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Unicity of Entire Functions Concerning Their Shifts and Derivatives

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Abstract

In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If \(f'(z)\) and \(f(z+c)\) share a, b IM, then \(f'(z)\equiv f(z+c)\). This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178, 2020).

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Acknowledgements

Our first version of the theorem is that f is of finite order. We would like to thank the anonymous referee for pointing out that our theorem is also valid for entire functions of hyper-order less than 1.

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  1. Department of Mathematics, Hangzhou Dianzi University, Hangzhou, 310012, China

    Xiaohuang Huang & Mingliang Fang

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  1. Xiaohuang Huang
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  2. Mingliang Fang
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Correspondence to Mingliang Fang.

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Communicated by Ilpo Laine.

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This paper is supported by the Natural Science Foundation of ZhejiangProvince(LY21A010012).

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Huang, X., Fang, M. Unicity of Entire Functions Concerning Their Shifts and Derivatives. Comput. Methods Funct. Theory 21, 523–532 (2021). https://doi.org/10.1007/s40315-020-00358-1

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