Research Article: Entire functions sharing a small function with their two difference operators

Date Published: August 1, 2017

Publisher: Springer International Publishing

Author(s): Feng Lü, Yanfeng Wang, Junfeng Xu.

http://doi.org/10.1186/s13662-017-1281-4

Abstract

In this article, we deduce a uniqueness result of entire functions that share a small entire function with their two difference operators, generalizing some previous theorems of (Farissi et al. in Complex Anal. Oper. Theory 10:1317-1327, 2015, Theorem 1.1) and (Chen and Li in Adv. Differ. Equ. 2014:311, 2014, Theorem 1.1) by omitting the assumption that the shared small entire function is periodic.

Partial Text

Nevanlinna theory of value distributions is concerned with the density of points where a meromorphic function takes a certain value in the complex plane. Nowadays, there has been recent interest in connections between the Nevanlinna theory and the difference operator. In addition, many papers have been devoted to the investigation of the uniqueness problems related to meromorphic functions and their shifts or their difference operators and one got a lot of results (see, e.g., [3–8]).

In this section, we state some results that we employ in our proofs.

Note that documentclass[12pt]{minimal}
usepackage{amsmath}
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begin{document}$f(z)$end{document}f(z) is a nonconstant entire function of finite order. Then documentclass[12pt]{minimal}
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begin{document}$Delta_{c}f(z)$end{document}Δcf(z) and documentclass[12pt]{minimal}
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begin{document}$Delta^{2}_{c}f(z)$end{document}Δc2f(z) are also two entire functions of finite order.

If documentclass[12pt]{minimal}
usepackage{amsmath}
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begin{document}$a(z)$end{document}a(z) is a constant, then it follows from Theorem C that documentclass[12pt]{minimal}
usepackage{amsmath}
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begin{document}$f(z)equiv Delta_{c} f(z)$end{document}f(z)≡Δcf(z). In the following, we assume that documentclass[12pt]{minimal}
usepackage{amsmath}
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begin{document}$a(z)$end{document}a(z) is a nonconstant entire function.

 

Source:

http://doi.org/10.1186/s13662-017-1281-4

 

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