Meromorphic functions that share two sets
type: Item
instance of: scholarly article
title: Meromorphic functions that share two sets
zbMATH Open document ID: 0891.30019
published in: Kodai Mathematical Journal
publication date: 1998-01-12T00:00:00Z
review text: Let \(f\) be a meromorphic function in \(\mathbb{C}\) and \(S\) be a subset of \(\widehat \mathbb{C}\). We define by \(E_f(s): =\cup_{a\in S} \{z\in\mathbb{C} |f(z)- a=0\}\) where each zero of \(f-a\) will be counted with multiplicities. We denote by \(\overline E_f(s)\) the set of all distinct points of \(E_f(s)\). One says the meromorphic functions \(f\) and \(g\) share the set \(S\) \(CM\) if \(E_f(s) =E_g(s)\) and they share \(S\) \(IM\) if \(\overline E_f(s)= \overline E_g(s)\). With \(w:= \cos(2\pi/n) +i\sin (2\pi/n)\), \(n\in\mathbb{N}\), we define \(S_1:= \{1,w, \dots, w^{n-1}\}\) and denote by \(S_2:= \{\infty\}\) and \(S_3:=\{0\}\). Then the authors prove the following theorems: Let \(f\) and \(g\) be meromorphic in the plane. Suppose that \(f\) and \(g\) fulfill one of the following conditions (1) \(f\) and \(g\) share \(S_1 CM\) and \(S_2\) \(IM\) with \(n\geq 6\) or (2) \(f\) and \(g\) share \(S_1\) \(IM\) and \(S_2\) \(CM\) with \(n\geq 10\) or (3) \(f\) and \(g\) share \(S_1\) and \(S_2\) \(IM\) with \(n\geq 11\) then either \(f=tg\) or \(f\cdot g=t\) where \(t^n=1\). With these results they improve earlier results of F. Gross, C. F. Osgood, H.-X. Yi, G. D. Song, N. Li and others.
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 1089816
zbMATH Keywords: sharing values
zbMATH Keywords: uniqueness theorems for meromorphic functions
author: Hongxun Yi
author: Lianzhong Yang
reviewed by: Günter Frank
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043751
OpenAlex ID: W2364353819
cites work: Q5723337
cites work: On the Distribution of Values of Meromorphic Functions
cites work: Meromorphic functions that share three sets
cites work: Meromorphic Functions that Share Three or Four Values
cites work: Q3957198
cites work: Q3031221
cites work: Q4728585
cites work: Q3995922
cites work: Meromorphic functions covering certain finite sets at the same points
cites work: Q3989548
cites work: Unicity theorems for entire functions
cites work: Uniqueness theorems for meromorphic functions. II
cites work: Q4714891
cites work: Meromorphic functions that share one or two values
DOI: 10.2996/KMJ/1138043751
description: scientific article; zbMATH DE number 1089816