Avoidance criteria for normality of quasiregular mappings

TL;DR

This paper extends Lappan's normality criteria for meromorphic functions avoiding each other to the setting of quasiregular mappings in Euclidean space, providing new criteria for normal families.

Contribution

The paper generalizes Lappan's results on normality and avoidance criteria from meromorphic functions to quasiregular mappings in higher dimensions.

Findings

Established an analogue of Lappan's normality result for quasiregular mappings.

Derived new avoidance criteria for normal quasiregular families.

Extended several of Lappan's theorems to the quasiregular setting in inity space.

Abstract

Peter Lappan in [9] proved that for each $n\in \mathbb{N}=\{1,2,3,\dots\}$, let $f_{1,n}, f_{2,n}$ and $f_{3,n}$ be three continuous functions on $\mathbb{D}:=\{z\in \mathbb{C} : |z| < 1\}$ such that for each $j=1,2,3,$ the sequence $(f_{j,n})$ converges locally uniformly to a function $f_j$ on $\mathbb{D}$. Suppose that the three functions $f_1, f_2,$ and $f_3$ avoid each other on $\mathbb{D}$. Let $\mathcal{F} =(g_n)$ be a sequence of meromorphic functions in $\mathbb{D}$ with the property that for each $n$, the four functions $g_n, f_{1,n}, f_{2,n},$ and $f_{3,n}$ avoid each other, then $\mathcal{F}$ is normal. We present here an analogue of this result in the setting of quasiregular mappings. We also obtain analogues of a few other results by Peter Lappan in [9] to quasiregular setting in the Euclidean space $\mathbb{R}^n$ for normal families and normal quasiregular mappings.