On the fixpoints of composite meromorphic functions and generalizations
type: Item
instance of: scholarly article
title: On the fixpoints of composite meromorphic functions and generalizations
zbMATH Open document ID: 0863.30035
DOI: 10.1007/BF02790204
author: Chungchun Yang
author: Jian-Hua Zheng
published in: Journal d'Analyse Mathématique
publication date: 1996-09-15T00:00:00Z
review text: The main purpose of this paper is to give lower bounds for the counting function of the zeros of meromorphic functions of the form \(f(g) - \gamma\). Here \(\gamma \) is nonconstant and satisfies \(T(r, \gamma) = o(T(r,g))\). We mention only two results. If \(f\) is a rational function of degree \(m\geq 1\) and \(g\) is a transcendental meromorphic function, then \[ \liminf_{r \to \infty,r \notin E} {N \biggl(r,1/ \bigl(f(g) - \gamma \bigr) \biggr) \over T(r,g)} \geq m- 2+ \delta (\infty,f) + \delta (0,f) \] for some set \(E\) of finite measure (Corollary 3.3). If \(f\) and \(g\) are transcendental entire functions and \(\gamma\) is a (nonconstant) polynomial, then \[ \lim_{r \to \infty,r \in I} {N\biggl(r+ \eta,1/ \bigl(f(g) - \gamma \bigr) \biggr) \over T(r,g)} = \infty \] for some set \(I\) of logarithmic density one. Here \(\eta = Kr \nu(r,g)^{-\delta}\) with certain constants \(K\) and \(\delta\), and \(\nu(r,g)\) is the central index (Theorem 4.2). The paper also contains growth estimates of canonical products which are used in the proofs, but which are also of independent interest.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 927195
zbMATH Keywords: fixpoint
zbMATH Keywords: factorization
zbMATH Keywords: value distribution
zbMATH Keywords: meromorphic functions
zbMATH Keywords: entire functions
reviewed by: Walter Bergweiler
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02790204
OpenAlex ID: W2082718059
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description: scientific article; zbMATH DE number 927195