Higher-Dimensional Generalizations of Some Theorems on Normality of Meromorphic Functions

Abstract

In [5], Grahl and Nevo obtained a significant improvement for the well-known normality criterion of Montel. They proved that for a family of meromorphic functions $\mathcal{F}$ in a domain $\mathit{D}\subset \mathbb{C}$, and for a positive constant ε, if for each $\mathit{f}\in \mathcal{F}$ there exist meromorphic functions ${\mathit{a}}_{\mathit{f}},{\mathit{b}}_{\mathit{f}},{\mathit{c}}_{\mathit{f}}$ such that f omits ${\mathit{a}}_{\mathit{f}},{\mathit{b}}_{\mathit{f}},{\mathit{c}}_{\mathit{f}}$ in D and

$$min\{\mathit{\rho }({\mathit{a}}_{\mathit{f}}(\mathit{z}),{\mathit{b}}_{\mathit{f}}(\mathit{z})),\mathit{\rho }({\mathit{b}}_{\mathit{f}}(\mathit{z}),{\mathit{c}}_{\mathit{f}}(\mathit{z})),\mathit{\rho }({\mathit{c}}_{\mathit{f}}(\mathit{z}),{\mathit{a}}_{\mathit{f}}(\mathit{z}))\}\ge \mathit{\epsilon }$$

for all $\mathit{z}\in \mathit{D}$, then $\mathcal{F}$ is normal in D. Here, ρ is the spherical metric in $\hat{\mathbb{C}}$. In this paper, we establish the high-dimensional versions for the above result and for the following well-known result of Lappan: A meromorphic function f in the unit disc $△:=\{\mathit{z}\in \mathbb{C}:|\mathit{z}|<1\}$ is normal if there are five distinct values ${\mathit{a}}_1,\dots ,{\mathit{a}}_5$ such that

$$sup\{(1-|\mathit{z}{|}^2)\frac{|{\mathit{f}}^{\prime }(\mathit{z})|}{1+|\mathit{f}(\mathit{z}){|}^2}:\mathit{z}\in {\mathit{f}}^{-1}\{{\mathit{a}}_1,\dots ,{\mathit{a}}_5\}\}<\infty .$$