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

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 ${\Delta }^{p}f=0.$ 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.

19 pages; This is a 2009 preprint of the authors; Accepted from "Filomat"