An entire function sharing a polynomial with its derivatives
type: Item
instance of: scholarly article
title: An entire function sharing a polynomial with its derivatives
zbMATH Open document ID: 1388.30036
author: Indrajit Lahiri
author: Imrul Kaish
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2017-11-01T00:00:00Z
review text: The paper deals with unicity theorems for entire functions. Let \(f\) be an entire function and \(a=a(z)\) a polynomial of degree \(p\). We let \(E(a;f)\) denote the set of zeros of \(f-a\) counted with multiplicities. We also denote by \(\overline{E}(a;f)\) the set of distinct zeros of \(f-a\). In the case where \(f\) is a transcendental function, we set \(\text{deg}\,f=+\infty\). We define the counting function \(n_A(r, f, a)\) as the number of zeros of \(f\) in \(A\cap \left\{|z|<r \right\}\) with multiplicities, and let \[ N_A(r, f, a)=\int_0^r (n_A(t, f, a)-n_A(0, f, a))dt/t. \] Suppose that \(n \geq p+2\) is a positive integer. Let \(A=\overline{E}(a;f)\setminus \overline{E}(a;f')\) and \(B=\overline{E}(a;f')\setminus (\overline{E}(a;f^{(n)})\cap \overline{E}(a;f^{(n+1)}))\). Then the authors prove the following unicity theorem: If \[ N_A(r, a, f)+N_B(r, a, f')=S(r, f) \] and each common zero of \(f-a\) and \(f'-a\) has the same multiplicity, then \(f=\lambda e^z\) for some constant \(\lambda \neq 0\).
reviewed by: Yoshihiro Aihara
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 6802157
zbMATH Keywords: entire functions
zbMATH Keywords: uniqueness theorems
MaRDI profile type: Publication
full work available at URL: s40574-016-0077-x
OpenAlex ID: W2346751180
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DOI: 10.1007/S40574-016-0077-X
description: scientific article; zbMATH DE number 6802157