Mapping Properties of Some Univalent Functions

Abstract

Geometric Function Theory is a dynamic branch of complex analysis that investigates an- alytic and harmonic functions through their mapping behavior such as univalence, close- to-convexity, starlikeness, convexity, etc. Analytic and harmonic functions provide the essential framework for this theory. In the analytic setting, convolution and subordination theories have attracted considerable attention, as they offer powerful tools to generalize well-established classical results. In the harmonic context, particular emphasis is placed on normalized, sense-preserving univalent harmonic mappings, a subject that has seen major developments since the foundational work of Clunie and Sheil-Small in 1984. The construction of new univalent harmonic mappings and exploration of their mapping behaviors remains a primary research focus since then. newlineThis thesis examines the mapping properties of various classical polynomials associated with functions from popular classes of univalent functions, emphasizing their behavior under convolution and subordination. In the harmonic setting, the main goal is to construct new harmonic mappings and present significant results that contribute to enhancing understanding in this field. Additionally, it provides new insights into the relatively less examined area of subordination in harmonic mappings and thus indicating potential directions for future research. newline newline