Value distribution of the difference operator
type: Item
instance of: scholarly article
title: Value distribution of the difference operator
zbMATH Open document ID: 1173.30018
author: J. Martínez
published in: Archiv der Mathematik
publication date: 2009-06-16T00:00:00Z
review text: This paper is devoted to considering value distribution of difference operators of transcendental entire functions. The key result is the following theorem: Let \(f\) be a transcendental entire function of finite order, \(p\) a non-vanishing polynomial, \(c\) a non-zero complex number, and \(n\geq 2\). If \(\Delta_{c}f:=f(z+c)-f(z)\) is not vanishing identically, then \(f(z)^{n}\Delta_{c}f-p(z)\) has infinitely many zeros. It will be shown by simple examples that the assertion fails if \(f\) is of infinite order, resp. if \(n=1\). This theorem is an extension to a previous result due to \textit{I. Laine} and \textit{C.-C. Yang} in [Proc. Japan Acad., Ser. A 83, 148--151 (2007; Zbl 1153.30030)] .
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 39A05
zbMATH DE Number: 5565241
zbMATH Keywords: entire functions
zbMATH Keywords: difference operator
zbMATH Keywords: finite order
MaRDI profile type: Publication
full work available at URL: s00013-009-2895-x
OpenAlex ID: W2006577028
DOI: 10.1007/S00013-009-2895-X
description: scientific article; zbMATH DE number 5565241