Higher dimensional generalizations of some theorems on normality of meromorphic functions

Abstract

In [Israel J. Math, 2014], 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 $D\subset \mathbb{C},$ and for a positive constant $\epsilon$, if for each $f\in \mathcal{F}$ there exist meromorphic functions $a_f,b_f,c_f$ such that $f$ omits $a_f,b_f,c_f$ in $D$ and $$\min \{\rho (a_f(z),b_f(z)),\rho (b_f(z),c_f(z)),\rho (c_f(z),a_f(z))\}\ge \epsilon ,$$ for all $z\in D$, then $\mathcal{F}$ is normal in $D$. Here, $\rho$ 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 $△:=\{z\in \mathbb{C}:|z|<1\}$ is normal if there are five distinct values $a_1,\dots ,a_5$ such that $$\sup \{(1-|z{|}^2)\frac{|f^{\prime }(z)|}{1+|f(z){|}^2}:z\in f^{-1}\{a_1,\dots ,a_5\}\}<\infty .$$