Some further results on the unique range sets of meromorphic functions
type: Item
instance of: scholarly article
title: Some further results on the unique range sets of meromorphic functions
zbMATH Open document ID: 0849.30025
author: Ping Li
author: Chungchun Yang
published in: Kodai Mathematical Journal
publication date: 1996-11-10T00:00:00Z
review text: Let \(S\) be a subset of the complex plane \(\mathbb{C}\). For a non-constant meromorphic function \(f\) let \(E_f(S)= \bigcup_{n\in S} \{(z, p)\in \mathbb{C}\times N\mid f(z)= a\) with multiplicity \(p\}\). \(S\) is called a unique range set for entire functions (URSE) if for any pair of non-constant entire functions \(f\) and \(g\) the condition \(E_f(S)= E_g(S)\) implies \(f= g\). Unique range sets for meromorphic functions (URSM) are defined in a similar way. The existence of URSE with finitely many elements has been exhibited by \textit{H.-X. Yi} [Sci. China, Ser. A 38, No. 1, 8-16 (1995; Zbl 0819.30017)], thereby solving a question of \textit{F. Gross} [Complex analysis, Proc. Conf., Lexington 1976, Lect. Notes Math. 599, 51-67 (1977; Zbl 0357.30007)]. In the paper under review, the authors show that there exist URSE with 7 elements and URSM with 15 elements. The proof uses Nevanlinna's theorem on the so-called ``Borel-identities'' and a very careful analysis of the zeros and poles of the Wronskian determinant occurring in this theorem. Lower bounds for the cardinality of unique range sets are also given. To determine the exact value of the minimum cardinality of unique range sets is an interesting but still unsolved problem in value distribution theory.
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 866120
zbMATH Keywords: sharing values
zbMATH Keywords: unique range set
reviewed by: Martin Reinders
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043482
OpenAlex ID: W2048363337
cites work: Q4129688
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DOI: 10.2996/KMJ/1138043482
description: scientific article; zbMATH DE number 866120