Some generalizations of the unicity theory of Nevanlinna
type: Item
instance of: scholarly article
title: Some generalizations of the unicity theory of Nevanlinna
zbMATH Open document ID: 0788.30020
author: Nobushige Toda
published in: Proceedings of the Japan Academy. Series A
publication date: 1994-05-26T00:00:00Z
review text: In the present paper a generalization of R. Nevanlinna's five-point theorem is given: If \(f_ 1\) and \(f_ 2\) are transcendental meromorphic functions in the plane such that \(E(f_ 1=a_ j)=E(f_ 2=a_ j)\) for five distinct values \(a_ j\), then \(f_ 1=f_ 2\). Here, \(E(f=a)\) is simply the set of zeros of \(f-a\). This makes also sense if \(a(\not\equiv f)\) is also a meromorphic function. The generalization is as follows: If \(E(f_ 1=a_ j)=E(f_ 2=a_ j)\) for at least seven meromorphic functions \(a_ j\) satisfying \(T(r,a_ j)=O(\min (T(r,f_ 1),T(r,k)))\) then the same conclusion \(f_ 1=f_ 2\) holds. The same is true if seven is replaced by five, under the assumption that there exist two functions \(b_ 1\), \(b_ 2\) in the same class: \(T(r,b_ j)=O(\min (T(r,f_ 1),T(r,f_ 2))\) with large deficiency, \(\delta(f_ k,v_ k)>4/5\), \(k=1,2\).
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 488099
zbMATH Keywords: value sharing
zbMATH Keywords: deficient function
zbMATH Keywords: five-point theorem
reviewed by: Norbert Steinmetz
MaRDI profile type: Publication
full work available at URL: pjaa.69.61
OpenAlex ID: W2033266527
cites work: Q5723337
cites work: An extension of unicity theorem for meromorphic functions
cites work: Eine Verallgemeinerung des zweiten Nevanlinnaschen Hauptsatzes.
DOI: 10.3792/PJAA.69.61
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description: scientific article; zbMATH DE number 488099