Higher dimensional generalizations of some theorems on normality of meromorphic functions

TL;DR

This paper extends known theorems on the normality of meromorphic functions to higher dimensions, providing new criteria and generalizations for understanding when families of such functions are normal.

Contribution

It establishes higher-dimensional versions of key normality theorems for meromorphic functions, broadening their applicability and understanding.

Findings

Generalized normality criteria to higher dimensions

Extended Lappan's theorem to multidimensional settings

Provided conditions ensuring family normality in complex spaces

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 $$\min\{\rho(a_f(z),b_f(z)), \rho(b_f(z),c_f(z)), \rho(c_f(z),a_f(z))\}\geq \epsilon,$$ for all $z\in D$, then $\mathcal F$ is normal in $D$. Here, $\rho$ is the spherical metric in $\widehat{\mathbb C}$. 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 $\triangle:=\{z\in\mathbb C: |z|<1\}$ is normal if there are five distinct values $a_1,\dots,a_5$ such that $$\sup\{(1-|z|^2)\frac{ |f '(z)|}{1+|f(z)|^2}: z\in f^{-1}\{a_1,\dots,a_5\}\} < \infty.$$