Some properties of planar p-harmonic and log-p-harmonic mappings

A 2p-times continuously differentiable complex-valued function F = u+iv in a domain ? ? C is p-harmonic if F satisfies the p-harmonic equation ?pF = 0. We say that F is log-p-harmonic if logF 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 ? {1,..., 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.