On some results of Lahiri.
type: Item
instance of: scholarly article
title: On some results of Lahiri.
zbMATH Open document ID: 1045.30018
DOI: 10.1016/S0022-247X(03)00335-4
author: Hongxun Yi
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-09-07T00:00:00Z
review text: Let \(f(z)\) and \(g(z)\) be transcendental meromorphic functions in the complex plane. The author considers uniqueness problems of meromorphic functions with weighted sharing value introduced by \textit{I. Lahiri}, see e.g. [Kodai Math. J. 24, No. 3, 421--435 (2001; Zbl 0996.30020)]. A value \(a\in\widehat{\mathbb C}= \mathbb C\cup\{\infty\}\) is called an IM (ignoring multiplicities) shared value of \(f(z)\) and \(g(z)\), \(f(z)= a\) if and only if \(g(z)= a\). A value \(a\in \widehat{\mathbb C}\) is called a CM (counting multiplicities) shared value of \(f(z)\) and \(g(z)\) if \(f(z)\) and \(g(z)\) assume \(a\) at the same points with the same multiplicities. Let \(k\) be a positive integer. Denote by \(E_k(a,f)\) the set of all \(a\)-points of \(f(z)\) of multiplicity \(m\) is counted \(m\) times if \(m\leq k\) and \(k+1\) times if \(m> k\). If \(E_k(a, f)= E_k(a, g)\), then it is said that \(f(z)\) and \(g(z)\) share the value \(a\) with weighted \(k\). Regard the case \(f(z)\) and \(g(z)\) share \(a\)-points CM to \(k=\infty\) in the definition above. The author writes \(f(z)\) and \(g(z)\) share \((a,k)\) to mean that \(f(z)\) and \(g(z)\) share \(a\) with weighted \(k\). Denote by \(N_{k)}(r, 1/(f-a))\) the counting function of \(a\)-points of \(f(z)\) of multiplicity at most \(k\). The author shows that if \(f(z)\) and \(g(z)\) share \((0,1)\), \((\infty,0)\) and \((1,5)\) and if for some interval \(I\in \mathbb R\) \[ \lim_{r\to\infty}\, \sup_{r\in I}\;{N_{1)}(r, 1/f)+ 3\overline N(r,f)- (1/2)m(r, 1/(g- 1))\over T(r,f)}< \frac12, \] then either \(f(z)\equiv g(z)\) or \(f(z) g(z)\equiv 1\). This result is an improvement of the result due to \textit{I. Lahiri} [Nagoya Math. J. 161, 193--206 (2001; Zbl 0981.30023)].
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1976006
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1976006