On the value distribution of \(f +a(f\prime)^n\)
type: Item
instance of: scholarly article
title: On the value distribution of \(f +a(f\prime)^n\)
zbMATH Open document ID: 1158.30020
author: Lawrence Zalcman
author: Mingliang Fang
published in: Science in China. Series A
publication date: 2008-09-23T00:00:00Z
review text: Let \(f\) be a transcendental meromorphic function, and let \(a\) be a nonzero finite complex number. \textit{Y. Ye} [Chin. Ann. Math., Ser. B 15, No. 1, 75--80 (1994; Zbl 0801.30027)] proved that \(f+a(f')^n\) assumes every finite complex value infinitely often for each positive integer \(n\geq 3\). It is a question whether the result remains valid for \(n=2\). The authors give an affirmative answer for this question. The authors also consider a normality criterion corresponding to the result above. Let \({\mathcal F}\) be a family of meromorphic functions on the plane \(D\), let \(n\geq 2\) be a positive integer, and let \(a\neq 0\), \(b\) be complex numbers. If, for each \(f\in{\mathcal F}\), all zeros of \(f\) are multiple and \(f+a(f')^n\neq b\) on \(D\), then \({\mathcal F}\) is normal on \(D\). The tools for the proofs are the standard Nevanlinna theory and the rescaliug lemma due to the second author of this paper.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 5346104
zbMATH Keywords: meromorphic function
zbMATH Keywords: normal families
zbMATH Keywords: differential polynomial
zbMATH Keywords: Nevanlinna theory
MaRDI profile type: Publication
full work available at URL: s11425-008-0022-2
OpenAlex ID: W2468831892
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DOI: 10.1007/S11425-008-0022-2
description: scientific article; zbMATH DE number 5346104