On the zeros of certain homogeneous differential polynomials
type: Item
instance of: scholarly article
title: On the zeros of certain homogeneous differential polynomials
zbMATH Open document ID: 0818.30021
DOI: 10.1007/BF01188569
published in: Archiv der Mathematik
publication date: 1995-05-18T00:00:00Z
review text: Let \(f\) be a meromorphic function of finite order and let \(a\) be a complex number satisfying \(a\neq 1\) and \(a\neq (n+ 1)/n\) for all natural numbers \(n\). It is shown that if \(f(z) f''(z)- af'(z)^ 2\) has only finitely many zeros, then \(f\) has the form \(f(z)= R(z) \exp P(z)\) with a rational function \(R\) and a polynomial \(P\). If \(f(z) f''(z)- af'(z)^ 2\) has no zeros and if \(f\) is transcendental, then \(f\) has the form \(f(z)= \exp(\alpha z+ \beta)\) with constants \(\alpha\) and \(\beta\). The result is connected to earlier work of Hayman, Mues and Langley dealing with the case that \(a= 0\) or that \(f\) is entire.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F99
zbMATH DE Number: 756418
zbMATH Keywords: differential polynomial
zbMATH Keywords: exceptional value
zbMATH Keywords: asymptotic value
zbMATH Keywords: critical value
zbMATH Keywords: fixed point
zbMATH Keywords: iteration
zbMATH Keywords: dynamics
zbMATH Keywords: meromorphic function
author: Walter Bergweiler
reviewed by: Walter Bergweiler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01188569
OpenAlex ID: W1997027373
description: scientific article; zbMATH DE number 756418