Composite meromorphic functions and normal families

Abstract

We study the normality of families of meromorphic functions defined in terms of certain omitted functions. In particular, we prove the following results. Firstly, if $\mathcal{F}$ is a family of meromorphic functions in a domain D ⊂ ℂ, and a(z), b(z) and c(z) are distinct meromorphic functions in D and if, for all f ∈ $\mathcal{F}$ and all z ∈ D, f(z) ≠ a(z), f(z) ≠ b(z) and f(z) ≠ c(z), then $\mathcal{F}$ is normal in D. Secondly, letting R(w) be a rational function of degree greater than or equal to 3 and $\mathcal{F}$ be a family of functions meromorphic in a domain D ⊂ ℂ, if there exists a non-constant meromorphic function α(z) in D such that, for all f ∈ $\mathcal{F}$ and z ∈ D, R(f(z)) ≠ α(z), then $\mathcal{F}$ is normal in D.