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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.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Apr 21, 2017
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