On entire functions which share one small function CM with their \(k\)th derivative
type: Item
instance of: scholarly article
title: On entire functions which share one small function CM with their \(k\)th derivative
zbMATH Open document ID: 1074.30027
DOI: 10.1007/BF03323007
author: Amer H. H. Al-Khaladi
published in: Results in Mathematics
publication date: 2005-06-09T00:00:00Z
review text: The author proves the following result: Let \(f\) be a non-constant entire function such that its Nevanlinna's characteristic functions satisfy \(\overline {N} (r,\frac{1}{f^{(k)}})=S(r,f)\). If \(f\) and its \(k\)-th derivative \(f^{(k)}\) share a small meromorphic function \(a(\not\equiv 0,\infty)\) CM (counting multiplicity), then \[ f-a=(1-P_{k-1}/a)(f^{(k)}-a), \] where \(P_{k-1}\) is a polynomial of degree at most \(k-1\) such that \(1-P_{k-1}/a=e^\beta\) for an entire function \(\beta\). If \(k=1\) and if \(a\) is constant, this result is due to \textit{R. Brück} [Result. Math. 30, 21--24 (1996; Zbl 0861.30032)].
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 2174556
zbMATH Keywords: entire function
zbMATH Keywords: uniqueness theorem
zbMATH Keywords: Nevanlinna theory
MaRDI profile type: Publication
cites work: On entire functions which share one small function CM with their first derivative
cites work: On entire functions which share one value CM with their first derivative
cites work: Q5723337
description: scientific article; zbMATH DE number 2174556