A generalization of Gu’s normality criterion

Abstract

Let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$, $k\in\mathbf{N}$ and $\mathcal{H}$ be a normal family of meromorphic functions on $D$ such that 0 is not in $\mathcal{H}$ and $\mathcal{H}$ has no sequence that converges to 0 or $\infty$ spherically locally uniformly on $D$. If for every $f\in\mathcal{F}$, $f(z)\neq 0$, and there exists an $h_{f}\in \mathcal{H}$ such that $f^{(k)}(z)\neq h_{f}(z)$ at every $z\in D$, then the family $\mathcal{F}$ is normal on $D$. This generalizes Gu’s well-known normality criterion. It is interesting that the condition $f(z)\neq 0$ cannot be replaced by that all zeros of $f$ have large multiplicities, at least $k+3$ for instance.