Unicity theorems for meromorphic functions sharing four small functions
type: Item
instance of: scholarly article
title: Unicity theorems for meromorphic functions sharing four small functions
zbMATH Open document ID: 0946.30019
author: Katsuya Ishizaki
author: Nobushige Toda
published in: Kodai Mathematical Journal
publication date: 2000-10-17T00:00:00Z
review text: Let \(f_1\), \(f_2\) be transcendental meromorphic functions on the complex plane \(\mathbb{C}\). The paper is a voluminous work which discusses the equality of \(f_1\) and \(f_2\) under different sets of conditions. In addition to the standard functions of the Nevanlinna theory, the bulk of the discussions involve the ``smallness w.r.t. \(f_j\)' in terms of the classes (1) \(s(f_j)\) of all real functions \(p\) on \((0,+\infty)\) such that \(p(r)/T(r,f_j)\to 0\) as \(r\to+\infty\) possibly excluding a set of finite linear measure and (2) \(S(f_j)\) of all meromorphic functions `\(a\)' on \(\mathbb{C}\) such that \(T(.,a)\in s(f_j)\) \((j= 1,2)\). Most of the results have the common assumption that \(f_1\) and \(f_2\) ``share'' four distinct functions `\(a\)' of \(S(f_1)\cap S(f_2)\) in the sense that \(\{z\in \mathbb{C}: f_j(z)= a(z)\}\) is the same for \(j= 1,2\). An illustration of the extra conditions (taken from cor. 1) is indicated in the abstract and it is the following: \[ \overline N(.,f_j)\leq u_j T(.,f_j)+ p_j\quad\text{for some }p_j\in s(f_j),\quad u_j\in [0,1/19)\quad\text{for }j= 1,2. \] Assumptions, in which five functions of \(S(f_1)\cap S(f_2)\) are shared and in which four constants of the extended complex plane are shared, (by \(f_1\) and \(f_2\)), are also discussed.
reviewed by: J. Gopala Krishna
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1286738
zbMATH Keywords: unicity theorems
zbMATH Keywords: sharing small functions
zbMATH Keywords: meromorphic functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043945
OpenAlex ID: W2064576383
cites work: Uniqueness of entire functions sharing four small functions
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cites work: An extension of unicity theorem for meromorphic functions
cites work: Some generalizations of the unicity theory of Nevanlinna
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DOI: 10.2996/KMJ/1138043945
description: scientific article; zbMATH DE number 1286738