The Miniowitz and Vuorinen theorems for the mappings with non-bounded characteristics

TL;DR

This paper investigates classes of mappings with unbounded quasiconformal characteristics, establishing conditions for their normality and growth behavior near points, extending classical theorems to broader mapping classes.

Contribution

It extends the Miniowitz and Vuorinen theorems to mappings with non-bounded characteristics, providing new criteria for normality and growth estimates.

Findings

Mappings with non-bounded characteristics have logarithmic growth near points.

Normality conditions are established for mappings omitting certain value sets.

Theorems are generalized to higher-dimensional spaces.

Abstract

The present paper is devoted to the study of classes of mappings with non--bounded characteristics of quasiconformality. It is proved that the normal families of mappings distorting the families of mappings in ${\Bbb R}^n$ by special way, have the logarithmic order of growth in the neighborhood of every point. There are proved some sufficient conditions of normality of such mappings $f:D\rightarrow \bar{{\Bbb R}^n},$ $n\ge 2,$ omitting the values of some set $E_f$ with some constraints of the type $c(E_f)\ge \delta,$ $\delta\in {\Bbb R},$ where $c(\cdot)$ is the special set function.