A note on normality of meromorphic functions

Abstract

Let $\mathcal F$ be a family of all functions $f$ meromorphic in a domain $D\subset\Bbb C$, for which, all zeros have multiplicity at least $k$, and $f(z)=0\Leftrightarrow f^{(k)}(z) = 1\Rightarrow |f^{(k+1)}(z)|\le h$, where $k\in\Bbb N$ and $h\in\Bbb R^+$ are given. Examples show that $\mathcal F$ is not normal in general (at least for $k=1$ or $k=2$). The example we give for $k = 1$ shows that a recent result of Y. Xu [5] is not correct. However, we prove that for $k\not=2$, there exists a positive integer $K\in\Bbb N$ such that the subfamily $\mathcal G =\{ f\in\mathcal F:\ \text{all possible poles of}\ f\ \text{in}\ D\ \text{have multiplicity at least}\ K\}$ of $\mathcal F$ is normal. This generalizes our result in [1]. The case $k = 2$ is also considered.