Pub Date : 2005-02-10DOI: 10.1080/02781070412331329403
P. R. Brown
A numerical method is developed for computing the accessory parameter of the Schwarzian derivative of the univalent mappings onto certain circular arc polygons with orthogonal boundary arcs. The geometrical properties of these polygons are investigated rigorously. An application of this method makes it possible to compute numerically the hyperbolic metric density function for some multiply-connected planar domains; in particular the covering map of the plane minus a lattice of discs.
{"title":"Mapping onto circular arc polygons","authors":"P. R. Brown","doi":"10.1080/02781070412331329403","DOIUrl":"https://doi.org/10.1080/02781070412331329403","url":null,"abstract":"A numerical method is developed for computing the accessory parameter of the Schwarzian derivative of the univalent mappings onto certain circular arc polygons with orthogonal boundary arcs. The geometrical properties of these polygons are investigated rigorously. An application of this method makes it possible to compute numerically the hyperbolic metric density function for some multiply-connected planar domains; in particular the covering map of the plane minus a lattice of discs.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"15 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120862533","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 : 2005-01-15DOI: 10.1080/02781070412331329412
Yu-Can Zhu, Mingsheng Liu †
In this article, we give some distortion theorems for biholomorphic convex mappings in Banach spaces and Hilbert spaces. In particular, we prove that the conjecture of Hamada and Kohr is true in Banach spaces.
{"title":"Distortion theorems for biholomorphic convex mappings in Banach spaces*","authors":"Yu-Can Zhu, Mingsheng Liu †","doi":"10.1080/02781070412331329412","DOIUrl":"https://doi.org/10.1080/02781070412331329412","url":null,"abstract":"In this article, we give some distortion theorems for biholomorphic convex mappings in Banach spaces and Hilbert spaces. In particular, we prove that the conjecture of Hamada and Kohr is true in Banach spaces.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122518078","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 : 2005-01-15DOI: 10.1080/02781070412331328585
P. Kargaev, K. Evgeny
Let a real function v be subharmonic in the plane and harmonic outside and . Assume that v belongs to the so-called subharmonic counterpart of the Cartright class of the entire functions. For such a function we obtain identities and estimates in terms of the Dirichlet integral.
{"title":"Identities for the dirichlet integral of subharmonic functions from the cartright class","authors":"P. Kargaev, K. Evgeny","doi":"10.1080/02781070412331328585","DOIUrl":"https://doi.org/10.1080/02781070412331328585","url":null,"abstract":"Let a real function v be subharmonic in the plane and harmonic outside and . Assume that v belongs to the so-called subharmonic counterpart of the Cartright class of the entire functions. For such a function we obtain identities and estimates in terms of the Dirichlet integral.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125332012","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 : 2004-12-15DOI: 10.1080/02781070412331316920
Wei Han-bai
In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.
{"title":"On the Douglas–Dirichlet functional and harmonic quasiconformal mappings","authors":"Wei Han-bai","doi":"10.1080/02781070412331316920","DOIUrl":"https://doi.org/10.1080/02781070412331316920","url":null,"abstract":"In this article, we discuss the minimal mappings of Douglas–Dirichlet functional and harmonic quasiconformal mappings, and solve the uniqueness problem of harmonic quasiconformal mappings posed by Shibata.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132890112","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 : 2004-12-15DOI: 10.1080/02781070412331327883
George Chailos
Let be a simply connected domain and let M be a connected subset of its boundary of positive Lebesque measure. With X we denote a separable Hilbert space or the space of bounded linear functionals on . We set f to be an X-valued holomorphic function, and with we denote the class of X-valued holomorphic functions on which belong to the Hardy class near the set M. In our main result, we show that if f belongs to , then f is representable by a Carleman type formula, and conversely, if f is representable by a Carleman type formula, and in some sense has an analytic continuation across M, then f belongs to . Furthermore we show that in general .
{"title":"Vector and operator-valued holomorphic functions representable by Carleman type formulas","authors":"George Chailos","doi":"10.1080/02781070412331327883","DOIUrl":"https://doi.org/10.1080/02781070412331327883","url":null,"abstract":"Let be a simply connected domain and let M be a connected subset of its boundary of positive Lebesque measure. With X we denote a separable Hilbert space or the space of bounded linear functionals on . We set f to be an X-valued holomorphic function, and with we denote the class of X-valued holomorphic functions on which belong to the Hardy class near the set M. In our main result, we show that if f belongs to , then f is representable by a Carleman type formula, and conversely, if f is representable by a Carleman type formula, and in some sense has an analytic continuation across M, then f belongs to . Furthermore we show that in general .","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124157323","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 : 2004-12-15DOI: 10.1080/02781070410001701074
T. Alzahary, Hong X. Yi †
In this article, we investigate the problem due to I. Lahiri [I. Lahiri (2001). Weighted value sharing and uniqueness of meromorphic functions. Complex Variables Theory Appl., 46, 241–253.]. Also we prove some uniqueness theorems of meromorphic functions which improve some earlier results.
{"title":"Weighted value sharing and a question of I. Lahiri","authors":"T. Alzahary, Hong X. Yi †","doi":"10.1080/02781070410001701074","DOIUrl":"https://doi.org/10.1080/02781070410001701074","url":null,"abstract":"In this article, we investigate the problem due to I. Lahiri [I. Lahiri (2001). Weighted value sharing and uniqueness of meromorphic functions. Complex Variables Theory Appl., 46, 241–253.]. Also we prove some uniqueness theorems of meromorphic functions which improve some earlier results.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"81 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120873639","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 : 2004-12-15DOI: 10.1080/02781070412331320457
Fred Brackx †, H. Schepper
In Brackx et al., 2004 (F. Brackx, R. Delanghe and F. Sommen (2004). Spherical means and distributions in Clifford analysis. In: Tao Qian, Thomas Hempfling, Alan McIntosch and Frank Sommen (Eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebra, Trends in Mathematics, pp. 65–96. Birkhäuser, Basel.), some fundamental higher dimensional distributions have been reconsidered within the framework of Clifford analysis. Here, the Fourier transforms of these distributions are calculated, revealing a.o. the Fourier symbols of some important translation invariant (convolution) operators, which can be interpreted as members of the considered families. Moreover, these results are the incentive for calculating the Fourier symbols of some differential operators which are at the heart of Clifford analysis, but do not show the property of translation invariance and hence, can no longer be interpreted as convolution operators.
在Brackx等人，2004 (F. Brackx, R. Delanghe and F. Sommen, 2004)。Clifford分析中的球面均值和分布。见:陶谦，托马斯·亨普林，艾伦·麦金托什和弗兰克·索曼(编)，《分析和几何的进展:使用克利福德代数的新发展》，《数学趋势》，第65-96页。Birkhäuser, Basel.)，一些基本的高维分布已经在Clifford分析的框架内重新考虑。在这里，计算这些分布的傅里叶变换，揭示一些重要的平移不变量(卷积)算子的傅里叶符号，这些算子可以被解释为所考虑的家族的成员。此外，这些结果是计算某些微分算子的傅里叶符号的动机，这些微分算子是Clifford分析的核心，但没有显示平移不变性，因此不能再被解释为卷积算子。
{"title":"On the Fourier transform of distributions and differential operators in Clifford analysis","authors":"Fred Brackx †, H. Schepper","doi":"10.1080/02781070412331320457","DOIUrl":"https://doi.org/10.1080/02781070412331320457","url":null,"abstract":"In Brackx et al., 2004 (F. Brackx, R. Delanghe and F. Sommen (2004). Spherical means and distributions in Clifford analysis. In: Tao Qian, Thomas Hempfling, Alan McIntosch and Frank Sommen (Eds.), Advances in Analysis and Geometry: New Developments Using Clifford Algebra, Trends in Mathematics, pp. 65–96. Birkhäuser, Basel.), some fundamental higher dimensional distributions have been reconsidered within the framework of Clifford analysis. Here, the Fourier transforms of these distributions are calculated, revealing a.o. the Fourier symbols of some important translation invariant (convolution) operators, which can be interpreted as members of the considered families. Moreover, these results are the incentive for calculating the Fourier symbols of some differential operators which are at the heart of Clifford analysis, but do not show the property of translation invariance and hence, can no longer be interpreted as convolution operators.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116807868","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 : 2004-11-15DOI: 10.1080/02781070412331310939
A. Escassut, Eberhard Mayerhofer †
Let h be a complex meromorphic function. The problem of decomposing h in two different ways, P (f) and Q(g) with f, g two other meromorphic functions and P, Q polynomials, was studied by C.-C. Yang, P. Li and H.K. Ha. Here we consider the problem when we replace the polynomials P, Q by rational functions F, G. Let deg(F ) be the maximum degree of numerator and denominator of F. Assume some zeros c 1, … ,c k of satisfy a pack of five conditions particularly involving G(d,) ≠ F(c j ) and D(d) ≠ 0 for every zero d of , with G = C/D, (j = 1,…,k). First, we show that if f, g are entire functions such that F(f) = G(g), then k deg (G) ≤ deg(F). Now, let u be the number of distinct zeros of the denominator of G and assume that meromorphic functions f, g satisfy F(f) = G(g), then k deg (G) ≤ deg (F) + kγ (D). When zeros c 1, …, c k of satisfy a stronger condition, then we show that k deg (G) ≤ deg (F) + k min (γ (C), γ (D)). E-mail: eberhard.mayerhofer@univie.ac.at
设h是一个复亚纯函数。c . c .研究了h用另外两个亚纯函数f, g和P, Q多项式以P (f)和Q(g)两种不同方式分解的问题。杨、李鹏及哈港强。这里我们考虑用有理函数F, G代替多项式P, Q的问题。设deg(F)是F的分子和分母的最大次。假设c 1，…，c k的某些0满足一组5个条件，特别涉及G(d，)≠F(c j)和d (d)≠0，其中G = c / d， (j = 1，…，k)。首先，我们证明了如果f, g是使f (f) = g(g)的完整函数，则k度(g)≤度(f)。现在，设u为G的分母的不同零的个数，并假设亚纯函数f, G满足f (f) = G(G)，则k deg (G)≤deg (f) + kγ (D)。当0 c 1，…，ck满足一个更强的条件时，则证明k deg (G)≤deg (f) + k min (γ (c)， γ (D))。电子邮件:eberhard.mayerhofer@univie.ac.at
{"title":"Rational decompositions of complex meromorphic functions","authors":"A. Escassut, Eberhard Mayerhofer †","doi":"10.1080/02781070412331310939","DOIUrl":"https://doi.org/10.1080/02781070412331310939","url":null,"abstract":"Let h be a complex meromorphic function. The problem of decomposing h in two different ways, P (f) and Q(g) with f, g two other meromorphic functions and P, Q polynomials, was studied by C.-C. Yang, P. Li and H.K. Ha. Here we consider the problem when we replace the polynomials P, Q by rational functions F, G. Let deg(F ) be the maximum degree of numerator and denominator of F. Assume some zeros c 1, … ,c k of satisfy a pack of five conditions particularly involving G(d,) ≠ F(c j ) and D(d) ≠ 0 for every zero d of , with G = C/D, (j = 1,…,k). First, we show that if f, g are entire functions such that F(f) = G(g), then k deg (G) ≤ deg(F). Now, let u be the number of distinct zeros of the denominator of G and assume that meromorphic functions f, g satisfy F(f) = G(g), then k deg (G) ≤ deg (F) + kγ (D). When zeros c 1, …, c k of satisfy a stronger condition, then we show that k deg (G) ≤ deg (F) + k min (γ (C), γ (D)). E-mail: eberhard.mayerhofer@univie.ac.at","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"26 6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126629281","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 : 2004-11-15DOI: 10.1080/02781070412331298598
Shi Jihuai, L. Hua
In this paper, it is proved that the sufficiently separated sequences are interpolating sequences for when f is a Bloch function. In other words, for a sufficiently separated sequence {ak } and each bounded sequence {ck }, there exists at least one Bloch function f (z) such that .
{"title":"Interpolating sequences for the fractral derivatives of Bloch functions in several variables","authors":"Shi Jihuai, L. Hua","doi":"10.1080/02781070412331298598","DOIUrl":"https://doi.org/10.1080/02781070412331298598","url":null,"abstract":"In this paper, it is proved that the sufficiently separated sequences are interpolating sequences for when f is a Bloch function. In other words, for a sufficiently separated sequence {ak } and each bounded sequence {ck }, there exists at least one Bloch function f (z) such that .","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114104790","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 : 2004-11-15DOI: 10.1080/02781070412331298543
G. Kresin, V. Maz'ya
We obtain sharp estimates of by the Lp -norm of on the circle , where , and α is a real valued function on DR . Here f is an analytic function in the disc whose real part is continuous on , ω is a real constant, and is orthogonal to some continuous function Φ on the circle . We derive two types of estimates with vanishing and nonvanishing mean value of Φ. The cases Φ = 0 and Φ = 1 are discussed in more detail. In particular, we give explicit formulas for sharp constants in inequalities for with p = 1, 2, ∞. We also obtain estimates for in the class of analytic functions with two-sided bounds of . As a corollary, we find a sharp constant in the upper estimate of by which generalizes the classical Carathéodory–Plemelj estimate with p=∞.
{"title":"Sharp pointwise estimates for analytic functions by the L p -norm of the real part","authors":"G. Kresin, V. Maz'ya","doi":"10.1080/02781070412331298543","DOIUrl":"https://doi.org/10.1080/02781070412331298543","url":null,"abstract":"We obtain sharp estimates of by the Lp -norm of on the circle , where , and α is a real valued function on DR . Here f is an analytic function in the disc whose real part is continuous on , ω is a real constant, and is orthogonal to some continuous function Φ on the circle . We derive two types of estimates with vanishing and nonvanishing mean value of Φ. The cases Φ = 0 and Φ = 1 are discussed in more detail. In particular, we give explicit formulas for sharp constants in inequalities for with p = 1, 2, ∞. We also obtain estimates for in the class of analytic functions with two-sided bounds of . As a corollary, we find a sharp constant in the upper estimate of by which generalizes the classical Carathéodory–Plemelj estimate with p=∞.","PeriodicalId":272508,"journal":{"name":"Complex Variables, Theory and Application: An International Journal","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121310017","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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