Properties of Some Classes of Planar Harmonic and Planar Biharmonic Mappings

Abstract

The aim of this paper is to investigate some properties of planar harmonic and biharmonic mappings. First, we use the Schwarz lemma and the improved estimates for the coefficients of planar harmonic mappings to generalize earlier results related to Landau’s constants for harmonic and biharmonic mappings. Second, we obtain a new Landau’s Theorem for a certain class of biharmonic mappings. At the end, we derive a relationship between the images of the linear connectivity of the unit disk \({\mathbb{D}}\) under the planar harmonic mappings \({f=h+\overline{g}}\) and under their corresponding analytic counterparts F = h − g.