An addition to the Tumura-Clunie theorem
type: Item
instance of: scholarly article
title: An addition to the Tumura-Clunie theorem
zbMATH Open document ID: 1153.30029
author: Jürgen Grahl
published in: Results in Mathematics
publication date: 2008-09-04T00:00:00Z
review text: Let \(P\) be an arbitrary differential polynomial of degree at most \(n-1\), with constant coefficients and let \(f\) be an entire function. In the paper under review, the author exploits \textit{X.-H. Hua}'s extension [Complex Variables, Theory Appl. 16, No. 1, 69--77 (1991; Zbl 0744.30023)] of the Tumura-Clunie theorem [c.f. \textit{J. Clunie}, J. Lond. Math. Soc. 37, 17--27 (1962; Zbl 0104.29504)], and uses some general results about differential polynomials, to show that if the entire function \(\psi:=f^n+P[f]\) does not vanish in \(\mathbb C\), then \(f\) itself has a Picard exceptional value and satisfies certain differential equations. Furthermore, under additional assumptions (cf. Theorem 1.6), \(f\) has the form \(f(z)=\exp(az+b)\). The author also shows by means of examples that a similar result need not hold for meromorphic functions.
reviewed by: George Csordas
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 5320903
zbMATH Keywords: differential polynomial
zbMATH Keywords: Picard type theorem
zbMATH Keywords: differential equation
MaRDI profile type: Publication
full work available at URL: s00025-007-0272-2
OpenAlex ID: W2044971744
DOI: 10.1007/S00025-007-0272-2
description: scientific article; zbMATH DE number 5320903