Nonexceptional Functions and Normal Families of Zero-free Meromorphic Functions

Abstract

Let $k$ be a positive integer, let $\mathcal F$ be a family of zero-free meromorphic
functions in a domain $D$, all of whose poles are multiple, and let $h$ be a meromorphic
function in $D$, all of whose poles are simple, $h\not\equiv0, \infty$. If for each $f\in\mathcal F$,
$f^{(k)}(z)-h(z)$ has at most $k$ zeros in $D$, ignoring multiplicities,
then $\mathcal F$ is normal in $D$. The examples are
provided to show that the result is sharp.