Volume 36 • Number 3 • 2013   • Some Properties of Planar $p$-Harmonic and log-$p$-Harmonic Mappings P. Li, S. Ponnusamy and X. Wang Abstract. A $2p$-times continuously differentiable complex-valued function $F=u+iv$ in a domain $\Omega \subseteq\mathbb{C}$ is {\it $p$-harmonic} if $F$ satisfies the $p$-harmonic equation $\Delta ^{p}F=0$. We say that $F$ is {\it $\log$-$p$-harmonic} if $\log F$ is $p$-harmonic. In this paper, we investigate several basic properties of $p$-harmonic and $\log$-$p$-harmonic mappings. In particular, we discuss the problem of when the composite mappings of $p$-harmonic mappings with a fixed analytic function are $q$-harmonic, where $q\in \{1,\ldots, p\}$. Also, we obtain necessary and sufficient conditions for a function to be $p$-harmonic (resp. $\log$-$p$-harmonic). We study the local univalence of $p$-harmonic and $\log$-$p$-harmonic mappings, and in particular, we obtain two sufficient conditions for a function to be a locally univalent $p$-harmonic or a locally univalent $\log$-$p$-harmonic. The starlikeness of $\log$-$p$-harmonic mappings is considered. 2010 Mathematics Subject Classification: Primary: 30C65, 30C45; Secondary: 30C20 Full text: PDF