Composite Holomorphic Functions and Normal Families

We study the normality of families of holomorphic functions. We prove the following result. Let alpha(z), a(i)(z), i = 1, 2, ... , p, be holomorphic functions and F a family of holomorphic functions in a domain D, P(z, w) := (w - a(1)(z))(w - a(2)(z)) ... (w - a(p)(z)), p >= 2. If P(w) omicron f(z) and P(w) omicron g(z) share alpha(z) IM for each pair f(z), g(z) is an element of F and one of the following conditions holds: (1) P(z(0), z) - alpha(z(0)) has at least two distinct zeros for any z(0) is an element of D; (2) there exists z(0) is an element of D such that P(z(0), z) - alpha(z(0)) has only one distinct zero and alpha(z) is nonconstant. Assume that beta(0) is the zero of P(z(0), z) - alpha(z(0)) and that the multiplicities l and k of zeros of f(z) - beta(0) and alpha(z) - alpha(z(0)) at z(0), respectively, satisfy k not equal lp, for all f(z) is an element of 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).