Subordination of $p$-harmonic mappings

Abstract

A $2p$ $(p\geq 1)$ times continuously differentiable complex-valued function $F=u+iv$ in a domain $D\subseteq \mathbb{C}$ is $p$-{\it harmonic} if $F$ satisfies the $p$-harmonic equation $\Delta^p F=\Delta(\Delta^{p-1})F=0$, where $\Delta$ represents the complex Laplacian operator $$\Delta=4\frac{\partial^{2}}{\partial z\partial\bar{z}}:=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}.$$ In this paper, the main aim is to investigate the subordination of $p$-harmonic mappings. First, the characterization for $p$-harmonic mappings to be subordinate are obtained. Second, we get two results on the relation of integral means of subordinate $p$-harmonic mappings. Finally, we discuss the existence of extreme points for subordination families of $p$-harmonic mappings. Two sufficient conditions for $p$-harmonic mappings to be extreme points of the closed convex hulls of the corresponding subordination families are established.