Normal families of holomorphic functions and multiple values

Abstract

Let $\mathcal{F}$ be a family of holomorphic functions defined in $D \subset C$, and let $ k, m, n, p $ be four positive integers with $ \frac{k+p+1}{m}+\frac{p+1}{n} < 1 $. Let $\psi (\not \equiv 0, \infty )$ be a meromorphic function in $ D $ and which has zeros only of multiplicities at most $ p $. Suppose that, for every function $ f \in \mathcal{F} $, (i) $ f $ has zeros only of multiplicities at least $ m $; (ii) all zeros of $ f^{(k)}-\psi(z) $ have multiplicities at least $ n $; (iii) all poles of $ \psi $ have multiplicities at most $ k $, and (iv) $ \psi(z) $ and $ f(z) $ have no common zeros, then $\mathcal{F}$ is normal in $ D $.