On Montel's theorem and Yang's problem
type: Item
instance of: scholarly article
title: On Montel's theorem and Yang's problem
zbMATH Open document ID: 1068.30025
author: Yan Xu
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-04-28T00:00:00Z
review text: The author proves the following normality criterion. Let \(\mathcal F\) be a family of functions meromorphic in the domain \(D\subset\mathbb C\), and let \(\psi(\not\equiv 0)\) be a function meromorphic in \(D\). If for every function \(f\in\mathcal F\), (1) \(f\) has only multiple zeros; (2) the poles of \(f\) have multiplicity at least 3; (3) at the common poles of \(f\) and \(\psi\), the multiplicity of \(f\) does not equal the multiplicity of \(\psi\); (4) \(f(z)\not=\psi(z)\), then \(\mathcal F\) is normal in \(D\). This gives a partial answer to a problem of L. Yang and generalizes Montel's theorem. The author also gives some examples to show the sharpness of our result. It seems that the condition (4) should be written as \(f(z)-\psi(z)\not=0\), as the the condition (4) implies \(f\) and \(\psi\) have no common pole.
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 2162035
zbMATH Keywords: meromorphic function
zbMATH Keywords: normal family
zbMATH Keywords: fix-point
zbMATH Keywords: exceptional function
reviewed by: Jianming Chang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.10.066
OpenAlex ID: W2055086953
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DOI: 10.1016/J.JMAA.2004.10.066
description: scientific article; zbMATH DE number 2162035