On monomials and Hayman's problem
type: Item
instance of: scholarly article
title: On monomials and Hayman's problem
zbMATH Open document ID: 0781.30027
author: Zhanliang Zhang
author: Wei Li
published in: Kodai Mathematical Journal
publication date: 1993-09-07T00:00:00Z
review text: The following problem is still open: Let \(f\) be a transcendental meromorphic function in the plane. It is true that, for \(n=1\), \(\psi=f^ nf'\) assumes every value, except possibly zero, infinitely often? In case \(n\geq 2\) this is known to be true, and even in the case \(\psi=f^ n(f')^{n_ 1}\cdots\bigl(f^{(k)}\bigl)^{n_ k}\), \(\max n_ j\geq 1\). In the paper under review it is shown, that \(\psi\) assumes every nonzero value infinitely often, provided \(n\geq 1\), \(n_ 1\geq 1\) and \(f\) ahs only `few' zeros. This additional hypothesis makes the reviewer's proof [Math. Z. 176, 255-264 (1981; Zbl 0466.30026)] work also in the case \(n=1\).
reviewed by: Norbert Steinmetz
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 409641
zbMATH Keywords: differential polynomial
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039703
OpenAlex ID: W1999404909
cites work: Q5723337
cites work: Picard values of meromorphic functions and their derivatives
cites work: Q5540608
cites work: Über ein Problem von Hayman
cites work: Deficiencies of monomials
cites work: Ein Malmquistscher Satz für algebraische Differentialgleichungen zweiter Ordnung. (A theorem of Malmquist for algebraic differential equations of second order)
cites work: Über die Nullstellen von Differentialpolynomen
DOI: 10.2996/KMJ/1138039703
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description: scientific article; zbMATH DE number 409641