Some Normality Criteria for Families of Meromorphic Functions

Abstract

Let $k$ be a positive integer and $\cal F$ be a family of meromorphic functions in a domain $D$ such that for each $f\in{\cal F}$, all poles of $f$ are of multiplicity at least 2, and all zeros of $f$ are of multiplicity at least $k+1$. Let $a$ and $b$ be two distinct finite complex numbers. If for each $f\in{\cal F}$, all zeros of $f^{(k)}-a$ are of multiplicity at least 2, and for each pair of functions $f,\,g\in{\cal F}$, $f^{(k)}$ and $g^{(k)}$ share $b$ in $D$, then $\cal F$ is normal in $D$.