Executive Summary : | The purpose of this project is two-fold. In the first part of the proposal we consider research problems on harmonic mappings, more specifically one harmonic meromorphic mappings wth nonzero pole. Researchers clarify here that harmonic mappings in the complex plane are one-to-one complex-valued harmonic functions whose real and imaginary parts are not necessarily conjugate. These functions are natural generalizations of conformal mappings and they play important role in parametrizing minimal surfaces. Because of the later reason, many differential geometers considered studying harmonic mappings. In 1984, J. Clunie and T. Shiel-Small pointed out that many classical results of conformal mappings have clear analogues for harmonic mappings and since then complex analyst started developing interest in this area of research. This is the main reason to consider some interesting research problems on Harmonic mappings. In this proposal we have incorporated mainly four interesting problems on these mappings. In the second part of this proposal we consider problems on Bloch constants. This problem originated in the year 1926. From the work of G. Valiron, A. Bloch found that every nonconstant entire function has holomorphic branches of the inverse in arbitrarily large Euclidean discs. From this observation of Bloch, E. Landau defined Landau and Bloch constants. Since conformation of the Bieberbach conjecture by Louis de Branges, this outstanding open problem in classical complex analysis is probably the determination of the precise value of the Bloch constants and Landau constants for holomorphic mappings defined on the unit disc. This is the main reason to consider Bloch constant problems for various classes of functions (analytic/meromorphic). |