On the \(k\)th derivative of meromorphic functions with zeros of multiplicity at least \(k+1\)
type: Item
instance of: scholarly article
title: On the \(k\)th derivative of meromorphic functions with zeros of multiplicity at least \(k+1\)
zbMATH Open document ID: 1151.30024
author: Shahar Nevo
author: Xiaojun Liu
author: Xuecheng Pang
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-10-06T00:00:00Z
review text: The well known ``Hayman's alternative'' is the starting point of this paper. Namely, it is known that if \(f(z)\) is a transcendental meromorphic function in the plane, then either \(f(z)\) assumes every finite value infinitely often, or every derivative of \(f(z)\) assumes every finite nonzero value infinitely often. I state one of the main results. Let \(f(z)\) be a transcendental meromorphic function on \(\mathbb C\) of finite order \(\rho(f)\), and \(a(z)= P(z)\exp(Q(z))\not\equiv 0\), where \(P\) and \(Q\) are polynomials, let also \(k\geq 2\) be an integer. Suppose that (C1) all zeros of \(f\) have multiplicity at least \(k+1\), except possibly many, and (C2) \(\limsup_{r\to\infty} (\frac{T(r,a)}{T(r,f)}+ \frac{T(r,f)}{T(r,a)})= \infty\). Then the function \(f^{(k)}-a(z)\) has infinitely many zeros. Moreover, in the case that \(\rho(f)\in\mathbb N\), the result holds with condition (C2) only. The main tools in this paper are the Nevanlinna theory, Zalcman's lemma and Phragmen-Lindelöf principle.
reviewed by: Katsuya Ishizaki
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 5349113
zbMATH Keywords: meromorphic function
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: Zalcman lemma
zbMATH Keywords: Phragmen-Lindelöf principle
zbMATH Keywords: Heyman alternative
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.07.019
OpenAlex ID: W1991887113
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DOI: 10.1016/J.JMAA.2008.07.019
description: scientific article; zbMATH DE number 5349113