The uniqueness of analytic functions on annuli sharing some values
type: Item
instance of: scholarly article
title: The uniqueness of analytic functions on annuli sharing some values
zbMATH Open document ID: 1254.30028
DOI: 10.1155/2012/896596
author: Hongyan Xu
author: Zu-Xing Xuan
published in: Abstract and Applied Analysis
publication date: 2012-09-07T00:00:00Z
review text: Let \(f\) and \(g\) be analytic functions in \(A = \{ z \mid 1/R_0 < |z| < R_0 \}\), where \(1 < R_0 \leq + \infty\), and let \(a_j\), \(j = 1, 2, 3, 4\), be distinct complex numbers. If \(f\) and \(g\) share \(a_1\) and \(a_2\) CM in \(A\), \(f = a_3\) implies \(g = a_3\) in \(A\), \(f = a_4\) implies \(g = a_4\) in \(A\), and \(f\) is transcendental or admissible in \(A\), then the authors prove that \(f(z) \equiv g(z)\) holds in \(A\). If CM is replaced by IM, the authors show that \(f \equiv g\) in \(A\) or \(f \equiv (a_3 g - a_1 a_2)/(g-a_4)\), and \(a_1 + a_2 = a_3 + a_4\), and \(a_3\) and \(a_4\) are exceptional values of \(f\) and \(g\) in \(A\), respectively. These results improve a theorem of \textit{T.-B. Cao, H.-X. Yi} and \textit{H.-Y. Xu} [Comput. Math. Appl. 58, No. 7, 1457--1465 (2009; Zbl 1189.30065)].
reviewed by: Linda R. Sons
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6078830
zbMATH Keywords: analytic functions
zbMATH Keywords: shared values
zbMATH Keywords: uniqueness
zbMATH Keywords: annuli
Wikidata QID: Q58697042
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2012/896596
OpenAlex ID: W2097249611
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description: scientific article; zbMATH DE number 6078830