On some properties of solutions of the $p$-harmonic equation

TL;DR

This paper explores the geometric properties and variability regions of p-harmonic functions in the unit disk, establishing new bounds and the existence of Landau constants for specific classes of these functions.

Contribution

It introduces new results on convexity, starlikeness, and variability regions of p-harmonic mappings, including the existence of Landau constants for a particular class of p-harmonic functions.

Findings

Established convexity and starlikeness properties of p-harmonic mappings.

Proved the existence of Landau constants for certain p-harmonic function classes.

Provided explicit upper bounds for Bloch norms of bi- and tri-harmonic mappings.

Abstract

A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $\Omega\subseteq\mathbb{C}$ is \textit{p-harmonic} if $f$ satisfies the $p$-harmonic equation $\Delta ^pf=0.$ In this paper, we investigate the properties of $p$-harmonic mappings in the unit disk $|z|<1$. First, we discuss the convexity, the starlikeness and the region of variability of some classes of $p$-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form $Df=zf_{z}-\barzf_{\barz}$, where $f$ is $p$-harmonic in $|z|<1$. Also, we discuss the region of variability for certain $p$-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.