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 Omega subset of C is p-harmonic if F satisfies the p-harmonic equation Delta F-P = 0. We say that F is 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 is an element of {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.