On functional equations for meromorphic functions and applications
type: Item
instance of: scholarly article
title: On functional equations for meromorphic functions and applications
zbMATH Open document ID: 1384.30004
author: Ha Huy Khoai
author: Vu Hoai An
author: Pham Ngoc Hoa
published in: Archiv der Mathematik
publication date: 2017-11-30T00:00:00Z
review text: Let \(P\) and \(Q\) be polynomials. The authors consider a functional equation of the form \(P(f)=Q(g)\), where \(f\), \(g\) are meromorphic functions in the complex plane. In terms of Nevanlinna theory, the research has been continued since [\textit{G. Barsegian} (ed.) et al., Value distribution theory and related topics. Boston, MA: Kluwer Academic Publishers (2004; Zbl 1081.30001)]. They call a polynomial of the form \(z^n+az^{n-m}+b\) Yi's polynomial, where \(a\) and \(b\) are non-zero complex numbers. This type of polynomial possesses connections with the study of the uniqueness problem for meromorphic functions. One of the main results for the functional equation \(P(f)=Q(g)\) with Yi's polynomial is the following. Let \(n\), \(m\) be positive integers, \(c\), \(d\), \(e\), \(u\), \(v\), \(t\in\mathbb C\) be non-zero constants, and let \(n\geq 2m+4\), and either \(m\geq 2\), \((n,m)=1\), or \(m\geq 4\). Suppose that \((f,g)\) is a non-constant meromorphic solution of the equation \[ cf^n+df^{n-m}+e=ug^n+vg^{n-m}+t. \] Then there exists a non-zero constant \(h\) such that \(g=hf\). The authors also discuss an analogue of Ritt's decomposition theorem for a class of polynomials of Fermat-Waring type in meromorphic functions.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6813953
zbMATH Keywords: functional equations
zbMATH Keywords: meromorphic functions
zbMATH Keywords: Ritt's decompositions
MaRDI profile type: Publication
full work available at URL: s00013-017-1093-5
OpenAlex ID: W2751294297
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DOI: 10.1007/S00013-017-1093-5
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description: scientific article; zbMATH DE number 6813953