On the univalence of polyharmonic mappings

TL;DR

This paper investigates the conditions under which polyharmonic mappings are univalent in linearly connected domains, exploring the relationship between the univalence of the mappings and their harmonic components, and introducing concepts like stable univalence.

Contribution

It provides new insights into the univalence criteria for polyharmonic mappings and examines the roles of harmonic components and stability concepts in univalence.

Findings

Univalence of polyharmonic mappings relates to that of their harmonic parts.

Conditions for stable univalence are established.

Connections between logpolyharmonic mappings and univalence are discussed.

Abstract

A 2p-times continuously differentiable complex valued function $f = u + iv$ in a simply connected domain is polyharmonic (or p-harmonic) if it satisfies the polyharmonic equation $\Delta^pF = 0$ . Every polyharmonic mapping f can be written as $f(z) =\sum_{k}^{p} |z|^{2(p-1)}G_{p-k+1}(z)$ where each $G_{p-k+1}$ is harmonic. In this paper we investigate the univalence of polyharmonic mappings on linearly connected domains and the relation between univalence of f(z) and that of $G_p(z)$. The notions of stable univalence and logpolyharminc mappings are also considered.