On some properties of solutions of the biharmonic equation

A 2p-times continuously differentiable complex-valued function f = u+ iv in a simply connected domainΩ ⊆ C is p-harmonic if f satisfies the p-harmonic equation ∆p f = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form Df = z fz − z fz, where f is p-harmonic in |z| < 1. Also, we discuss the region of variability for certain p-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for biand tri-harmonic mappings.