Pub Date : 2005-12-01DOI: 10.1080/02781070500327832
H. Begehr, G. Harutjunjan
The aim of this article is to give explicit representations for solutions of the Robin boundary value problem for the Cauchy-Riemann operator [image omitted]. In the homogeneous cases we investigate the Robin boundary condition in a more general form. Finally, we give solutions of the corresponding higher-order operators.
{"title":"Robin boundary value problem for the Cauchy-Riemann operator","authors":"H. Begehr, G. Harutjunjan","doi":"10.1080/02781070500327832","DOIUrl":"https://doi.org/10.1080/02781070500327832","url":null,"abstract":"The aim of this article is to give explicit representations for solutions of the Robin boundary value problem for the Cauchy-Riemann operator [image omitted]. In the homogeneous cases we investigate the Robin boundary condition in a more general form. Finally, we give solutions of the corresponding higher-order operators.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127486200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-12-01DOI: 10.1080/02781070500259969
T. Alzahary
This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.
{"title":"Small functions and weighted sharing three values","authors":"T. Alzahary","doi":"10.1080/02781070500259969","DOIUrl":"https://doi.org/10.1080/02781070500259969","url":null,"abstract":"This article studies the problem of the uniqueness of meromorphic functions that weighted sharing three values which improve some results given by Yi [Theorem 4, Yi, H.X., 1995, Unicity theorems for meromorphic functions that share three values. Kodai Mathematical Journal, 18, 300-314] and Ueda [Ueda, H., 1983, Unicity theorems for meromorphic or entire functions II. Kodai Mathematical Journal, 6, 26-36] and other authors. An application of these new results, if f and g are two distinct nonconstant meromorphic functions sharing 0, 1 and CM, and a is a nonconstant rational function, then N2)(r,1/(g-a))= [image omitted] An example shows that the latter result is not true for some transcendental small functions of f and g.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129439427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-12-01DOI: 10.1080/02781070500284306
L. M. Hai, Nguyen Van Khue
The aim of this article is to introduce the notions of weak tautness and hyperconvexity of a domain in a Banach space and to establish the relation between them. Moreover, some conditions under which a balanced domain in a Hilbert space and a Hartogs domain in a Banach space are hyperconvex are also given.
{"title":"Weak tautness and hyperconvexity in Hilbert spaces","authors":"L. M. Hai, Nguyen Van Khue","doi":"10.1080/02781070500284306","DOIUrl":"https://doi.org/10.1080/02781070500284306","url":null,"abstract":"The aim of this article is to introduce the notions of weak tautness and hyperconvexity of a domain in a Banach space and to establish the relation between them. Moreover, some conditions under which a balanced domain in a Hilbert space and a Hartogs domain in a Banach space are hyperconvex are also given.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115015496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-12-01DOI: 10.1080/02781070500303247
E. Ballico
Let X be a smooth and connected Riemann surface of genus g 0 and f: X P1, h: X P1 non-constant meromorphic functions on X. Fix an integer n 4 and assume the existence of n distinct points a1, , an P1 such that [image omitted] (set-theoretically) for every i. Here we prove that either f = h or [image omitted].
设X是X上的g属和f: X P1, h: X P1的光滑连通黎曼曲面。固定一个整数n 4，并假设存在n个不同的点a1，，和P1，使得[图像省略](集合理论)对每一个i。这里我们证明了f = h或[图像省略]。
{"title":"Meromorphic functions on compact Riemann surfaces and value sharing","authors":"E. Ballico","doi":"10.1080/02781070500303247","DOIUrl":"https://doi.org/10.1080/02781070500303247","url":null,"abstract":"Let X be a smooth and connected Riemann surface of genus g 0 and f: X P1, h: X P1 non-constant meromorphic functions on X. Fix an integer n 4 and assume the existence of n distinct points a1, , an P1 such that [image omitted] (set-theoretically) for every i. Here we prove that either f = h or [image omitted].","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"153 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116327906","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-12-01DOI: 10.1080/02781070500327576
K. Avetisyan
We introduce fractional -integro-differentiation for functions holomorphic in the upper half-plane. It gives us a tool to construct Cauchy-Bergman type kernels associated with the weights Some estimates of the kernels enable us to obtain reproducing integral formulas for Bergman spaces with general weights which may decrease to zero with arbitrary rate near the origin. Accordingly, such Bergman functions have arbitrary growth near the real axis.
{"title":"Integral representations in general weighted Bergman spaces","authors":"K. Avetisyan","doi":"10.1080/02781070500327576","DOIUrl":"https://doi.org/10.1080/02781070500327576","url":null,"abstract":"We introduce fractional -integro-differentiation for functions holomorphic in the upper half-plane. It gives us a tool to construct Cauchy-Bergman type kernels associated with the weights Some estimates of the kernels enable us to obtain reproducing integral formulas for Bergman spaces with general weights which may decrease to zero with arbitrary rate near the origin. Accordingly, such Bergman functions have arbitrary growth near the real axis.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129547432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-11-01DOI: 10.1080/02781070500277672
Ó. Blasco, M. Lindstróm, J. Taskinen
We study analytic mappings φ : Bn → Bm and the corresponding analytic composition operators Cφ : f → f ◦ φ. Here n,m ∈ N and Bn is the unit ball of C. In the one complex variable case n = m = 1, D := B1, the investigation of composition operators from the Bloch space B(D) into BMOA(D) has only recently taken place. Boundedness and compactness of Cφ : B(D) → BMOA(D), Cφ : B0(D) → VMOA(D) and Cφ : B(D) → VMOA(D) has been studied in [SZ] by Smith and Zhao and by Makhmutov and Tjani in [MT]. Madigan and Matheson [MM] proved that Cφ is always bounded on B(D). Moreover, [MM] contains a characterization of symbols φ inducing compact composition operators on B(D) and B0(D). The essential norm of a composition operator from B(D) into Qp(D) was computed in [LMT]. In the case of several complex variables, Ramey and Ullrich [RU] have studied the case mentioned in the beginning: their result states that if φ : Bn → D is Lipschitz, then Cφ : B(D) → BMOA(Bn) is well defined, and consequently bounded by the closed graph theorem. Our results below are, of course, more general. The case of Cφ : B(Bn) → B(Bn) was considered by Shi and Luo [SL], where they proved that Cφ is always bounded and gave a necessary and sufficient condition for Cφ to be compact. Our main result states that if φ : Bn → Bm satisfies a very mild regularity condition, then the boundedness of Cφ : B(Bm) → BMOA(Bn) is characterized by the fact that dμφ(z) = (1−|z|2)|Rφ(z)|2 (1−|φ(z)|2)2 dA(z) is a Carleson measure (see notations below). Similarly, a corresponding o–growth condition characterizes the compactness. Let N := {1, 2, 3, . . . }. For z, w ∈ C let 〈z, w〉 = ∑n i=1 ziwi denote the complex inner product on C and |z| = 〈z, z〉1/2. The radial derivative operator is denoted by R; so, if f : Bn → C is analytic, then
{"title":"Bloch-to-BMOA compositions in several complex variables","authors":"Ó. Blasco, M. Lindstróm, J. Taskinen","doi":"10.1080/02781070500277672","DOIUrl":"https://doi.org/10.1080/02781070500277672","url":null,"abstract":"We study analytic mappings φ : Bn → Bm and the corresponding analytic composition operators Cφ : f → f ◦ φ. Here n,m ∈ N and Bn is the unit ball of C. In the one complex variable case n = m = 1, D := B1, the investigation of composition operators from the Bloch space B(D) into BMOA(D) has only recently taken place. Boundedness and compactness of Cφ : B(D) → BMOA(D), Cφ : B0(D) → VMOA(D) and Cφ : B(D) → VMOA(D) has been studied in [SZ] by Smith and Zhao and by Makhmutov and Tjani in [MT]. Madigan and Matheson [MM] proved that Cφ is always bounded on B(D). Moreover, [MM] contains a characterization of symbols φ inducing compact composition operators on B(D) and B0(D). The essential norm of a composition operator from B(D) into Qp(D) was computed in [LMT]. In the case of several complex variables, Ramey and Ullrich [RU] have studied the case mentioned in the beginning: their result states that if φ : Bn → D is Lipschitz, then Cφ : B(D) → BMOA(Bn) is well defined, and consequently bounded by the closed graph theorem. Our results below are, of course, more general. The case of Cφ : B(Bn) → B(Bn) was considered by Shi and Luo [SL], where they proved that Cφ is always bounded and gave a necessary and sufficient condition for Cφ to be compact. Our main result states that if φ : Bn → Bm satisfies a very mild regularity condition, then the boundedness of Cφ : B(Bm) → BMOA(Bn) is characterized by the fact that dμφ(z) = (1−|z|2)|Rφ(z)|2 (1−|φ(z)|2)2 dA(z) is a Carleson measure (see notations below). Similarly, a corresponding o–growth condition characterizes the compactness. Let N := {1, 2, 3, . . . }. For z, w ∈ C let 〈z, w〉 = ∑n i=1 ziwi denote the complex inner product on C and |z| = 〈z, z〉1/2. The radial derivative operator is denoted by R; so, if f : Bn → C is analytic, then","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132073884","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-11-01DOI: 10.1080/02781070500260017
Jinman Kim, M. W. Wong
We give conditions on the functions and u on [image omitted] such that if u is given by the convolution of and u, then u is harmonic on [image omitted].
我们给出了函数和u在[图像省略]上的条件，如果u由和u的卷积给出，则u在[图像省略]上是调和的。
{"title":"Invariant mean value property and harmonic functions","authors":"Jinman Kim, M. W. Wong","doi":"10.1080/02781070500260017","DOIUrl":"https://doi.org/10.1080/02781070500260017","url":null,"abstract":"We give conditions on the functions and u on [image omitted] such that if u is given by the convolution of and u, then u is harmonic on [image omitted].","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127564368","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-11-01DOI: 10.1080/02781070500324300
Min Ho Lee
We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.
{"title":"Jacobi-like forms, differential equations, and Hecke operators","authors":"Min Ho Lee","doi":"10.1080/02781070500324300","DOIUrl":"https://doi.org/10.1080/02781070500324300","url":null,"abstract":"We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"142 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121001359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-11-01DOI: 10.1080/02781070500283944
G. Oros
In this article we shall extend the nonlinear superordination [image omitted] to the superordination of the form [image omitted]
在这篇文章中，我们将把非线性先验[图像省略]扩展到形式的先验[图像省略]
{"title":"On a first order nonlinear differential superordination","authors":"G. Oros","doi":"10.1080/02781070500283944","DOIUrl":"https://doi.org/10.1080/02781070500283944","url":null,"abstract":"In this article we shall extend the nonlinear superordination [image omitted] to the superordination of the form [image omitted]","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121036185","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2005-11-01DOI: 10.1080/02781070500278274
L. Solanilla, Wilson Rivera
In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.
{"title":"Canonical Euler-Lagrange equations and Jacobi's theorem on regular surfaces","authors":"L. Solanilla, Wilson Rivera","doi":"10.1080/02781070500278274","DOIUrl":"https://doi.org/10.1080/02781070500278274","url":null,"abstract":"In this article we establish conditions under which canonical variables can be defined for a variational problem defined on a geometric (compact) surface. Also, we show the form the corresponding Euler-Lagrange equations assume once we rewrite them in terms of such canonical variables. Furthermore, we prove a version of Jacobi's theorem generalizing the univariate standard version of this theorem. The main results are applied to the conformal Gauss curvature functional.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122386044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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