The uniqueness of an entire function sharing a small entire function with its derivatives
type: Item
instance of: scholarly article
title: The uniqueness of an entire function sharing a small entire function with its derivatives
zbMATH Open document ID: 1173.30019
author: Jun Wang
author: Xiao-Min Li
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-05-19T00:00:00Z
full work available at URL: https://eudml.org/doc/55208
review text: The essential result in this paper is the following improvement to an earlier result due to \textit{G. G. Gundersen} and \textit{L.-Z. Yang} in [J. Math. Anal. Appl., 223, 88--95 (1998; Zbl 0911.30022)]: Let \(f\) be a non-constant entire function of finite lower order \(\mu(f)\), \(a\) be a non-vanishing entire function such that \(\rho(a)<\mu(f)\) and \(n\) be a natural number. If \(f\), \(f^{(n)}\), and \(f^{(n+1)}\) share \(a\) IM, while \(f^{(n)}\), \(f^{(n+1)}\) share \(a\) CM, then \(f=f'\). Actually, this result is being deduced here from preparatory order considerations of certain linear differential equations in the complex plane, typically of type \(f^{(k)}-a_{1}= (f-a_{2})e^{Q}\), where \(a_{1},a_{2}\) are entire functions of order \(<1\), and \(Q\) is a polynomial. In this situation, it will be proved for the hyper-order of \(f\) that \(\mu_{2}(f)=\rho_{2}(f)=\)deg \(Q\). This result for the hyper-order is more or less well known, while the assertion for the lower hyper-order seems to be new. The proofs in the paper essentially are based on a Wiman--Valiron type reasoning. (Reviewers remark: Some parts of the proofs could have been made shorter.)
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 34M10
zbMATH DE Number: 5555205
zbMATH Keywords: entire functions
zbMATH Keywords: order of growth
zbMATH Keywords: shared values
zbMATH Keywords: uniqueness theorems
MaRDI profile type: Publication
OpenAlex ID: W1985826268
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DOI: 10.1016/J.JMAA.2008.12.067
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description: scientific article; zbMATH DE number 5555205