Composite meromorphic functions and normal families

Abstract

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and ${\mathcal{F}}$ a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair ${f(z),g(z)\in \mathcal{F}}$ and one of the following conditions holds: (1) P(z) − α(z ₀) has at least three distinct zeros for any ${z_{0}\in D}$; (2) There exists ${z_{0}\in D}$ such that P(z) − α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) − α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) − β ₀ and α(z) − α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all ${f(z)\in\mathcal{F}}$. Then ${\mathcal{F}}$ is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.