Composite Holomorphic Functions and Normal Families

Abstract

We study the normality of families of holomorphic functions. We prove the following result. Letα(z),  ai(z),  i=1,2,…,p,be holomorphic functions andFa family of holomorphic functions in a domainD,P(z,w):=(w-a1(z))(w-a2(z))⋯(w-ap(z)),  p≥2. IfPw∘f(z)andPw∘g(z)shareα(z)IM for each pairf(z),  g(z)∈Fand one of the following conditions holds: (1)P(z0,z)-α(z0)has at least two distinct zeros for anyz0∈D; (2) there existsz0∈Dsuch thatP(z0,z)-α(z0)has only one distinct zero andα(z)is nonconstant. Assume thatβ0is the zero ofP(z0,z)-α(z0)and that the multiplicitieslandkof zeros off(z)-β0andα(z)-α(z0)atz0, respectively, satisfyk≠lp, for allf(z)∈F, thenFis normal inD. In particular, the result is a kind of generalization of the famous Montel's criterion. At the same time we fill a gap in the proof of Theorem 1.1 in our original paper (Wu et al., 2010).