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Research articles

ScienceAsia 46 (2020): 227-234 |doi: 10.2306/scienceasia1513-1874.2020.027


Normal families of meromorphic functions which share a set


Jinhua Cai, Fanning Meng, Jia Xie, Wenjun Yuan*

 
ABSTRACT:     In this paper, by using the Nevanlinna’s value distribution theory and the method of Zalcman-Pang, it investigates the normality of a family of meromorphic functions, denoted by  , defined in a domain D, which concerns the conditions for each ƒ ∈  : (i) E(S1, ƒ ) = E (S2, ( ƒ (k))q); (ii) both zeros and poles of ƒ −a have multiplicities at least k (> 2 or  2) and k + 1, respectively, or k ( 4) and k − 1, respectively, where k and q are positive integers, a is any finite complex number, S1 = {a1, a2, a3} and S2 = {b1, b2, b3} are made up of finite complex numbers. The conclusion still holds if condition (ii) is replaced by the assumption that zeros of ƒ − ai have multiplicities at least k, where k  1 and i = 1, 2, 3.

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a School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006 China

* Corresponding author, E-mail: wjyuan1957@126.com

Received 8 Jul 2019, Accepted 9 Apr 2020