Normal families and shared values of holomorphic functions
type: Item
instance of: scholarly article
title: Normal families and shared values of holomorphic functions
zbMATH Open document ID: 1117.30025
author: Jiangtao Li
author: Hongxun Yi
published in: Applied Mathematics. Series B (English Edition)
publication date: 2006-12-05T00:00:00Z
review text: Let \(\mathbb C\) be the complex plane, and let \(f\) be a meromorphic function in a domain \(D\subset\mathbb C\). For \(a\in\mathbb C \), set \({\overline E}_f(a)=\{\,z\in\mathcal D:f(z)=a\,\}\). Let \(\mathcal F\) be a family of meromorphic functions in a domain \(\mathcal D\subset\mathbb C\). Then \(\mathcal F\) is said to be normal in \(\mathcal D\) if every sequence \(\{f_n\}\subseteq\mathcal F\) contains a subsequence which converges spherically uniformly on compact subsets of \(\mathcal D\). The main theorem in the paper under review is the following: Let \(\mathcal F\) be a family of holomorphic functions in a domain \(\mathcal D\), let \(k\) be a positive integer, \(a\), \(b(\neq 0)\), \(c(\neq0)\) and \(d\) be finite complex numbers. If, for each \(f\in\mathcal F\), all zeros of \(f(z)-d\) have multiplicity at least \(k\), \({\overline E}_f(0)\subseteq{\overline E}_{f^{(k)}}(a)\), and \({\overline E}_{f^{(k)}}(b)\subseteq{\overline E}_g(c)\), then \(\mathcal F\) is normal in \(\mathcal D\). This result extends the well-known normality criterion of C. Miranda and improves some results due to \textit{Y. Xu} [Indian J. Pure App. Math., 30, No. 3, 287--293 (1999; Zbl 0922.30028)], \textit{H. H. Chen} and \textit{M. L. Fang} [J. Math. Anal. Appl., 260, No. 1, 124--132 (2001; Zbl 1020.30035)], \textit{X. C. Pang} [Analysis 22, No. 2, 175--182 (2002; Zbl 1030.30030] and \textit{M. L. Fang} and \textit{Y. Xu} [Israel J. Math. 129, 125--141 (2002; Zbl 1007.30025)]. Further, some examples are provided to show that the obtained results are sharp.
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5077209
zbMATH Keywords: holomorphic function
zbMATH Keywords: normal family
zbMATH Keywords: shared value
reviewed by: Rauno Aulaskari
MaRDI profile type: Publication
full work available at URL: s11766-003-0010-9
OpenAlex ID: W2238706784
cites work: Meromorphic functions that share finite values with their derivative
cites work: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen
cites work: Sharing values and normality
cites work: Normality criteria concerning sharing values
cites work: Q3148718
cites work: Shared values and normal families of meromorphic functions
cites work: Normal families of holomorphic functions and shared values
cites work: Sur un nouveau critère de normalité pour les familles de fonctions holomorphes
cites work: Normal Families and Shared Values
cites work: Improvement of Marty's criterion and its application
cites work: The spherical derivative of integral and meromorphic functions
cites work: On the singularities of the inverse to a meromorphic function of finite order
cites work: On the zeros of certain homogeneous differential polynomials
cites work: Normality and exceptional values of derivatives
cites work: Normal families: New perspectives
DOI: 10.1007/S11766-003-0010-9
Recommended article: Normal families of holomorphic functions and shared values
Recommended article: Normal families and shared value of holomorphic function
Recommended article: NORMAL FAMILIES AND SHARED HOLOMORPHIC FUNCTIONS
Recommended article: Q5455619
Recommended article: Q4822399
Recommended article: Q4574865
Recommended article: Normal families of holomorphic functions and multiple values
Recommended article: Normal families of holomorphic functions concerning sharing a polynomial
Recommended article: Normal families and shared values of meromorphic functions
description: scientific article; zbMATH DE number 5077209