On meromorphic functions sharing five one-point or two-point sets IM
type: Item
instance of: scholarly article
title: On meromorphic functions sharing five one-point or two-point sets IM
zbMATH Open document ID: 1205.30032
author: Manabu Shirosaki
published in: Proceedings of the Japan Academy. Series A
publication date: 2010-04-08T00:00:00Z
review text: In this paper, a meromorphic function means a function meromorphic in the whole complex plane~\(\mathbb{C}\). The author studies such possible non-constant meromorphic functions that attain given one-point or two-point sets in the extended complex plane \(\bar{\mathbb{C}}:=\mathbb{C}\cup\{\infty\}\) at the same points. As the main theorem of this paper, Theorem~1, the author shows that one of the two functions must be a Möbius transformation of the other if certain mutually distinct \textit{five} one-point or two-point sets in~\(\bar{\mathbb{C}}\) are shared by such functions \textbf{IM} (ignoring multiplicities). Note that two meromorphic functions, say \(f\) and \(g\), are said to share a finite set \(S \subset \bar{\mathbb{C}}\) \textbf{IM} when the preimages of~\(S\) by~\(f\) and by~\(g\) coincide in~\(\mathbb{C}\). The term \textbf{IM} is replaced by \textbf{CM} (counting multiplicities) if, in addition, the following holds: if \(a, b \in S\) and \(z_0\) is an \(a\)-point of~\(f\) of multiplicity~\(p\) and a \(b\)-point of~\(g\) of multiplicity~\(q\), then \(p=q\). Therefore, all the five sets in Theorem~1 are shared by the pair of functions not only \textbf{IM} but also \textbf{CM}, naturally. The celebrated \textit{five-point theorem} of R. Nevanlinna states that if two non-constant meromorphic functions share distinct \textit{five} one-point sets \textbf{IM}, then they must be identical. On the other hand, his \textit{four-point theorem} states that if \(f\) and \(g\) are meromorphic functions sharing distinct \textit{four} one-point sets \textbf{CM}, then \(f\) is a Möbius transformation of \(g\). This is no longer true for functions sharing \textit{four} values \textbf{IM} only, as \textit{G. G. Gundersen} [J. Lond. Math. Soc. 20, 456--466 (1979; Zbl 0413.30025)] showed by an interesting counterexample, that is, a pair of meromorphic functions sharing certain \textit{four} distinct values \textbf{IM} (and never \textbf{CM} in each) which cannot be related …
reviewed by: Kazuya Tohge
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5690866
zbMATH Keywords: uniqueness theorem
zbMATH Keywords: sharing sets
zbMATH Keywords: Nevanlinna theory
MaRDI profile type: Publication
full work available at URL: pjaa.86.6
OpenAlex ID: W1999374192
cites work: Q5723337
cites work: A new characterization of collections of two-point sets with the uniqueness property
cites work: On meromorphic functions sharing two one-point sets and two two-point sets
cites work: Meromorphic functions covering certain finite sets at the same points
DOI: 10.3792/PJAA.86.6
description: scientific article; zbMATH DE number 5690866