Normal families of meromorphic functions whose poles are locally uniformly discrete

Abstract

Let $h$ be a positive number, and let $a(z)$ be a function holomorphic and zero-free on a domain $D$. Let $\mathcal{F}$ be a family of meromorphic functions on $D$ such that for every $f\in\mathcal{F}$, $f(z)=0\Rightarrow f'(z)=a(z)$ and $f'(z)=a(z)\Rightarrow{|f''(z)|\leq{h}}$. Suppose that each pair of functions $f$ and $g$ in $\mathcal{F}$ have the same poles. Then $\mathcal{F}$ is normal on $D$.