On the theorem of Tumura-Clunie
type: Item
instance of: scholarly article
title: On the theorem of Tumura-Clunie
zbMATH Open document ID: 0678.30020
author: Hongxun Yi
published in: Kodai Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let f be meromorphic in the plane. A meromorphic function g is called ``small'' (with respect to f) if its Nevanlinna characteristic satisfies \(T(r,g)=S(r,f)\). A differential polynomial Q[f] is a finite sum of monomials \(a(z)(f)^{j_ 0}(f')^{j_ 1}...(f^ n)^{j_ n},\) a ``small''. The numbers \(\gamma_ Q=\max (j_ 0+j_ 1+...+j_ n)\) and \(\Gamma_ Q=\max (j_ 0+2j_ 1+...+(n+1)j_ n)\) are called the degree and the weight of Q, respectively. Generalizing a theorem of Tumura, which was first proved by \textit{J. Clunie} [J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.295)] and, in a refined form, by \textit{W. K. Hayman} [Meromorphic functions (1964; Zbl 0149.030)] the author proves the following Theorem. Let Q be a differential polynomial of degree \(\leq n-2\) and set \[ F=f^ n+a_{n-1}f^{n-1}+Q[f]. \] Then, either \(F=(f+a_{n- 1}/n)^ n\) or \[ 2T(r,f)\leq \bar N(r,1/F)+\alpha \bar N(r,f)+\bar N(1,(f+a_{n-1}/n)^{-1})+S(r,f), \] where \(\alpha =\max (1,\Gamma_ Q+3-n)\).
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 4111018
zbMATH Keywords: theorem of Tumura-Clunie
zbMATH Keywords: differential polynomial
reviewed by: Norbert Steinmetz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138038989
OpenAlex ID: W2039479501
cites work: On Integral and Meromorphic Functions
cites work: Exceptional values of differential polynomials
cites work: Q5723337
cites work: The Theorem of Tumura-Clunie for Meromorphic Functions
cites work: On an extension of the theorem of Tumura-Clunie
cites work: Q5765892
cites work: On the Theorem of Tumura and Clunie
DOI: 10.2996/KMJ/1138038989
description: scientific article; zbMATH DE number 4111018