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Some results for normal families of meromorphic functions

In this paper, we will prove some results about normal families of meromorphic functions. These results are generalizations and improvement of some problems studied by W. Yuan et. al. in | Phạm Thị Tuyết Mai và Đtg Tạp chí KHOA HỌC & CÔNG NGHỆ 135(05): 85 - 90 SOME RESULTS FOR NORMAL FAMILIES OF MEROMORPHIC FUNCTIONS Pham Thi Tuyet Mai*, Nguyen Van Thin Department of Mathematics, Thai Nguyen University of Education, Luong Ngoc Quyen Street, Thai Nguyen city, Viet Nam Abstract In this paper, we will prove some results about normal families of meromorphic functions. These results are generalizations and improvement of some problems studied by W. Yuan et. al. in ([17]). Keywords: Normal family, Meromorphic function, share value. 1 Introduction In 2008, by ideas of sharing value, Zhang [13] proved that D be a domain in the complex plane C and F be a family of meromorphic functions in D. The family F is said to be normal in D, in the sense of Montel, if for any sequence {fv } ⊂ F, there exists a subsequence {fvi } such that {fvi } converges spherically locally uniformly in D, to a meromorphic function or ∞. Let The first, we introduce a normal criterion related to a Hayman normal conjecture [2] Theorem A. Let F be a family holomorphic (meromorphic) function defined in a D, n ∈ N, a 6= 0, b ∈ C. If f (z) + af n (z) − b 6= 0 for each f ∈ F and n ≥ 2(n ≥ 3), then F is normal in D. domain 0 Theorem B. Let F be a family holomorphic (meromorphic) function defined in a domain D, n be a positive integer, and b be two complex numbers such that a 6= 0. If n ≥ 4(n ≥ 2) and for every pair 0 n of functions f and g in F , f (z) − af (z) 0 n and g (z) − ag (z) share the value b, then F is normal in D. and In 2012, W. Yuan, Z. Li and J. Lin [17] improved Theorem B and obtained Theorem C. Let F be a family of meromorphic functions D, bers. If (k) (i) f − af n (ii) f results multiplicity less than Drasin [3] for n ≥ 3, are Pang [5] for Chen and Fang [7] for n = 2. due to n = 3, These re- sults for the meromorphic function case are Chen and Fang n ≥ 5, Pang[5] for [7] for n = 3 and n = 3, obtained indepen- given by Langley [10] .