COEFFICIENT ESTIMATES, RADIUS CONSTANTS AND SUBORDINATION OF CERTAIN ANALYTIC FUNCTIONS

Abstract

Univalent Function Theory (UFT) is a fascinating branch of Complex Analysis. The ele- gance of UFT lies in its ability to derive significant insights from relatively simple geometric considerations. Owing to its strong focus on geometric interpretations, UFT constitutes a fundamental part of Geometric Function Theory (GFT), where such geometric prop- erties play a central and significant role. The thesis provides a comprehensive study of subclasses of analytic functions, offering sharp coefficient estimates, radius constants, and inclusion relations across various classes. It introduces novel techniques, such as convolution-defined results, differential subordinations, and new Ma-Minda subclasses, while generalizing and extending known results in UFT. Chapter 2 introduces a unified class of analytic functions, providing sharp estimates for initial coefficients and the Fekete- Szegö functional, as well as results on convolution-defined classes and second order Hankel determinants for certain close-to-convex functions. Chapter 3 explores univalent as well as non-univalent analytic functions associated with a parabolic region, deriving radius constants (univalence, starlikeness) and sufficient conditions, supported by dia- grams. In Chapter 4, sharp radius problems for the class S ∗(β ) and a product function involving tilted Carathéodory functions are determined, obtaining sharp radius constants and generalizing earlier known results. Chapter 5 introduces a new Ma-Minda subclass S ∗ ρ , associated with the hyperbolic cosine function cosh √z, establishing inclusion rela- tions and sharp radius results in context of various analytic classes. Finally, Chapter 6 uses Briot-Bouquet differential subordination and admissibility conditions to derive suffi- cient conditions for functions in the class S ∗ ρ . Also applications are provided, supported by diagrams. This study offers significant insights into analytic function subclasses, ex- tending known results and introducing novel techniques in UFT.