Compositions of polyharmonic mappings

TL;DR

This paper investigates the conditions under which compositions of polyharmonic mappings preserve harmonicity levels in simply connected domains, providing necessary and sufficient criteria for such compositions.

Contribution

It establishes precise conditions for when the composition of a polyharmonic mapping with various classes of functions remains polyharmonic, advancing understanding of harmonic mappings.

Findings

Characterizes when compositions are l-harmonic for analytic functions

Identifies conditions for compositions with harmonic and q-harmonic mappings

Provides a comprehensive framework for polyharmonic composition analysis

Abstract

The paper is devoted to the study of compositions of polyharmonic mappings in simply connected domains. More precisely, we determine necessary and sufficient conditions of polyharmonic mapping $f$ such that $f\circ F$ (resp. $F\circ f$) is $l$-harmonic for any analytic function (or harmonic mapping but not analytic, or $q$-harmonic mapping but not $(q-1)$-harmonic) $F$.