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Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function

Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function In this paper, we study the value distribution of differential polynomial with the form $$f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)},$$ f n ( f n 1 ) ( t 1 ) ⋯ ( f n k ) ( t k ) , where f is a transcendental meromorphic function. Namely, we prove that $$f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)}-P(z)$$ f n ( f n 1 ) ( t 1 ) ⋯ ( f n k ) ( t k ) - P ( z ) has infinitely zeros, where P(z) is a nonconstant polynomial and $$n\in {\mathbb {N}},$$ n ∈ N , $$k, n_1, \dots , n_k, t_1, \dots , t_k$$ k , n 1 , ⋯ , n k , t 1 , ⋯ , t k are positive integer numbers satisfying $$n+\sum _{v}^{k}n_v\ge \sum _{v=1}^{k}t_v+3.$$ n + ∑ v k n v ≥ ∑ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Normal Criteria for Family Meromorphic Functions Sharing Holomorphic Function

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References (14)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-017-0492-x
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the value distribution of differential polynomial with the form $$f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)},$$ f n ( f n 1 ) ( t 1 ) ⋯ ( f n k ) ( t k ) , where f is a transcendental meromorphic function. Namely, we prove that $$f^n(f^{n_1})^{(t_1)}\dots (f^{n_k})^{(t_k)}-P(z)$$ f n ( f n 1 ) ( t 1 ) ⋯ ( f n k ) ( t k ) - P ( z ) has infinitely zeros, where P(z) is a nonconstant polynomial and $$n\in {\mathbb {N}},$$ n ∈ N , $$k, n_1, \dots , n_k, t_1, \dots , t_k$$ k , n 1 , ⋯ , n k , t 1 , ⋯ , t k are positive integer numbers satisfying $$n+\sum _{v}^{k}n_v\ge \sum _{v=1}^{k}t_v+3.$$ n + ∑ v k n v ≥ ∑ 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.

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Apr 21, 2017

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