Bulletin of the Belgian Mathematical Society - Simon Stevin

Subordination of $p$-harmonic mappings

J. Qiao and X. Wang

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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

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

First available in Project Euclid: 7 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

$p$-harmonic mapping subordination integral mean extreme point


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

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