## Bulletin of the Belgian Mathematical Society - Simon Stevin

- Bull. Belg. Math. Soc. Simon Stevin
- Volume 19, Number 1 (2012), 47-61.

### 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.

#### Article information

**Source**

Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 1 (2012), 47-61.

**Dates**

First available in Project Euclid: 7 March 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.bbms/1331153408

**Digital Object Identifier**

doi:10.36045/bbms/1331153408

**Mathematical Reviews number (MathSciNet)**

MR2782765

**Zentralblatt MATH identifier**

1224.30111

**Subjects**

Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Secondary: 30C20: Conformal mappings of special domains

**Keywords**

$p$-harmonic mapping subordination integral mean extreme point

#### Citation

Qiao, J.; Wang, X. Subordination of $p$-harmonic mappings. Bull. Belg. Math. Soc. Simon Stevin 19 (2012), no. 1, 47--61. doi:10.36045/bbms/1331153408. https://projecteuclid.org/euclid.bbms/1331153408