Meromorphic functions that share one or two values. II
type: Item
instance of: scholarly article
title: Meromorphic functions that share one or two values. II
zbMATH Open document ID: 0939.30020
author: Hongxun Yi
published in: Kodai Mathematical Journal
publication date: 2000-07-04T00:00:00Z
review text: For a meromorphic function \(h\), let \(\overline N_{(2}(r,h)\) denote the counting function of multiple poles of \(h\), each counted only once. Define \(N_2(r,h) = \overline N(r,h) + \overline N_{(2}(r,h)\) and \(N^*(r,h) = 2N_2(r,h) + 3\overline N(r,h)\). The author proved in [Complex Variables, Theory Appl. 28, 1-11 (1995; Zbl 0841.30027)], that either \(f\equiv g\) or \(fg\equiv 1\) whenever \(f, g\) are two nonconstant meromorphic functions sharing the value \(1\) CM and in a set \(I\subset [0, +\infty)\) of infinite linear measure, \[ \limsup_{r\to \infty, r\in I}\frac{N_2(r,1/f) + N_2(r,f) + N_2(r, 1/g) + N_2(r,g)}{\max(T(r,f), T(r,g))}<1. \] In this paper, the same conclusion will be proved for \(f, g\) sharing the value \(1\) IM, provided \[ \limsup_{r\to\infty, r\in I}\frac{N^*(r,1/f) + N^*(r,f) + N^*(r,1/g) + N^*(r,g)}{T(r,f) + T(r,g)}<1. \] If \(f,g\) share the values \(1\) and \(\infty\) IM and \[ \limsup_{r\to \infty, r\in I} \frac{N^*(r,1/f) + N^*(r,1/g) + 12\overline N(r,f)}{T(r,f) + T(r,g)}<1, \] then again \(f\equiv g\) or \(fg\equiv 1\). The proofs need a careful analysis of \(1\)-points of \(f\) and \(g\) with respect to different multiplicities combined with elementary Nevanlinna theory.
reviewed by: Ilpo Laine
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1367788
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138044046
OpenAlex ID: W1989081372
cites work: Q5723337
cites work: Meromorphic Functions that Share Three or Four Values
cites work: Q4294394
cites work: Meromorphic functions that share one or two values
cites work: Uniqueness of meromorphic functions and a question of c.c.yang
cites work: Q4714891
DOI: 10.2996/KMJ/1138044046
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description: scientific article; zbMATH DE number 1367788