Quasi-nearly subharmonic functions and quasiconformal mappings

TL;DR

This paper investigates the properties of quasi-nearly subharmonic functions under quasiregular mappings and characterizes quasiconformal mappings through their interactions with these functions.

Contribution

It establishes new results on the composition of quasi-nearly subharmonic functions with quasiregular mappings and provides criteria for quasiconformality based on these compositions.

Findings

Composition of quasi-nearly subharmonic functions with quasiregular mappings preserves quasi-nearly subharmonicity.

Characterization of quasiconformal mappings via their effect on quasi-nearly subharmonic functions.

Extension of results to regularly oscillating functions.

Abstract

We prove that the composition of a quasi-nearly subharmonic function and a quasiregular mappings of bounded multiplicity is quasi-nearly subharmonic. Also, we prove that if $u\circ f$ is quasi-nearly subharmonic for all quasi-nearly subharmonic $u$ and $f$ satisfies some additional conditions, then $f$ is quasiconformal. Similar results are further established for the class of regularly oscillating functions.