On a conjecture of W. Bergweiler
type: Item
instance of: scholarly article
title: On a conjecture of W. Bergweiler
zbMATH Open document ID: 1056.30032
author: Hongxun Yi
author: Weichuan Lin
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116443603
review text: \textit{W.Bergweiler} [Arch. Math., 64, 199--202 (1995; Zbl 0818.30021)] conjectured that if \(f\) is a transcendental meromorphic function which is not of the form \(f= \exp (\alpha z + \beta)\) and if \(a\neq 1\) and \(a\neq (n+1)/n\), then \(ff'' - a(f')^{2}\) has at least one zero, where \(n\) is any integer. In response to this conjecture the authors of the paper prove that if \(ff'' - a(f')^{2}\) has only finitely many zeros then \(f\) is of the form \(f = R\exp(P)\) for a rational function \(R\) and a polynomial \(P\), where \(a\neq 1, (n\pm 1)/n\) for any integer \(n\).
reviewed by: Indrajit Lahiri
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 2024720
zbMATH Keywords: Differential polynomial
zbMATH Keywords: Normal family
zbMATH Keywords: Meromorphic function
Wikidata QID: Q123196271
MaRDI profile type: Publication
OpenAlex ID: W2047916689
cites work: Picard values of meromorphic functions and their derivatives
cites work: Über eine Vermutung von Hayman
cites work: Proof of a Conjecture of Hayman Concerning f and f ″
cites work: Q5723337
cites work: Über die Nullstellen homogener Differentialpolynome
cites work: On the zeros of certain homogeneous differential polynomials
cites work: On differential polynomials, fixpoints and critical values of meromorphic functions
cites work: Q3907810
cites work: Normality and exceptional values of derivatives
cites work: On the singularities of the inverse to a meromorphic function of finite order
cites work: Iteration of a class of hyperbolic meromorphic functions
cites work: Q3154016
cites work: Normal Families and Shared Values
cites work: Q4274978
cites work: Meromorphe funktionen, die mit ihrer ersten und zweiten ableitung einen endlichen wert teilen
cites work: Meromorphic functions which share the value zero with their first two derivatives
DOI: 10.3792/PJAA.79.23
description: scientific article; zbMATH DE number 2024720