无零点亚纯函数族的正规与拟正规定则

Let $k,K$ be positive integers, φ(z)(otequiv0)$ be ananalytic function, and $mathcal{F}$ be a family of zero-free meromorphicfunctions on a domain $D$, all of whose poles are multiple. If for each$finmathcal{F}$, $f^{(k)}(z)-φ(z)$ has at most $K$ distinctzeros(ignoring multiplicity), then $mathcal{F}$ is quasinormalof order at most $u$ on $D$, where $u=[rac{K}{k+2}]$ is equal tothe largest integer not exceeding $rac{K}{k+2}$. In particular, if $K=k+1$, then $cal F$ is normal on $D$.