 [0;1]

> 0, !(f ;

= fg 2 C[0; 1] :

n kf

g 0 ; g 00

kgkC 2 [0;1] = g 0

2

C[0;1]

+ g 00

qn Approximation properties of Bn;

C[0;1]

(1 + )!(f ; ).

2 C[0; 1]g, Peetre’s K

gkC[0;1] + kgkC 2 [0;1]

where k:kC 2 [0;1] is the uniform norm on C 2 [0; 1]; de…ned by

)

o

:

n

qn From the de…nition (5) of Bn; n (f ; x) and the …rst equation in (4), we have qn Bn; (1; x) = 1: n

(6)

qn To construct our approximation theorem for the sequence fBn; n g, we need the following lemma. qn Lemma 2.1. If the operator Bn; qn Bn; (t; x) = n qn Bn; (t2 ; x) = n

n

is de…ned by (5), then

[n]qn [ n] 1 x+ ; (7) [n]qn + n 1 2 ([n]qn + n 1) [n]2qn [n]qn [ n ] [n]qn ([ n ] 1)(2 [ n ] 1) x2 + x+ :(8) 2 2 ([n]qn + n 1) ([n]qn + n 1) 6([n]qn + n 1)2

Proof. Let us consider the case when f is the function t 7! t; then qn Bn;

n

(t; x) =

=

n [ n] 1 [k]qn + i 1 X X Pk;n;qn (x) [ n] [n]qn + n 1 k=0 i=0 ( n X [k]qn 1 1 [ n ] [n]qn Pk;n;qn (x) [ n ] [n]qn + n 1 [n]qn k=0 ) n X 1 + ([ n ] 1) [ n ] Pk;n;qn (x) : 2 k=0

3 83

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

Using (4), we obtain [n]qn [n]qn + n

qn Bn; (t; x) = n

1

x+

Taking f : t 7! t2 , we …nd qn Bn; (t2 ; x) = n

n [ n] 1 1 X X [ n]

=

1 [

1

n ] ([n]qn

+

n ] ([n]qn + (

2 n ] [n]qn

[

+

n

1)2

n

1)2

1

1 [

[n]qn +

i=0

k=0

=

[k]qn + i

[

1

n

!2

n] 1 n [X X

k=0

[ n] 1 2 ([n]qn + n

[k]2qn + 2 [k]qn i + i2 Pk;n;qn (x)

i=0

n ] ([ n ]

k=0

1)(2 [ 6

n]

:

Pk;n;qn (x)

n X [k]2qn Pk;n;qn (x) + [ [n]2qn

n ] ([ n ]

1)

1)

n X

1) [n]qn

n X [k]qn Pk;n;qn (x) [n]qn k=0

)

Pk;n;qn (x)

k=0

From (4) it follows that qn (t2 ; x) = Bn; n

1 [

n ] ([n]qn

[ + =

1

[

+

2 n ] [n]qn

n

1)2

x(1 x) +[ [n]qn 1)(2 [ n ] 1) 6

x2 +

n ] ([ n ]

n ] ([ n ]

1) [n]qn x

[n]2qn [n]qn [ n ] [n]qn ([ n ] 1)(2 [ n ] 1) x2 + x+ : 2 2 ([n]qn + n 1) ([n]qn + n 1) 6([n]qn + n 1)2

Combining (6) and Lemma 2.1, we have the following main result. qn Theorem 2.2. For f 2 C[0; 1], the polynomials Bn; n (f ; x) converge uniformly to f (x) on [0; 1] as n ! 1.

Proof. From (6) - (8), we have qn (1; x) = 1; Bn; n qn Bn; (t; x) = n qn Bn; (t2 ; x) = n

[n]qn [ n] 1 x+ ; [n]qn + n 1 2 ([n]qn + n 1) [n]2qn [n]qn [ n ] [n]qn ([ n ] 1)(2 [ n ] 1) x2 + x+ 2 2 ([n]qn + n 1) ([n]qn + n 1) 6([n]qn + n 1)2

and we obtain qn Bn; (1; x) n

1

qn Bn; (t; x) n

x

qn Bn; (t2 ; x) n

x2

C[0;1] C[0;1]

C[0;1]

= 0; [n]qn [n]qn + n

1

[n]2qn [n]qn ([n]qn + n 1)2 4 84

1 +

n

[n]qn

1 +

;

1 2n n + 2 3 [n]q [n]qn n

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

from condition lim

n

n!1 [n]qn

= 0; we get qn Bn; (tr ; x) n

lim

n!1

xr

C[0;1]

= 0;

r = 0; 1; 2:

The proof of uniform convergence is then completed by applying the well-known Korovkin theorem (see [1]).

3

Rate of convergence

Theorem 3.1. For any f 2 C[0; 1], qn Bn; (f ; x) n

0

2! @f ;

f (x)

q

4

2 n

+ [n]qn

[n]qn

1 A

where !(f ; ) is the modulus of continuity as given in De…nition 1.3. P[ n ] 1 P f (x)Pk;n;qn (x) = f (x), express the di¤erProof. Using the relation [ 1n ] nk=0 i=0 qn ence between Bn; n (f ; x) and f as qn (f ; x) Bn; n

" n [ n] 1 [k]qn + i 1 X X f (x) = f( ) [ n] [n]qn + n 1

#

f (x) Pk;n;qn (x)

i=0

k=0

and so qn Bn;

n

(f ; x)

n [ n] 1 [k]qn + i 1 X X f( ) [ n] [n]qn + n 1

f (x)

k=0

Letting y =

[k]qn + i [n]qn +

)!(f ; ). Thus jf (y) qn Bn; (f ; x) n

f (x)

n

1

and jy

f (x)j 1 [

n]

1+

!(f ; )

f (x) Pk;n;qn (x)

(9)

i=0

xj = jy xj

n [ X k=0

, we have jf (y)

f (x)j

!(f ;

)

(1 +

!(f ; ), and hence, by (6) n]

1

0

X B @1 +

[k]qn +i [n]qn + n 1

x

i=0

!(f ; ) 8 2 > n [ n] 1 < 1 1 X X 1+ 4 > [ n] : k=0 i=0

[k]qn + i [n]qn +

n

1

1

C A Pk;n;qn (x) !2

x

31=2 9 > = Pk;n;qn (x)5 > ;

where we have invoked the Cauchy-Schwartz inequality. Expanding the squared term and making use of (6), (7) and (8), we obtain qn Bn; (f ; x) n

f (x)

1 !(f ; ) 1 + " [n]2qn [n]qn [ n ] [n]qn ([ n ] 1)(2 [ n ] 1) x2 + x+ 2 2 ([n]qn + n 1) ([n]qn + n 1) 6([n]qn + n 1)2 ) 1=2 [n]qn [ n] 1 2x x+ + x2 : (10) [n]qn + n 1 2 ([n]qn + n 1) 5 85

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

Now, since 0

x

1 and 1

[

n]

n,

deduce n [ n] 1 1 X X [ n]

k=0 i=0 [n]2qn [n]qn

=

([n]qn + +

=

n ] [n]qn + n 1)2

([n]qn

[ [n]qn

( n 1)2 [n]qn 2 [ x + ([n]qn + n 1)2 2 n

([n]qn +

n

4

2 n

n

+ [n]qn

[n]2qn

n]

1)2 =

+ [n]qn in (10), we [n]qn

Pk;n;qn (x) !

1 1

n

+ [n]qn + ([n]qn +

!

n n

x+ [

([ n ] 1)(2 [ n ] 1) 6([n]qn + n 1)2

n] 1)2

1

n

2 2 n + [n]qn n + ([n]qn + n 1)2 3([n]qn +

+

2 n

4

+ 1 x2

1

n] +

=

([ n ] 1)(2 [ n ] 1) 6([n]qn + n 1)2

x+

[ n ] + [n]qn + n [ n ]2 2 x + x + 1)2 ([n]qn + n 1)2 3([n]qn + n

2 n

([n]qn +

x

1

n

2[n]qn [n]qn + n

1)2

n

[

!2

[k]qn + i [n]qn +

by choosing

p

1)2

1)2

n

2

(11)

so that qn Bn;

n

(f ; x)

f (x)

2 ! f;

p

4

!

2 n

+ [n]qn [n]qn

as claimed. Theorem 3.2. If f is Hölder continuous on [0; 1] with exponent f 2 LipM ( ; [0; 1]) with 0 < 1, then ! p 4 2n + [n]qn qn M Bn; n (f ; x) f (x) [n]qn

2 (0; 1], denoted

Proof. From (9) we have qn (f ; x) Bn; n

n [ n] 1 [k]qn + i 1 X X f( ) [ n] [n]qn + n 1

f (x)

f (x) Pk;n;qn (x)

i=0

k=0

n [ n] 1 [k]qn + i M X X [ n] [n]qn + n 1

x Pk;n;qn (x)

i=0

k=0

by the Hölder condition. Application of Hölder’s inequality gives 9 ( 8 )2 2 1 n] n [ X n 2 =2 X < X [k] + i M q n qn Bn; n (f ; x) f (x) x Pk;n;qn (x) Pk;n;qn (x) ; [ n] : [n]qn + n 1 i=0

k=0

= M

8 n 0 and 0
0 such that !(f ; h)

Lemma 4.3. For x 2 (0; 1), the following estimate holds: 1

qn (f ; x) Dqn Bn; n

1

x

!(f ; ) [n]qn +

K2 h .

1

(13)

where !(f ; :) is the modulus of continuity as given in De…nition 1.3. Proof . Taking the qn qn (f ; x) Dqn Bn; n

=

=

derivative of (5) with respect to x:

n [ n] 1 [k]qn + i 1 X X n f( ) [ n] [n]qn + n 1 k

1 [

n]

k=0

i=0

n [ X

n]

k=0

(

X

1

f(

i=0

[k]qn + i [n]qn + nY k 1

k

Dqn x

=

1

n

qn

n k

)

nY k 1

k

+ x Dqn

s=0

n

(f ; x)

=

n [ n] 1 [k]qn + i 1 X X n f( ) [ n] [n]qn + n 1 k qn k=0 i=0 ( nY k 1 [k]qn xk 1 (1 qns+1 x) [n n [ n] 1 [k]qn + i [n]qn X X n f( ) [ n] [n]qn + n 1 k k=1

[n]qn [ n] =

[n]qn [ n]

k=0

n X1 [ k=0

1 1

i=0

n X1 [

n]

X

1

f(

i=0

n]

X

1

(

f(

i=0

n

[k]qn + i [n]qn +

k

1

n

[k + 1]qn + i [n]qn +

1

k

x qn

qns x)

(1

qns x)

!)

s=0

k

k]qn x

1

n

nY k 1

n

)

)

(1

k 1

x qn

f(

)

qns x)

(1

nY k 1

(1

qns+1 x)

(1

qns x)

s=0

1 k

nY k 1 s=1

Using properties of q binomial coe¢ cients, we write Dqn

(1

s=0

s=0

qn Bn;

!

qn

qns+1 x)

(1

xk

Dqn

nY k 1

xk qn

nY k 1 s=1

[k]qn + i [n]qn +

n

1

)

)

qns x)

s=1

Upon taking absolute values of both sides and using the modulus of continuity, we obtain qn Dqn Bn; (f ; x) n

n 1 [ n] 1 [k + 1]qn + i [n]qn X X f( ) [ n] [n]qn + n 1 k=0

i=0

n

1 k 8 88

k

x qn

nY k 1 s=1

(1

f(

[k]qn + i [n]qn +

qns x)

n

1

)

:

BUYUKYAZICI: q-BERNSTEIN POLYNOMIALS

Qn

k 1 (1 s=1

since 0 < x; qns < 1; we have (1 qns x) < 1. Therefore, we can write Qn k 1 Q 1 (1 qns x) 1 1 x ns=0k 2 (1 qns x): Hence, we have s=0 1 x Dqn

qn Bn;

n

n 1 [ n] 1 [k + 1]qn + i [n]qn 1 X X ) f( [ n] 1 x [n]qn + n 1

(f ; x)

n 1 [ n] 1 [k + 1]qn [n]qn 1 X X ! f; [ n] 1 x [n]qn +

[n]qn + !

[k]qn 1 n

i=0

k=0

[k]qn + i

f(

i=0

k=0

qns x) =

Pk;n

n

1

) Pk;n

1;qn (x)

1;qn (x)

n 1

[n]qn X qnk ! f; 1 x [n]qn +

=

k=0

Since 0 < qnk < 1 and

n

Pk;n

1

n

1;qn (x):

1 > 0, we get n 1

[n]qn X 1 1 1+ 1 x [n]qn +

qn (f ; x) Dqn Bn; n

k=0

n

!(f ; )Pk;n

1

1;qn (x)

n 1

X [n]qn 1 1 !(f ; ) 1+ 1 x [n]qn

Pk;n

1;qn (x)

k=0

1

1

x

!(f ; ) [n]qn +

1

:

So the proof is completed. Theorem 4.4. If f 2 C[0; 1] is such that qn Bn; (f ; x) n

C ; [n]qn

f (x)

for some positive constant C and exponent 0
0 and 0 ≤ c − r (2.1) a+b   ad(T x, T y) + b d(x, T x) + d(y, T y) + c[d(y, T x) + d(x, T y)] ≤ sd(x, y) + rd(x, T 2 x) (2.2) hold for all x, y ∈ X. Then, T has at least one fixed point. Proof. Let x0 ∈ X be arbitrary . Define a sequence {xn } in the following way: xn+1 := T xn

n = 0, 1, 2, ...

(2.3)

When we substitute x = xn and y = xn+1 on the inequality (2.2), it implies that   ad(T xn , T xn+1 ) + b d(xn , T xn ) + d(xn+1 , T xn+1 ) + 4c[d(xn+1 , T xn ) + d(xn , T xn+1 )] ≤ sd(xn , xn+1 ) + rd(xn , T 2 xn ) (2.4) 2000 Mathematics Subject Classification. 47H10,54H25,46J10, 46J15. Key words and phrases. Fixed Point Theorem, Cone Metric Spaces, TVS-Cone Metric Spaces. 1

92

2

ERDAL KARAPINAR

for all a, b, c, s, r that satisfy (2.1). Due to (2.3), the statement (2.4) turns into   ad(xn+1 , xn+2 ) + b d(xn , xn+1 ) + d(xn+1 , xn+2 ) + c[d(xn+1 , xn+1 ) + d(xn , xn+2 )] ≤ sd(xn , xn+1 ) + rd(xn , xn+2 ). (2.5) By a simple calculation, one can get (a + b)d(xn+1 , xn+2 ) + (c − r)d(xn , xn+2 ) ≤ (s − b)d(xn , xn+1 )

(2.6)

which implies d(xn+1 , xn+2 ) ≤ kd(xn , xn+1 ) where k = inductively

s−b a+b .

(2.7)

Due to (2.1), we have 0 ≤ k < 1. Taking account of (2.7), we get

d(xn , xn+1 ) ≤ kd(xn−1 , xn ) ≤ k 2 d(xn−2 , xn−1 ) ≤ · · · ≤ k n d(x0 , x1 ).

(2.8)

To show that {xn } is a Cauchy sequence, assume n > m. Then by (2.8) and triangular inequality, one can obtain d(xn , xm ) ≤ d(xn , xn−1 ) + d(xn−1 , xn+1 ) + · · · + d(xm−1 , xm ) ≤ k n d(x0 , x1 ) + k n−1 d(x0 , x1 ) + · · · + k m d(x0 , x1 ) km ≤ 1−k d(x0 , x1 ).

(2.9)

which concludes that {xn } is a Cauchy sequence. Since (X, d) is complete, the sequence {xn } converges to some element of X, namely z. To show z is a fixed point of T , it is sufficient to substitute x = z and y = xn on the inequality (2.2). Indeed,   ad(T xn , T z) + b d(xn , T xn ) + d(z, T z) + c[d(z, T xn ) + d(xn , T z)] (2.10) ≤ sd(xn , z) + rd(xn , T 2 xn ) which is equivalent to   ad(xn+1 , T z) + b d(xn , xn+1 ) + d(z, T z)) + c[d(z, xn+1 ) + d(xn+1 , T z)] ≤ sd(xn , z) + rd(xn , xn+2 ). (2.11) Consequently, (a + b + c)d(T z, z) ≤ 0 as n → ∞. Thus, T z = z as a + b + c > 0.  3. A Fixed Point Theorem on TVS-Cone Metric Spaces Throughout this section, the pair (E, S) stands for a real Hausdorff locally convex topological vector space E with its generating system of semi-norms S. A nonempty subset P of E is called a cone if P + P ⊂ P , λP ⊂ P for λ ≥ 0 and P ∩ (−P ) = {0}. The cone P will be assumed to be closed as well. For a given cone P , a partial ordering (denoted by ≤: or ≤P ) with respect to P can be defined as follow: x ≤ y if and only if y − x ∈ P . The notation x < y indicates that x ≤ y and x 6= y. Analogously, x 0).

A NEW NON-UNIQUE FIXED POINT THEOREM

3

Definition 2. (See [6], [9], [10]) For c ∈ intP , the nonlinear scalarization function φc : E → R is defined by φc (y) = inf{t ∈ R : y ∈ tc − P }, for all y ∈ E.

Lemma 3. (See [6], [9], [10]) For each t ∈ R and y ∈ E, the following are satisfied: (i) φc (y) < t ⇔ y ∈ tc − intP , (ii) if y1 ∈ y2 + P , then φc (y2 ) ≤ φc (y1 ), (iii) φc (y1 + y2 ) ≤ φc (y1 ) + φc (y2 ), for all y1 , y2 ∈ E. Definition 4. Let X be a non-empty set. and E as usual a Hausdorff locally convex topological space. Suppose a vector-valued function p : X × X → E satisfies: (M 1) 0 ≤ p(x, y) and p(x, y) = 0 if and only if x = y, for all x, y ∈ X, (M 2) p(x, y) = p(y, x) for all x, y ∈ X (M 3) p(x, y) ≤ p(x, z) + p(z, y), for all x, y, z ∈ X. Then, p is called TVS-cone metric on X, and the pair (X, p) is called a TVS-cone metric space (in short, TVS-CMS). Remark 5. In [12], the authors considered E as a real Banach space in the definition of TVS-CMS. Thus, a cone metric space (in short, CMS) in the sense of Huang and Zhang [12] is a special case of TVS-CMS. Lemma 6. (See [10]) Let (X, p) be a TVS-CMS. Then, dp : X × X → [0, ∞) defined by dp = φc ◦ p is a metric. Remark 7. Since a cone metric space (X, p) in the sense of Huang and Zhang [12], is a special case of TVS-CMS, then dp : X × X → [0, ∞) defined by dp = φc ◦ d is also a metric. Definition 8. (See [10]) Let (X, p) be a TVS-CMS, x ∈ X and {xn }∞ n=1 a sequence in X. (i) {xn }∞ n=1 TVS-cone converges to x ∈ X whenever for every 0 n0 . This means p(xn , x) → 0 in (E, S). Conversely, suppose that p(xn , x) → 0 in (E, S). For c ∈ E with c >> 0 find δ > 0 and ρ ∈ S such that q(b) < δ implies b n0 and so p(xn , x) n0 . Therefore xn → x in (X, p). (ii) The proof is similar to that in (i).  Lemma 15. (see [2]) Let (X, p) be a TVS-cone metric space over a normal cone of a locally convex space (E, S), where S is the family of semi-norms defining the locally convex topology. Let {xn } and {yn } be two sequences in X and xn → x, yn → y. Then p(xn , yn ) → p(x, y) in (E, S). ε Proof. Let ε > 0 and q ∈ S be given. Choose c ∈ E with c >> 0 such that q(c) < . 6 From xn → x and yn → y, find n0 such that for all n > n0 , p(xn , x) 0 and 0 ≤ c − r (3.2) 0≤ a+b   ap(T x, T y) + b p(x, T x) + p(y, T y) + c[p(y, T x) + p(x, T y)] ≤ sp(x, y) + rp(x, T 2 x) (3.3) hold for all x, y ∈ X. Then, T has at least one fixed point. Proof. Construct a sequence {xn } as in the proof of Theorem 1. Analogously one can get that {xn } is a Cauchy sequence. Due to the completeness of (X, p), {xn } converges to some point in (X, p), namely z. Regarding Lemma 14 and Lemma 15, one can get the analogy of (2.11) and (2.10) as in the proof of Theorem 1.  Remark 17. Theorem 16 stay valid if one change TVS-cone metric space with cone metric space. Remark 18. In [10], it is concluded that some fixed point theorems on usual metric spaces and on TVS-cone metric spaces (in particular, cone metric spaces) are equivalent by using Lemma 3, Lemma 6, Lemma 9 and Remark 10. Note that the analogy of condition (3.3) can not be obtained by applying the nonlinear scalarization mapping (see Definition 2 and Lemma 3). Thus the technique that was used in [10] is not applicable for Theorem 16. References [1] Abdeljawad T., Karapnar, E. A gap in the paper ”A note on cone metric fixed point theory and its equivalence” [Nonlinear Anal. 72(5), (2010), 2259-2261], Gazi University Journal of Science 24(2), 233-234 (2011) [2] Abdeljawad, T.: Completion of Locally Convex Cone Metric Spaces and Some Fixed point Theorems, Hacettepe Journal of Mathematics and Statistics (to appear). [3] Achari, J.: On Ciric’s non-unique fixed points, Mat. Vesnik, 13 (28)no. 3, 255–257 (1976). [4] Abdeljawad, T. and Karapınar, E.: Quasicone Metric Spaces and Generalizations of Caristi Kirk’s Theorem. Fixed Point Theory Appl. (2009), doi:10.1155/2009/574387. [5] Aliprantis, C. D. and Tourky, R.: Cones and Duality, American Mathematical Society (2007). [6] Chen,G.Y., Huang,X.X., Yang, X.Q.: Vector Optimization, Springer-Verlag, Berlin, Heidelberg, Germany, 2005. ´ c, L. B.: On some maps with a nonunique fixed point. Publ. Inst. Math., 17, 52–58 (1974). [7] Ciri´ [8] Ciric, L.B., Jotic, N.: A Further Extension of Maps With Non-uniqueFixed Points, Mat. Vesnik , 50(1-2), 1–4 (1998).

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[9] Du Wei-Shih: On some nonlinear problems induced by an abstract maximal element principle J. Math. Anal. Appl., ,(347)(2), 391–399 (2008) [10] Du Wei-Shih: A note on cone metric fixed point theory and its equivalence Nonlinear Analysis, 72(5), 2259–2261 (2010). [11] Gupta, S., Ram, B.: Non-unique fixed point theorems of Ciric type, (Hindi) Vijnana Parishad Anusandhan Patrika 41(4), 217–231(1998). [12] Huang Long-Guang, Zhang Xian: Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332, 1468–1476 (2007). [13] Karapınar, E.: Fixed Point Theorems in Cone Banach Spaces Fixed Point Theory Appl. (2009), Article ID 609281, 9 pages, doi:10.1155/2009/609281 [14] Karapınar, E.: ”Couple Fixed Point on Cone Metric Spaces”, Gazi University Journal of Science, 24(1),51-58(2011). [15] Karapınar, E.: ”Some Fixed Point Theorems on the Cone Banach Spaces” Proc. of 7 ISAAC Congress, World Scientific, 606-612 (2010). [16] Karapınar, E.:”Some Nonunique Fixed Point Theorems of ?iri? type on Cone Metric Spaces” Abstr. Appl. Anal., vol. 2010, Article ID 123094, 14 pages (2010), doi:10.1155/2010/123094 [17] Karapınar, E.: ”Couple Fixed Point Theorems For Nonlinear Contractions in Cone Metric Spaces” Comput. Math. Appl. 59( 12), , Pages 3656–3668 (2010) [18] Pachpatte, B. G.: On Ciric type maps with a nonunique fixed point, Indian J. Pure Appl. Math., 10(8), 1039–1043 (1979). [19] Zeqing Liu, Li Wang, Shin Min Kang, Yong Soo Kim: On Nonunique Fixed Point Theorems, Applied Mathematics E-Notes,8, 231–237 (2008). erdal karapınar, ˙ Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey E-mail address: [email protected] E-mail address: [email protected]

97

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,98-101,COPYRIGHT 2012 EUDOXUS PRESS,LLC

A simple proof of the Nadler’s contraction principle M. Eshaghi Gordji, H. Baghani and S. Davoudi

Department of Mathematics, Semnan University, P.O. Box 35195-363, Semnan, Iran Center of Excellence in Nonlinear Analysis and Applications (CENAA), Semnan University, Iran E-mail:[email protected], [email protected], samaneh− 84− [email protected]

Abstract: We give a simple proof of the Nadler’s contraction theorem. Moreover, we find the sufficient condition for the uniqueness of fixed points of set–valued contractions. 2000 Mathematics Subject Classification: 54H25 Keywords: Fixed point, Complete metric space, Set–valued contraction 1. Introduction In 1922, the Polish mathematician Stefan Banach [1] proved a theorem which ensures, under appropriate conditions, the existence and uniqueness of a fixed point. Nadler in 1969 proved a multi–valued extension of the Banach contraction theorem. Many fixed point theorems have been proved by various authors as generalizations to the Nadlers theorem where the contractive nature of the map is weakened along with some additional requirements. Let (X, d) be a metric space and let CB(X) denotes the collection of all nonempty closed bounded subsets of X. For A, B ∈ CB(X) and x ∈ X, define D(x, A) := inf{d(x, a); a ∈ A} and H(A, B) := max{sup D(a, B), sup D(b, A)}. a∈A

b∈B

It is easy to see that H is a metric on CB(X). Moreover, (CB(X), H) is complete metric space if (X, d) is a complete metric space. H is called the Hausdorff metric. Note that a point p ∈ X is said to be a fixed point of multi–valued mapping T : X → CB(X) if p ∈ T (p) [2]. In this paper, we will find out the sufficient conditions for the existence and uniqueness of a fixed point for set–valued contraction mapping T : X → CB(X).

98

2

A simple proof of the Nadler’s contraction principle 2. Main results

Fundamental contraction Inequality: If T : X → CB(X) is a contraction with constant L < 1 and inf {d(x1 , y) + d(y, z)} ≤ D(x1 , T x1 ) + D(T x1 , z),

y∈T x1

(2.1)

for all x1 , x2 ∈ X, z ∈ T x2 , then d(x1 , x2 ) ≤

1 (D(x1 , T x1 ) + D(x2 , T x2 )) 1−L

(2.2)

for all x1 , x2 ∈ X. Proof. It follows from triangle inequality that d(x1 , x2 ) ≤ d(x1 , y) + d(y, z) + d(z, x2 ) for all x1 , x2 , y, z ∈ X. Then by (2.1), we have d(x1 , x2 ) ≤ inf {d(x1 , y) + d(y, z)} + d(z, x2 ) y∈T x1

≤ D(x1 , T x1 ) + D(z, T x1 ) + d(z, x2 ) for all x1 , x2 , z ∈ X. On the other hand by (2.1), we obtain that inf {D(z, T x1 ) + d(z, x2 )} ≤ inf {d(y, z) + d(z, x2 )}

z∈T x2

z∈T x2

≤ D(y, T x2 ) + D(x2 , T x2 ) for all x1 , x2 ∈ X, and all y ∈ T X1 . Therefore, we have d(x1 , x2 ) ≤ D(x1 , T x1 ) + D(y, T x2 ) + D(x2 , T x2 ) ≤ D(x1 , T x1 ) + H(T x1 , T x2 ) + D(x2 , T x2 ) ≤ D(x1 , T x1 ) + Ld(x1 , x2 ) + D(x2 , T x2 ) and d(x1 , x2 ) ≤

1 (D(x1 , T x1 ) + D(x2 , T x2 )) 1−L

for all x1 , x2 ∈ X.



Theorem 2.1. Let (X, d) be a complete metric space. If T : X → CB(X) is a contraction with constant L < 1, then T has a fixed point. Moreover, if T satisfies (2.1), then the fixed point of T is unique. Proof. Let x0 ∈ X be arbitrary. Choose an element x1 ∈ X such that x1 ∈ T x0 . If x0 = x1 , then x0 is a fixed point of T , and the proof of theorem is complete. Therefore, we suppose that x0 6= x1 . Let L1 ∈ (L, 1) be arbitrary. Since D(x1 , T x1 ) ≤ H(T x0 , T x1 ) < L1 d(x0 , x1 ), then there exists a x2 ∈ T x1 such that d(x1 , x2 ) ≤ L1 d(x0 , x1 ).

99

M. Eshaghi Gordji, H. Baghani and S. Davodi

3

Continuing this process and having chosen xn in X, we obtain xn+1 in X such that xn+1 ∈ T xn and xn 6= xn+1 such that d(xn , xn+1 ) ≤ L1 d(xn−1 , xn ) for all n ≥ 2. Hence, 1 (d(xn , xn+1 ) + d(xm , xm+1 )) 1−L (2.3) 1 ≤ (Ln1 d(x0 , x1 ) + Lm d(x , x )) 0 1 1 1−L for all m > n. By taking n, m → ∞ in above inequality, it follows that {xn } is a Cauchy sequence. By completeness of X, there exists some point z ∈ X such that xn → z. It follows that D(z, T z) = D( lim xn+1 , T z) = lim D(xn+1 , T z) d(xn , xm ) ≤

n→∞

n→∞

≤ lim H(T xn , T z) ≤ lim L(d(xn , z)) n→∞

n→∞

≤ lim d(xn , z) = 0. n→∞

This means that D(z, T z) = 0. In the other words z ∈ T z and z is a fixed point of T . Now suppose that T satisfies the condition (2.1). We show that z is the unique fixed point of T . Let r ∈ X be another fixed point of T , that is r ∈ T r. From (2.2), we have 1 d(z, r) ≤ (D(z, T z) + D(r, T r)) = 0 1−L therefore z = r.  In inequality (2.3) of Theorem 2.1, if m → ∞, then we have Ln1 d(xn , z) ≤ d(x0 , x1 ). 1−L The importance of this latter inequality is as follows. Suppose we are willing to accept an error of , i.e., instead of the actual fixed point of T we will be satisfied with a point xn satisfying d(xn , z) < , and suppose also that we start our iteration at some point x0 in X. Since we want d(xn , z) < , we just have to pick N so large that LN 1 1−L d(x0 , x1 )

< . Now the quantity A = d(x0 , x1 ) is something that we can compute after the first iteration and we can then compute how large N has to be by taking the log of the above inequality and solving for N (remembering that log(L) and log(L1 ) are negative). The result is: Stopping rule : If A = d(x0 , x1 ) and N>

log() + log(1 − L) − log(A) log(L1 )

then d(xN , z) < . From a practical programming point of view, this inequality allows us to express our iterative algorithm with a for loop rather than a while loop, but it has another interesting interpretation. Suppose we take  = 10−m in our stopping rule m inequality. What we see is that the growth of N with m is a constant plus | log(L 1 )| more iteration steps. Stated a little differently, if we need N iterative steps to get m

100

4

A simple proof of the Nadler’s contraction principle

decimal digits of precision, then we need another N to double the precision to 2m digits. S Example 2.2. Let X = { 12 , 14 , ..., 21n , ...} {0, 1}, d(x, y) = |x − y| for all x, y ∈ X. Define mapping F : X → CB(X) as ( 1 { 2n+1 } x = 21n , n = 0, 1, ..., F (x) = {0} x = 0. One can show that F satisfies the conditions of Theorem 2.1, and 0 is the unique fixed point of F . References [1] S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux´equations int´egrales. Fund. Math. 3 (1922), 133-181. [2] S. B. Nadler, Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475-488.

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JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,102-107,COPYRIGHT 2012 EUDOXUS PRESS,LLC

Some New Ostrowski Type Inequalities W. T. Sulaiman Department of Computer Engineering College of Engineering University of Mosul, Iraq. Abstract 2000 Mathematics Subject classification : 26D10, 26D15. Key words : Ostrowski' inequality, Integral inequality.

1. Introduction A. M. Ostrowski proved the following interesting and useful inequality (see[1, p.468]) b ⎛ 1 (x − a +2 b )2 ⎞ 1 ⎜ + ⎟ ′ f ( t ) dt (1) < ⎜ 4 (b − a) 2 ⎟ (b − a ) f ∞ , b − a ∫a ⎝ ⎠ for all x ∈ [a, b] , where f : [a, b] ⊆ ℜ → ℜ is continuous on [a, b] and differentiable on (a, b) whose derivative f ′ : (a, b) → ℜ is bounded on (a, b) , that is f ′ ∞ = sup f ′( x) < ∞ .

f ( x) −

x∈( a ,b )

The object of this paper is to present many new inequalities similar to Ostrowskis inequality.

2. New Results We state and prove the following Theorem 2.1 Let f , g : [a, b] ⊆ ℜ → ℜ be continuous on [a, b] and differentiable on (a, b) whose derivative f ′, g ′ : (a, b) → ℜ are bounded on (a, b) , that is f ′ ∞ = sup f ′( x) < ∞ , g ′ ∞ = sup g ′( x) < ∞ . x∈( a ,b )

x∈( a ,b )

Suppose further that f and g are non-decreasing. Then b

f ( x) g ( x) −

1 f (t ) g (t ) dt b − a ∫a ≤

f ( x) g ′

∞

+ g ( x) f ′

b−a

Proof. b

f ( x) g ( x) −

1 f (t ) g (t ) dt b − a ∫a b

=

1 ( f ( x) g ( x) − f (t ) g (t )) dt b − a ∫a

102

∞

⎛ ( x − a ) 2 + ( x − b) 2 ⎜⎜ 2 ⎝

⎞ ⎟⎟ ⎠

(2)

SULAIMAN: OSTROWSKI INEQUALITY

b x

≤

1 b − a ∫a

′

∫ ( f (u ) g (u )) du dt t

b x

=

1 b − a ∫a

∫ ( f (u ) g ′(u ) + f ′(u ) g (u )) du dt t

f ( x) g ′

≤

∞

+ g ( x) f ′

b

∞

b−a

= =

f ( x) g ′

∞

a

+ g ( x) f ′

∞

b−a f ( x) g ′

∞

∫ x − t dt

+ g ( x) f ′

∞

b−a

b ⎛x ⎞ ⎜ ∫ ( x − t ) dt + ∫ (t − x) dt ⎟ ⎜ ⎟ x ⎝a ⎠ 2 2 ⎛ ( x − a ) + ( x − b) ⎞ ⎜⎜ ⎟⎟ . 2 ⎝ ⎠

Theorem 2.2. Let f , g : [a, b] ⊆ ℜ → ℜ be continuous on [a, b] and differentiable on (a, b) whose derivative f ′, g ′ : (a, b) → ℜ are bounded on (a, b) , that is f ′ ∞ = sup f ′( x) < ∞ , g ′ ∞ = sup g ′( x) < ∞ . x∈( a ,b )

x∈( a ,b )

Suppose further

sup

f ( x) − f ′( x) g 2 ( x)

x∈( a ,b )

= M,

sup

g ( x) − g ′( x)

x∈( a ,b )

g 2 ( x)

= N,

Then b N g′ ∞ + M f ′ 1 f ( x) f (t ) − dt ≤ ∫ g ( x) b − a a g (t ) b−a

∞

⎛ ( x − a ) 2 + ( x − b) 2 ⎜⎜ 2 ⎝

⎞ ⎟⎟. ⎠

Proof. b b ⎛ f ( x) f (t ) ⎞ f ( x) 1 f (t ) 1 ⎜ ⎟ dt − = − dt ∫ ∫ g ( x) b − a a g (t ) b − a a ⎜⎝ g ( x) g (t ) ⎟⎠ ′ b x ⎛ f (u ) ⎞ 1 ⎟ du dt ⎜ = b − a ∫a ∫t ⎜⎝ g (u ) ⎟⎠

′ ⎛ f (u ) ⎞ ∫t ⎜⎜⎝ g (u ) ⎟⎟⎠ du dt

b x

1 ≤ b − a ∫a

⎛ g (u ) f ′(u ) − f (u ) g ′(u ) ⎞ ⎟⎟ du dt g 2 (u ) ⎠ t

b x

=

1 b − a ∫a

=

1 b − a ∫a

∫ ⎜⎜⎝

⎛ f ′(u )( g (u ) − g ′(u ) ) + g ′(u ) ( f ′(u ) − f (u ) ) ⎞ ⎟⎟ du dt g 2 (u ) ⎠ t

b x

∫ ⎜⎜⎝

103

(3)

SULAIMAN: OSTROWSKI INEQUALITY

⎛ f ′(u ) g (u ) − g ′(u ) + g ′(u ) f ′(u ) − f (u ) ⎞ ⎟ du dt ⎟ g 2 (u ) t ⎠ b N g′ ∞ + M f ′ ∞ ≤ ∫a x − t dt b−a b x

≤

=

1 b − a ∫a

∫ ⎜⎜⎝

N g′

+M f′

∞

b−a

∞

⎛ ( x − a ) 2 + ( x − b) 2 ⎜⎜ 2 ⎝

⎞ ⎟⎟ . ⎠

Theorem 2.3. Let g : [a, b] ⊆ ℜ → ℜ be continuous on [a, b] and differentiable on (a, b) whose derivative g ′ : (a, b) → ℜ are bounded on (a, b) , that is g ′ ∞ = sup g ′( x) < ∞ , x∈( a ,b )

and let f : g ([a, b]) → ℜ be also continuous and has a bounded derivative, that is f ′ ∞ = sup f ′( x) < ∞ . x∈g ([ a ,b ] )

Then fοg ( x) −

b f ′ ∞ g′ 1 fοg (t ) dt ≤ ∫ b−a a b−a

∞

⎛ ( x − a ) 2 + ( x − b) 2 ⎜⎜ 2 ⎝

⎞ ⎟⎟ . ⎠

(4)

Proof.

fοg ( x) −

b

b

1 1 ( fοg ( x) − fοg (t )) dt fοg (t ) dt = ∫ b−a a b − a ∫a b x

≤

1 b − a ∫a

′

∫ ( fοg (u )) du dt t

b x

=

≤ =

=

1 b − a ∫a

f′

∞

∫ f ′( g (u ) g ′(u ) du dt t

g′

b x

∞

b−a f′

∞

g′

a t

b

∞

b−a

f′

∞

g′

b−a

∫ ∫ du dt ∫ x − t dt a

∞

⎛ ( x − a ) 2 + ( x − b) 2 ⎜⎜ 2 ⎝

⎞ ⎟⎟ . ⎠

Theorem 2.4. Let f : [a, b] ⊆ ℜ → ℜ is continuous on [a, b] and differentiable on (a, b) whose derivative f ′ : (a, b) → ℜ is bounded on (a, b) , that is f ′ ∞ = sup f ′( x) < ∞ . x∈( a ,b )

Let 0 < a < b. Then

104

SULAIMAN: OSTROWSKI INEQUALITY

b b

1 (b − a ) 2

f ( xy) −

∫ ∫ f (st ) dsdt ≤ a a

((b − a) x( ( y − a )

2

)

(

+ ( y − b) 2 + (b 2 − a 2 ) ( x − a ) + ( x − b) 2 2

)) 2(bf−′ a) ∞

2

(5)

.

Proof.

1 f ( xy) − (b − a) 2 = =

≤ =

b b

1 (b − a ) 2

b b xy

f′

(b − a ) 2 f′

∫ ∫ ∫ f ′(u ) du dsdt a a st

b b xy

∫ ∫ ∫ f ′(u ) du dsdt a a st

∫ ∫ xy − st dsdt a a

b b

∞

(b − a ) 2

f′

a a

b b

∞

∫ ∫ xy − xt + xt − st dsdt a a

b b

∞

(b − a ) 2

=

a a

∫ ∫ ( f ( xy) − f (st )) dsdt

f′

≤

∫ ∫ f (st ) dsdt

1 (b − a) 2

1 (b − a ) 2

≤

b b

∫ ∫ ( x y − t + t x − s )dsdt a a

b ⎛ ⎛b ⎞⎛ b ⎞⎞ ⎜ ((b − a) x ) y − t dt + ⎜ t dt ⎟ ⎜ x − t dt ⎟ ⎟ 2 ⎜ ∫ ∫ ∫ ⎜ ⎟⎜ ⎟⎟ (b − a) ⎝ a ⎝a ⎠⎝ a ⎠⎠

∞

(

(

)

(

≤ (b − a ) x ( y − a ) + ( y − b) 2 + (b 2 − a 2 ) ( x − a ) + ( x − b) 2 2

2

)) 2(bf−′ a) ∞

2

.

Theorem 2.5. Let f : [a, b] ⊆ ℜ → ℜ is continuous on and bounded on [a, b] that is f

∞

= sup f ( x) < ∞ . x∈[ a ,b ]

Define F by x

F ( x) = ∫ f (t ) dt .

(6)

a

Then

(

)

b f ∞ 1 F ( x) − f ( t ) dt ≤ ( x − a ) 2 + ( x − b) 2 . ∫ b−a a 2(b − a )

Proof. b

F ( x) −

b

1 1 (F ( x) − F (t )) dt f (t ) dt = ∫ b−a a b − a ∫a

105

(7)

SULAIMAN: OSTROWSKI INEQUALITY

b x

=

1 F ′(u ) dudt b − a ∫a ∫t

=

1 f (u ) dudt b − a ∫a ∫t

b x

b

f

≤

b − a ∫a

=

(( x − a) 2(b − a )

∞

f

x − t dt

∞

2

)

+ ( x − b) 2 .

Theorem 2.6. Let f , g : [a, b] ⊆ ℜ → ℜ be continuous on [a, b] and differentiable on (a, b) whose derivative f ′, g ′ : (a, b) → ℜ are bounded on (a, b) , that is f ′ ∞ = sup f ′( x) < ∞ , g ′ ∞ = sup g ′( x) < ∞ . x∈( a ,b )

x∈( a ,b )

Then b

2 f ( x) g ( x) −

b

g ( x) f ( x) f (t ) dt − g (t ) dt ∫ b−a a b − a ∫a

≤

g ( x) f ′

∞

+ f ( x) g ′

∞

2(b − a )

(( x − a)

2

)

+ ( x − b) 2 .

(8)

Proof. b

2 f ( x) g ( x) −

b

g ( x) f ( x) f (t ) dt − g (t ) dt ∫ b−a a b − a ∫a

b

b

=

g ( x) ( f ( x) − f (t ) ) dt + f ( x) ∫ (g ( x) − g (t )) dt ∫ b−a a b−a a

=

g ( x) f ( x) f ′(u ) du dt + g ′(u ) du dt ∫ ∫ b−a a t b − a ∫a ∫t

b x

≤

=

b x

g ( x) f ′

∞

+ f ( x) g ′

b

∞

b−a

g ( x) f ′

∞

+ f ( x) g ′

2(b − a )

∫ x − t dt a

∞

(( x − a)

2

)

+ ( x − b) 2 .

Theorem 2.7. Let f , g : [a, b] ⊆ ℜ → ℜ be continuous on [a, b] and differentiable on (a, b) whose derivatives f ( m +1) , g ( n +1) : (a, b) → ℜ are bounded on (a, b) , that is

f ( m +1)

∞

= sup f ( m+1) ( x) < ∞ , g ( n +1) x∈( a ,b )

Then

106

∞

= sup g ( n +1) ( x) < ∞ . x∈( a ,b )

SULAIMAN: OSTROWSKI INEQUALITY

b

f ( m ) ( x) g ( n ) ( x) +

(

)

1 f ( m ) (t ) g ( n ) (t ) − f ( m ) ( x) g ( n ) (t ) − g ( n ) ( x) f ( m ) (t ) dt b − a ∫a ≤

f

( m +1) ∞

g ( n +1)

∞

3(b − a )

(( x − a)

3

+ ( x − b) 3 ).

(9) Proof. b

f ( m ) ( x) g ( n ) ( x) +

b

(

1 = f b − a ∫a

(m)

(

1 f ( m ) (t ) g ( n ) (t ) − f b − a ∫a

)

1 ( f ( m ) ( x) − f b − a ∫a

≤

1 f ( m +1) (u ) du ∫ g ( n +1) (u ) dudt ∫ ∫ b−a a t t

b

≤ =

g ( n +1)

( m +1) ∞

(t ) )(g ( n ) ( x) − g ( n ) (t ) )dt

b

∞

b−a f

(m)

x

b x

∞

)

( x) g ( n ) (t ) − g ( n ) ( x) f ( m ) (t ) dt

( x) g ( n ) ( x) + f ( m ) (t ) g ( n ) (t ) − f ( m ) ( x) g ( n ) (t ) − g ( n ) ( x) f ( m ) (t ) dt

=

f ( m +1)

(m)

g ( n +1)

3(b − a )

∫ (x − t)

2

dt

a

∞

(( x − a)

3

+ ( x − b) 3 ).

References [1] D. S. Mitrinović, J. Pečarić and A. M. Fink, Inequalities involving functions and Their Integrals and derivatives, Kluwer Academic publishers, Dordrecht, 1991.

107

JOURNAL OF APPLIED FUNCTIONAL ANALYSIS, VOL.7, NO’S 1-2,108-117,COPYRIGHT 2012 EUDOXUS PRESS,LLC

Logarithmic order and type on some weighted function spaces A. El-Sayed Ahmed Taif University , Faculty of Science Mathematics Department Box 888 El-Hawiyah, El-Taif 5700, Saudi Arabia and Sohag University, Faculty of Science, Department of Mathematics, 82524Sohag- Egypt e-mail: [email protected] A. Kamal The High Institute of Computer Science, Al-Kawser city at Sohag Egypt

Abstract We study the space QK,ω (p, q), of analytic functions on the unit disk in terms of a q+2

nondecreasing function K. The relation between QK,ω (p, q) and Bω p spaces, which have attracted considerable attention, is given by studying the growth order of K. The counterpart Q# K,ω (p, q) of QK,ω (p, q) for the meromorphic case is also considered and investigated. We note that some characterizations of Q# K,ω (p, q) and QK,ω (p, q) are different.

1

Introduction

Let D = {z : |z| < 1} be the open unit disk in the complex plane C. Recall that the well known Bloch space (cf. [4]) is defined as follows: B = {f : f analytic in D and sup(1 − |z|2 )|f 0 (z)| < ∞}, z∈D

the little Bloch space B0 (cf. [4]) is a subspace of B consisting of all f ∈ B such that lim (1 − |z|2 )|f 0 (z)| = 0.

|z|→1−

The Dirichlet space is defined by Z 2

|f 0 (z)| dA(z) < ∞},

D = {f : f analytic in D and D

where dA(z) is the Euclidean area element dxdy. Let 0 < q < ∞. Then the Besov-type spaces ( ) Z  q−2 q q 0 2 2 2 B = f : f analytic in D and sup |f (z)| 1 − |z| (1 − |ϕa (z)| ) dA(z) < ∞ a∈D

D

are introduced and studied intensively by Stroethoff (cf. [13]). Here, ϕa (z) stands for the M¨ obius transformation of D given by a−z , where a ∈ D. ϕa (z) = 1−a ¯z AMS: Primary 30D45 , Secondary 46E15. Key words and phrases: QK,ω (p, q) spaces, Q# K,ω (p, q) spaces and α-weighted Bloch space.

1108

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

2 In 1994, Aulaskari and Lappan [4] introduced a class of holomorphic functions, the so called Qp -spaces as follows: ( ) Z 2 p 0 Qp = f : f analytic in D and sup |f (z)| g (z, a)dA(z) < ∞ , a∈D

where 0 < p < ∞ and the weight function

D

¯z 1 − a g(z, a) = log a−z

is defined as the composition of the M¨ obius transformation ϕa and the fundamental solution of the two-dimensional real Laplacian. The weight function g(z, a) is actually Green’s function in D with pole at a ∈ D. For a point a ∈ D and 0 < r < 1, the pseudo-hyperbolic disk D(a, r) with pseudo-hyperbolic center a and pseudo-hyperbolic radius r is defined by D(a, r) = ϕa (rD). The pseudo-hyperbolic disk D(a, r) is also an Euclidean disk: its Euclidean center and Euclidean radius are 2 )r (1−r 2 )a and (1−|a| , respectively (see [13]). Let A denote the normalized Lebesgue area measure on D, and for 1−r 2 |a|2 1−r 2 |a|2 a Lebesgue measurable set K1 ⊂ D, denote by |K1 | the measure of K1 with respect to A. It follows immediately that: |D(a, r)| =

(1 − |a|2 )2 2 r . (1 − r2 |a|2 )2

Let K : [0, ∞) → [0, ∞) be a nondecreasing function. For 0 < p < ∞, −2 < q < ∞, we say that a function f analytic in D belongs to the space QK (p, q) (cf. [19]), if Z  q p kf kpQK (p,q) = sup |f 0 (z)| 1 − |z|2 K(g(z, a))dA(z) < ∞. a∈D

D

Using the above mentioned function K, several authors have been studied some classes of holomorphic and meromorphic function spaces (see [2, 7, 8, 10, 11, 15, 16, 17, 18, 19, 20] and others). Now, given a reasonable function ω : (0, 1] → (0, ∞), the weighted Bloch space Bω (see [6]) is defined as the set of all analytic functions f on D satisfying (1 − |z|)|f 0 (z)| ≤ Cω(1 − |z|),

z ∈ D,

for some fixed C = Cf > 0. In the special case where ω ≡ 1, Bω reduces to the classical Bloch space B. Here, the word ”reasonable” is a non-mathematical term; it was just intended to mean that the ”not too bad” and the function satisfy some natural conditions. Now, we introduce the following definitions: Definition 1.1 For a given reasonable function ω : (0, 1] → (0, ∞) and for 0 < α < ∞. An analytic function f on D is said to belong to the α−weighted Bloch space Bωα if kf kBωα = sup z∈D

(1 − |z|)α 0 |f (z)| < ∞. ω(1 − |z|)

Definition 1.2 For a given reasonable function ω : (0, 1] → (0, ∞) and for 0 < α < ∞. An analytic function f α on D is said to belong to the little weighted Bloch space Bω,0 if α kf kBω,0 = lim

|z|→1−

(1 − |z|)α 0 |f (z)| = 0. ω(1 − |z|)

Throughout this paper and for some techniques we consider the case of ω 6≡ 0. The logarithmic order (log-order) of the function K(r) is defined as ρ = lim sup r→∞

ln+ ln+ K(r) , ln r

+

where ln x = max{ln x, 0}. If 0 < ρ < ∞, the logarithmic type (log-type) of the function K(r) is defined as σ = lim sup r→∞

ln+ K(r) . rρ

Note that if f is an entire function, then the growth order of f is just the log-order of M (r), the maximum modulus function of f. 109

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

3 Definition 1.3 [10, 11] For a nondecreasing function K : [0, ∞) → [0, ∞), 0 < p < ∞, −2 < q < ∞ and for a given reasonable function ω : (0, 1] → (0, ∞), an analytic function f in D is said to belong to the space QK,ω (p, q) if   q Z (1 − |z|) K g(z, a) p kf kpQK,ω (p,q) = sup |f 0 (z)| dA(z) < ∞. ω p (1 − |z|) a∈D D Definition 1.4 For a given reasonable function ω : (0, 1] → (0, ∞), and let 0 < p < ∞, −2 < q < ∞, 0 < s < ∞ and an analytic function f in D is said to belong to the spaces Fω (p, q, s) if Z g s (z, a) p kf kpFω (p,q,s) = sup |f 0 (z)| (1 − |z|2 )q p dA(z) < ∞. ω (1 − |z|) a∈D D Moreover, if

Z p

|f 0 (z)| (1 − |z|2 )q

lim

|a|→1−

D

g s (z, a) dA(z) = 0, − |z|)

ω p (1

then f ∈ Fω,0 (p, q, s). Since every M¨ obius map ϕ can be written as ϕ(z) = eiθ ϕa (z), where θ is real. We assume throughout the paper that Z1

1 (1 − r2 )−2 K(log )rdr < ∞. r

0

We can define an auxiliary function as follows: ϕK (s) = sup 0 0. Now, since we assume that ω is non-decreasing, then we obtain that Z 2 q 1 p (1 − |z| ) |f 0 (z)| p dA(z) kf kpQK,ω (p,q) ≥ K(log ) r E(a,r) ω (1 − |z|) Z K(log r1 )(1 − r)p (1 − |a|)q p ≥ |f 0 (z)| dA(z) p ω ((1 − r)(1 − |a|)) E(a,r)

Since |f 0 (z)|p is a subharmonic function, then Z p |f 0 (z)|p dA(z) ≥ |E(a, r)| |f 0 (a)|p = r2 (1 − |a|)2 |f 0 (a) . E(a,r)

Then we obtain kf kpQK,ω (p,q)

≥

K(log r1 )(1 − r)p−1 (1 − |a|)q+2 0 |f (a)|p ω p ((1 − r)(1 − |a|))

≥

C1

K(log r1 )(1 − r)p (1 − |a|)q+2 0 |f (a)|p ω p ((1 − |a|))

which implies that kf kpQK,ω (p,q)

kf kp q+2 ≤ Bω

C1 (1 − r)p−1 K(log r1 )

p

.

(1)

Our proposition is therefore established. Next, we give the following proposition. Proposition 2.2 If the log-order ρ and the log-type σ of a nondecreasing function K(r) satisfy one of the following conditions: (i) ρ > 1 (ii) ρ = 1 and 0 < σ < ∞, then kf kpQK,ω (p,q) ⊂ kf kp q+2 , where 0 < p < ∞, −2 < q < ∞ and ω : (0, 1] → (0, ∞). Bω

p

Proof: By Proposition 2.1, it suffices to show that each non-constant weighted α-Bloch function f can not belong to the spaces QK,ω (p, q). In fact, if either the log-order ρ of K(r) is greater than 0, or the log-order ρ of K(r) equals 1 and the log-type σ of K(r) is greater than 2, then there exists a sequence {rn } with rn → ∞ as n → ∞ such that lim

n→∞

or

ln+ ln+ K(rn ) =ρ>1 ln rn

(2)

ln+ K(rn ) =λ>0 rn

(3)

σ = lim

n→∞

In the case (2) or (3), we obtain K(rn ) = const. eλrn Let f be a non-constant weighted α-Bloch function. Then lim

n→∞

kf kp q+2 = sup{ Bω

p

z∈D

However, by (1) and (4) we have kf kpQK,ω (p,q)

(1 − |z|2 )q 0 |f (z)|p : z ∈ D} 6= 0. ω p (1 − |z|) Z

q K(g(z, a)) dA(z) ω p (1 − |z|) a∈D D 1 ≥ πkf kp q+2 (1 − tn )p K(log ) 6→ ∞. t p n B

= sup

|f 0 (z)|

p



ω

Hence, f ∈ QK,ω (p, q). 111

1 − |z|2

(4)

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

5 Theorem 2.1 For each nondecreasing function K : [0, ∞) → [0, ∞) and ω : (0, 1] → (0, ∞), satisfying both of the following: Condition A. There exist a constant p > 1 such that lim

n→∞

K(r) = c 6= 0. rp

Condition B. The log-order ρ and the log-type σ satisfy one of the following cases: (i) 0 ≤ ρ < 1 (ii) ρ = 1 and 0 < σ < ∞. q+2

Then QK,ω (p, q) = Bω p . Proof: Let

K(r) =C= 6 0. rp for some p ∈ (1, ∞). Then there exists a fixed r1 ∈ (0, 1) such that lim

r→∞

K(r) c ≤ ≤ c + 1, 0 < r < r1 . 2 rp

(5)

We may choose r0 ∈ (0, 1) such that z ∈ D\D(a, r0 ) ⇒ g(z, a) = log

1 < r1 . |ϕa (z)|

(6)

Now we first suppose that f ∈ QK,ω (p, q) with Z  q K(g(z, a)) p sup |f 0 (z)| 1 − |z|2 dA(z) = C, ω p (1 − |z|) a∈D D and write

Z

 q (g(z, a))p p |f 0 (z)| 1 − |z|2 dA(z) ω p (1 − |z|) D Z q (g(z, a))p  p dA(z) = |f 0 (z)| 1 − |z|2 ω p (1 − |z|) D(a,r0 ) Z  q (g(z, a))p p |f 0 (z)| 1 − |z|2 + dA(z) ω p (1 − |z|) D\D(a,r0 ) = I1 + I2 . p+2 q

Since QK,ω (p, q) ⊂ Bω

(7)

from Proposition 2.1, we have Z  q (g(z, a))p p I1 = |f 0 (z)| 1 − |z|2 dA(z) ω p (1 − |z|) D(a,r0 ) Z  −2  1 p 1 − |z|2 dA(z) log ≤ kf kp q+2 ϕa (z) p D(a,r0 ) Bω Z r0  1 p r(1 − r2 )−2 log dr = 2πkf kp q+2 r p 0 B ω

=

2πkf kp q+2 I(r0 , p), Bω

(8)

p

where the integral I(r0 , p) < ∞ for 0 < r0 < 1 and 1 < p < ∞. On the other hand, by (5) and (6) Z  q (g(z, a))p p dA(z). |f 0 (z)| 1 − |z|2 I2 = ω p (1 − |z|) D\D(a,r0 ) Z  q K(g(z, a)) 2 p |f 0 (z)| 1 − |z|2 ≤ dA(z) c D\D(a,r0 ) ω p (1 − |z|) ≤

2C < ∞. c

(9)

Consequently, by (7), (8), and (9), Z  q (g(z, a))p p dA(z) ≤ sup{I1 + I2 } < ∞. sup |f 0 (z)| 1 − |z|2 ω p (1 − |z|) a∈D D 112

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

6 p+2

Thus f ∈ QK,ω (p, q). Hence QK,ω (p, q) ⊂ Bω q . Since f ∈ QK,ω (p, q), f must be a weighted Bloch function in D, and Z  q (1 − |ϕ (z)|2 )p p a dA(z) = C < ∞. sup |f 0 (z)| 1 − |z|2 ω p (1 − |z|) a∈D D It follows from (5) and (6) that Z |f 0 (z)|

p



1 − |z|2

 q K log

1 ϕa (z)



dA(z) ω p (1 − |z|) p  Z  q log ϕ 1(z) a p (c + 1) |f 0 (z)| 1 − |z|2 dA(z) ω p (1 − |z|) D\D(a,r0 )

J2 =

D\D(a,r0 )

≤ =

(c + 1)C.

(10)

Since f ∈ QK,ω (p, q), f must be weighted Bloch function in D. Similar to (8) we have   Z  q K log ϕ 1(z) a p 0 2 dA(z) J1 = |f (z)| 1 − |z| ω p (1 − |z|) D(a,r0 ) Z r0 1 (1 − r2 )−2 K(log )rdr ≤ 2πkf kp q+2 r p 0 B ω q+2 p

Now we show that the integral

R r0 0

Z

2πBω (1 − r02 )2

≤

r0 0

1 K(log )rdr r

(11)

K(log r1 )rdr in (11) is convergent. Setting t = log r1 , we have Z r0 Z +∞ K(t) 1 dt. J(K) = K(log )rdr = r e2t 0 t0

If K(t) satisfies condition (i), then for given ² > 0 with ρ + ² < 1, there exists t1 > t0 such that K(t) < etρ+² < et ,

t ≥ t1 .

Therefore, Z J(K)

= ≤

t1

t0 Z t1 t0

K(t) dt + e2t K(t) dt + e2t

Z

+∞

t1 Z +∞ t1

K(t) dt e2t 1 dt < ∞. e2t

(12)

If K(t) satisfies condition (ii), then for given ² > 0 with 0 < σ + 2ε < 2, there exists t2 > t0 such that K(t) < e(σ+²)t < e(2−ε)t ,

t ≥ t2 .

Thus Z J(K)

= Z ≤ Z =

t2 t0 t2

t0 t2 t0

K(t) dt + e2t K(t) dt + e2t K(t) dt + e2t

Z Z Z

+∞ t2 +∞ t2 +∞

K(t) dt e2t e(2−ε)t dt e2t e−εt dt < ∞.

t2

Therefore, by (10) and (11), we get Z  q K(g(z, a)) p sup |f 0 (z)| 1 − |z|2 dA(z) = sup{J1 + J2 } < ∞, ω p (1 − |z|) a∈D D a∈D which implies that f ∈ QK,ω (p, q). The proof is therefore completed. By the proof of Theorem 2.1, we have 113

(13)

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

7 Corollary 2.1 Let K : [0, ∞) → [0, ∞) be a nondecreasing function. If K(r) satisfies Condition A in Theorem q+2

2.1 for some p, 1 < p < ∞, then QK,ω (p, q) ⊂ Bω p . q+2

Remark 2.1 In the case p > 1, we know that QK,ω (p, q) = Bω p q+2 p

QK,ω (p, q) ⊂ Bω

(see [11]), but now we have always that

.

Corollary 2.2 Let K : [0, ∞) → [0, ∞) be a bounded and nondecreasing function. If lim

r→0

K(r) = C 6= ∞. rp

q+2

holds for some p > 1, then QK,ω (p, q) = Bω p .

3

Meromorphic classes Q# K,ω (p, q)

For a meromorphic function f a natural analogue of |f 0 (z)| is the spherical derivative f # (z) =

|f 0 (z)| . (1 + |f (z)|2 )

Corresponding to the definitions of the spaces Fω (p, q, s) and Fω,0 (p, q, s), we define the classes Fω# (p, q, s) and # Fω,0 (p, q, s) as follows: Definition 3.1 Let 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. A function f meromorphic in D is said to belong to the class Fω# (p, q, s) if Z  p  q g s (z, a) dA(z) < ∞. kf kp # = sup f # (z) 1 − |z|2 Fω p,q,s ω p (1 − |z|) a∈D D Moreover, if

Z 

kf kp #

Fω,0 p,q,s

then f ∈

= lim

|a|→1−

D

p  q g s (z, a) dA(z) = 0, f # (z) 1 − |z|2 ω p (1 − |z|)

Fω# (p, q, s).

# Therefore we define the classes Mω# (p, q, s) and Mω,0 (p, q, s) as follows.

Definition 3.2 Let 0 < p < ∞, −2 < q < ∞ and 0 ≤ s < ∞. A function f meromorphic in D is said to belong to the class Mω# (p, q, s) if Z  p  q (1 − |ϕ (z)|2 )s a dA(z) < ∞. kf kp # = sup f # (z) 1 − |z|2 Mω p,q,s ω p (1 − |z|) a∈D D Moreover, if kf kp

Z 

# Mω,0 p,q,s

= lim

|a|→1−

f # (z)

p 

1 − |z|2

D

q (1 − |ϕ (z)|2 )s a dA(z) = 0, ω p (1 − |z|)

then f ∈ Mω# (p, q, s). Let Nωα be the class of all normal functions in D. We recall that a function f meromorphic in D is called to be normal if and only if (1 − |z|2 )α # f (z) < ∞. sup z∈D ω(1 − |z|) We will need the following theorem in future: Theorem 3.1 [14, 15] Let 2 ≤ q < ∞, and let 0 < p < ∞ and let 0 < r < 1. Then a function f meromorphic in D is normal if and only if Z  q  q−2  p sup f # (z) 1 − |z|2 g(z, a) dA(z) < ∞. a∈D

D

Now, we give the following theorem. 114

AHMED, KAMAL: WEIGHTED FUNCTION SPACES

8 Theorem 3.2 Let 0 < p < ∞, 0 < q < ∞ and let 0 < s < ∞ and let 0 < r < 1. Then a function f meromorphic in D is normal if and only if Z  p  q (g(z, a))s dA(z) < ∞. Fω# (p, q, s)(f ) = sup f # (z) 1 − |z|2 ω p (1 − |z|) a∈D D Proof: The proof of this theorem is very similar to Theorem 3.1, so it will be omitted. In the corresponding way to the analytic case, we define the meromorphic classes Q# K,ω (p, q) as follows. Definition 3.3 Let K : [0, ∞) → [0, ∞) be a nondecreasing function. For 0 < p < ∞ and −2 < q < ∞ a function f meromorphic in D is said to belong to the classes Q# K,ω (p, q) if Z  p  q K(g(z, a)) dA(z) < ∞. (14) f # (z) 1 − |z|2 sup ω p (1 − |z|) a∈D D Remark 3.1 Similar to the analytic case, if we take ω ≡ 1 and K(t) = ts for 0 ≤ s < ∞, then Q# K,ω (p, q) = # p F (p, q, s) (see [15]), the corresponding meromorphic of F (p, q, s) spaces. If we take K(t) = t , q = 0 and ω ≡ 1, # # then Q# K,ω (p, q) = Qp . When ω ≡ 1 and p = 2 and q = 0, we obtain QK space (see [3, 7, 14]). Definition 3.4 [14] A function f meromorphic in D is said to be a spherical Bloch function, denoted by f ∈ B# , if there exists an r, 0 < r < 1, such that Z sup (f # (z))2 dA(z) < ∞. a∈D

D

It is easy to see that a normal function is a spherical Bloch function, that is, N ⊂ B # , but the converse is not true. A counterexample can be found in [9]. Proposition 3.1 For each nondecreasing function K : [0, ∞) → [0, ∞), the classes Q# K,ω (p, q) are subsets of the q+2

spherical Bloch classes B# ω p , where 0 < p < ∞. Proof: We can prove the proposition by making the obvious modifications to the proof of Proposition 2.1. Theorem 3.3 Let K : [0, ∞) → [0, ∞) be a bounded and nondecreasing function and let f is a normal function. If K(r) =c 0, there is N 2 N such that " . Since P is a normal cone with normal constant for all n N ,kG (xn ; xn ; x)k < 2K K, kG (x; xn ; x)k

K kG (x; xn ; xn ) + G (xn ; xn ; x)k = K (kG (x; xn ; xn )k + kG (xn ; xn ; x)k) " " + = ": < K 2K 2K From (G4), G (xn ; x; x) = G (x; xn ; x) so for any " > 0, there is N 2 N such that for all n N , kG (xn ; x; x)k < ". This means G (xn ; x; x) ! , as n ! 1. (3) =) (4) : Suppose that G (xn ; x; x) ! ,as n ! 1. If we choose a = x in (G5), then G (xm ; xn ; x) G (xm ; x; x) + G (x; xn ; x). Since G (xn ; x; x) ! , as n ! 1 and from (G4),G (xm ; x; x) ! and G (xn ; x; x) = G (x; xn ; x) ! . So for " any " > 0, there is N 2 N such that for all n; m N , kG (xm ; x; x)k < 2K , " kG (x; xn ; x)k < 2K . Since P is a normal cone with normal constant K, kG (xm ; xn ; x)k

K kG (xm ; x; x) + G (x; xn ; x)k = K (kG (xm ; x; x)k + kG (x; xn ; x)k) " " < K + = ": 2K 2K So for any " > 0, there is N 2 N such that for all n; m N , kG (xm ; xn ; x)k < ". This means G (xm ; xn ; x) ! , as m; n ! 1. (4) =) (1) : Suppose that G (xm ; xn ; x) ! , as m; n ! 1. From (G4),G (xm ; xn ; x) = G (x; xn ; xm ). So from Lemma 2, (xn ) is G-cone convergent to x. Lemma 4. Let (X; G) be a G-cone metric space, P be a normal cone with normal constant K. Let (xn ) be a sequence in X. If (xn ) is G-cone convergent to x and (xn ) is G-cone convergent to y, then x = y. That is the limit of (xn ) is unique.

120

4

DURAN TURKOGLU AND NURCAN BILGILI

Proof. From xn ! x and xn ! y (n; m ! 1), for any c 2 E with