Normal families and composite meromorphic functions

Abstract

In the paper,we study the normality of families of meromorphic functions.We prove the result: Letα(z) be a meromorphic function,F a family of meromorphic functions in a domain D,and R(z) be a rational function of degree at least 3.If R o f(z) and R o g(z) shareα(z) IM for each pair f(z),g(z)∈F and one of the following conditions holds:(1)R(z)-α(z_0) has at least three distinct zeros or poles for any z_0∈D;(2)There exists z_0∈D such that R(z)-α(z_0):=(z-β_0)~pH(z) has at most two distinct zeros(or poles)β_0 and suppose that the multiplicities l and k of zeros of f(z)-β_0 andα(z)-α(z_0) at z_0,respectively,satisfy k≠l|p|,where H(z) is a rational function and H(β_0)≠0,∞andα(z_0)∈C U {∞} andα(z) is nonconstant.Then F is normal in D.In particular,the result is a kind of generalization of the famous Montel's criterion.