On some Properties of Solutions of the $p$-Harmonic Equation

Sh. Chen, S. Ponnusamy, X. Wang

A $2p$-times continuously differentiable complex-valued function $f=u+iv$ in a simply connected domain $\Omega\subseteq\Bbb C$ is $p$-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 $Df=zf_z-\bar{z}f_\bar{z}$, 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.