Executive Summary : | This project aims to address fundamental problems in the theory of Harmonic Mappings, including those related to harmonic Hardy spaces and Bergman Spaces. The roots of complex-valued univalent harmonic mappings can be traced back to the work of Weierstrass and Enneper on minimal surfaces, which revealed the relation between minimal surfaces defined by isothermal parameters to univalent harmonic mappings. Douglas received Field's medal for solving Plateau's Problem on area minimizing surfaces over non convex domains in 1936. The project aims to obtain characterization for univalent harmonic mapping f, ensuring that the minimal surface obtained via Weierstrass-Enneper representation is globally area minimizing minimal surface. Clunie and Sheil-Small's landmark paper initiated the study of sense-preserving univalent harmonic mappings, obtaining sharp estimates and growth theorems for convex and close-to-convex harmonic mappings. Future research will focus on fundamental problems such as determining the growth of harmonic mappings, quasiconformal mappings, Taylor coefficients, and boundary behavior of harmonic mappings. Logarithmic coefficient problems will be studied, as well as geometry preserving polynomial approximation of harmonic mappings and Bohr phenomenon problems. Applications of Bohr phenomenon for operator valued functions will also be explored. Zeros of polynomials will be studied further, and integral means problems will be explored to understand boundary behavior of harmonic functions. As harmonic mappings are a solution to real-world problems such as fluid flow, heat distribution, and minimal surface problems, the project's results are expected to have applications in these fields. |