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arXiv · 2007.14583

A note on entire functions sharing a finite set with applications to difference equations

Molla Basir Ahamed
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Abstract

Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function $ f $ when sharing a set $ \mathcal{S}=\{α(z), β(z)\} $ of two entire functions $ α$ and $ β$ with $ \max\{ρ(α), ρ(β)\}<ρ(f) $ with its difference $ \mathcal{L}^n_c(f)=\sum_{j=0}^{n}a_jf(z+jc) $, then $ \mathcal{L}^n_c(f)\equiv f $, and more importantly certain form of the function $ f $ has been found. The results in this paper improve those given by \emph{k. Liu}, \emph{X. M. Li}, \emph{J. Qi, Y. Wang and Y. Gu} etc. Some constructive examples have been exhibited to show the condition $ \max\{ρ(α), ρ(β)\}<ρ(f) $ is sharp in our main result. Examples have been also exhibited to show that if $ CM $ sharing is replaced by $ IM $ sharing, then conclusion of the main results ceases to hold.

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Molla Basir Ahamed. 2020-07-29. A note on entire functions sharing a finite set with applications to difference equations. https://arxiv.org/abs/2007.14583

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