The uniqueness of meromorphic functions with their derivatives
type: Item
instance of: scholarly article
title: The uniqueness of meromorphic functions with their derivatives
zbMATH Open document ID: 0932.30027
author: Qingcai Zhang
published in: Kodai Mathematical Journal
publication date: 1999-03-28T00:00:00Z
review text: The paper deals with the question up to which class of functions a meromorphic function \(f\) in \(\mathbb{C}\) is determined if it shares with its derivative \(f'\) the value 1 \(CM\) (= counting multiplicity). This means that the preimages of 1 under \(f\) and \(f'\) are equal and the 1-points of \(f\) and \(f'\) have the same multiplicity. The main result obtained is that if in addition \[ \overline N(r,f)+N \left(r,{1\over f'}\right) <\bigl(\lambda+ o(1)\bigr) T(r,f') \] \((N,T\) from Nevanlinna-theory), for some \(\lambda\in(0,{1\over 2})\) then \[ f(z)=Ae^{cz}+ 1-{1\over c}, \quad A,c\in \mathbb{C}\setminus \{0\}. \] This generalizes a result of \textit{R. Brück} [Result Math. 30, No. 1-2, 21-24 (1996; Zbl 0861.30032)]. There are also obtained similar results if either \(f\) is entire or \(f'\) is replaced by some higher derivative \(f^{(k)}\).
reviewed by: G. Jank
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30D20
zbMATH DE Number: 1269766
zbMATH Keywords: Nevanlinna-theory
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043871
OpenAlex ID: W2051560493
cites work: Q5723337
cites work: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen
cites work: On entire functions which share one value CM with their first derivative
cites work: Uniqueness of meromorphic functions and a question of c.c.yang
DOI: 10.2996/KMJ/1138043871
description: scientific article; zbMATH DE number 1269766