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| Let $k,K$ be positive integers, $ arphi(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$finmathcal{F}$, $f^{(k)}(z)- arphi(z)$ 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 $D$. |
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| Keywords:meromorphic function, normal family, quasinormal family. |
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