Composite Holomorphic Functions and Normal Families

الملخص EN

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).