Sharing values and a problem due to C. C. Yang
type: Item
instance of: scholarly article
title: Sharing values and a problem due to C. C. Yang
zbMATH Open document ID: 0874.30024
published in: Pacific Journal of Mathematics
publication date: 1997-06-23T00:00:00Z
review text: Let \(f,g,a\) and \(b\) be meromorphic functions in the plane such that \(T(r,a)= S(r,f)\) and \(T(r,b)= S(r,g)\). If the two functions \(f-a\) and \(g-b\) have the same zeros with the same multiplicities, then one says: \(f\) and \(g\) share the pair \((a,b)CM\). In the paper under review the author proves: Let \(f,g,a,\alpha,b\) and \(\beta\) be meromorphic functions in the plane, where \(a\) and \(\alpha\) are small functions with respect to \(f\) and \(b\) and \(\beta\) are small functions with respect to \(g\), i.e. \(T(r,a)\), \(T(r,\alpha) =S(r,f)\) and \(T(r,b)\), \(T(r,\beta) =S(r,g)\) and \(a\neq \alpha\) and \(b\neq\beta\). Suppose that \(f\) and \(g\) share the pair \((a,b)CM\) and \(\delta (\alpha,f) +\delta(\beta,g) +\delta(\infty,f) +\delta (\infty,g)>3\). Then either \[ {f-\alpha \over a- \alpha}= {g-\beta \over b- \beta} \quad\text{or} \quad {f-\alpha \over a-\alpha} \cdot {g-\beta \over b-\beta} =1. \] He also shows by the following example that the number 3 in the above inequality is best possible: \[ f(z)= e^{2z}- Q(z)e^z, \quad g(z)=e^{2z}/ \left(e^z+ {P(z) \over Q(z)} \right), \] where \(P\) and \(Q\) are nonzero polynomials. Here \(f\) and \(g\) share the pair \((P,Q)CM\) and \(\alpha= \beta=0\).
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1025780
zbMATH Keywords: sharing values
zbMATH Keywords: sharing functions
author: Xinhou Hua
reviewed by: Günter Frank
MaRDI profile type: Publication
full work available at URL: pjm.1996.175.71
OpenAlex ID: W1557631377
DOI: 10.2140/PJM.1996.175.71
description: scientific article; zbMATH DE number 1025780