Composite Holomorphic Functions and Normal Families

Abstract

We study the normality of families of holomorphic functions. We prove the following result. Let α(z), ai(z), i = 1,2, …, p, be holomorphic functions and ℱ a family of holomorphic functions in a domain D, P(z, w): = (w − a1(z))(w − a2(z))⋯(w − ap(z)), p ≥ 2. If Pw∘f(z) and Pw∘g(z) share α(z) IM for each pair f(z), g(z) ∈ ℱ and one of the following conditions holds: (1) P(z0, z) − α(z0) has at least two distinct zeros for any z0 ∈ D; (2) there exists z0 ∈ D such that P(z0, z) − α(z0) has only one distinct zero and α(z) is nonconstant. Assume that β0 is the zero of P(z0, z) − α(z0) and that the multiplicities l and k of zeros of f(z) − β0 and α(z) − α(z0) at z0, respectively, satisfy k ≠ lp, for all f(z) ∈ ℱ, then ℱ is normal in D. 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).