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A note on entire functions sharing a finite set with applications to difference equations
- Publication Year :
- 2020
-
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}=\{\alpha(z), \beta(z)\} $ of two entire functions $ \alpha $ and $ \beta $ with $ \max\{\rho(\alpha), \rho(\beta)\}<\rho(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\{\rho(\alpha), \rho(\beta)\}<\rho(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.
- Subjects :
- Mathematics - Complex Variables
30D35
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2007.14583
- Document Type :
- Working Paper