Geometric properties of harmonic and polyharmonic mappings

This dissertation lies in the areas of Complex Analysis and Geometric Function Theory. Its subject concerns mainly the classes of harmonic and polyharmonic mappings as well as their relation to analytic functions. Harmonic mappings are defined as the solutions to the complex Laplacian equation. Since every analytic function solves the Laplacian equation, the harmonic mappings have been widely studied as generalisations of the analytic functions. Polyharmonic mappings form the wider class of functions which solve the p-Laplacian equation, for a certain positive p. In the first paper, we improve the harmonic analogue of the classical Bohr inequality, proved by Kayumov, Ponnusamy, and Shakirov. We show that anon-negative term can be added and still obtain the same Bohr radius. In the second paper, we examine how we can improve known results for analytic functions, given by R. Fournier and St. Ruscheweyh, for Bohr's phenomenon in more general domains. We also prove a sharp Bohr's inequality for harmonic mappings defined in disks larger than the unit disk. In the third paper, we study the behaviour of harmonic mappings, defined either in the unit disk or in the upper half-plane, near the boundary. We also find sufficient conditions in order for the boundary function of such a mapping to be continuous, based on a result of Bshouty, Lyzzaik, and Weitsman. In the fourth paper, we show that Radó's theorem remains true for the wider class of polyharmonic mappings. We also obtain results about the continuity of the boundary function of a polyharmonic mapping. At the end, we present a method for constructing a fully close-to-convex biharmonic mapping by using a convex harmonic one.