On the value distribution of \(f^ l(f^{(k)})^ n\)
type: Item
instance of: scholarly article
title: On the value distribution of \(f^ l(f^{(k)})^ n\)
zbMATH Open document ID: 0802.30029
author: Chungchun Yang
author: Chi Kong Tse
published in: Kodai Mathematical Journal
publication date: 1994-07-12T00:00:00Z
review text: The main result of the present paper is, in a simplified form, as follows: suppose \(f\) is a transcendental meromorphic function in \(\mathbb{C}\) and \(F = f^ \ell (f^{(k)})^ n - 1\) with \(\ell, k, n\) positive integers, \(\ell \leq 2\). Then (i) if \(\ell = 1\) and \(n > 9e + 1\) (hence \(n \geq 26)\) there exists a positive constant \(C\) and a set \(E \subseteq [0, \infty)\) of infinite measure such that \[ T(r,f^{(k)}) \leq C \overline N \left( r, {1 \over F} \right),\quad r \in E; \] (ii) if \(\ell = 2\) and \(k \geq 2\), \(n \geq 2\), then \[ T(r,f) \leq (2 + \eta) \overline N \left( r, {1 \over F} \right) + S(r,f) \] holds for any \(\eta > 0\). In particular, \(F\) has infinitely many zeros. [Reviewer's remark: The context is as follows: If \(f\) is transcendental, then it is known that \(f^ nf'\) assumes every value \(a \neq 0\) infinitely often, provided \(n \geq 1\) in the entire case and \(n \geq 2\) in the meromorphic case. The question whether \(n \geq 1\) suffices has been open for a long time and has only recently (1994) been settled by Bergweiler- Eremenko and Zalcman (preprints)].
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 605453
zbMATH Keywords: value distribution
reviewed by: Norbert Steinmetz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138039905
OpenAlex ID: W2048509479
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DOI: 10.2996/KMJ/1138039905
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description: scientific article; zbMATH DE number 605453