The value distribution of \(f^ n f'\)
type: Item
instance of: scholarly article
title: The value distribution of \(f^ n f'\)
zbMATH Open document ID: 0839.30026
published in: Science in China. Series A
publication date: 1996-06-25T00:00:00Z
review text: The Hayman's conjecture ``If \(f\) is a transcendental meromorphic function, then \(f^nf\) assumes every finite non-zero complex value infinitely often for any positive integer \(n\)'' has been completely proved. The authors prove (theorem 1) the Hayman's conjecture for \(n= 1\). Together with earlier investigations of other authors it proves the Hayman's conjecture. Other results are new criteria for the normality of the family of meromorphic functions (theorem 2, 3) and strengthen variants (it is proved an existence of Julia's ray) of the theorem 1 and the Hayman's conjecture (theorem 4, 5) under the assumptions \[ \varlimsup_{r\to \infty} {T(r, f)\over (\ln r)^2}= \infty,\quad \varlimsup_{r\to \infty} {T(r, f)\over (\ln r)^2}= \infty. \] The authors state that W. Bergweiler and A. Eremenko, and L. Zalcman also proved the Hayman's conjecture simultaneously and independently.
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 817801
zbMATH Keywords: normal family
zbMATH Keywords: Hayman's conjecture
zbMATH Keywords: meromorphic function
author: Huaihui Chen
author: Mingliang Fang
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 817801