Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function

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Abstract

In this paper, we study the value distribution of differential polynomial with the form \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)},\) where f is a transcendental meromorphic function. Namely, we prove that \(f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)}-P(z)\) has infinitely zeros, where P(z) is a nonconstant polynomial and \(n\in {\mathbb {N}},\) \(k, n_1, \dots , n_k, t_1, \dots , t_k\) are positive integer numbers satisfying \(n+\sum _{v}^{k}n_v\ge \sum _{v=1}^{k}t_v+3.\) Using it, we establish some normality criterias for family of meromorphic functions under a condition where differential polynomials generated by the members of the family share a holomorphic function with zero points. Our results generalize some previous results on normal family of meromorphic functions.

Keywords

Meromorphic function Normal family Nevanlinna theory 

Mathematics Subject Classification

Primary 30D35 30D45 

Notes

Acknowledgements

The author wishes to thank the managing editor and referees for their very helpful comments and useful suggestions.

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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of MathematicsThai Nguyen University of EducationThai Nguyen CityVietnam

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