Uniqueness of entire functions sharing two pairs of values with its difference operator

In this paper, we investigate the sharing values problem that entire function \(f(z)\) and its first order difference operator \(\Delta_{\eta}f(z)\) share two distinct pairs of finite values IM. We prove: Let \(f(z)\) be a non-constant entire function of hyper-order less than \(1\), let \(\eta\) be...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Author: Huang, XiaoHuang
Format: Article
Language:English
Subjects:
Complex numbers
Entire functions
Finite differences
Operators (mathematics)
Online Access:Get full text
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Summary:In this paper, we investigate the sharing values problem that entire function \(f(z)\) and its first order difference operator \(\Delta_{\eta}f(z)\) share two distinct pairs of finite values IM. We prove: Let \(f(z)\) be a non-constant entire function of hyper-order less than \(1\), let \(\eta\) be a non-zero complex number, and let \(a\) be a nonzero finite number. Then there exists no such entire function so that \( f(z)\) and \(\Delta_{\eta}f(z)\) share \((0,0)\) and \((a,-a)\) IM. Furthermore, using a result in Wang-Chen-Hu \cite{wch}, we obtain some uniqueness results that when \( f(z)\) and \(\Delta_{\eta}f(z)\) share \(a\neq0\) and \(-a\) IM.
ISSN:2331-8422