SUBORDINATION AND RADIUS RESULTS FOR CERTAIN ANALYTIC FUNCTIONS

Abstract

Geometric Function Theory (GFT), a branch of complex analysis, explores the interplay between ge- ometry and analysis, focusing on the qualitative properties of complex-valued functions. A pivotal area within GFT is the theory of univalent functions, where simple geometric considerations unveil their re- markable properties. Despite the existence of numerous significant studies, the seminal work of Ma and Minda, “A unified treatment of some special classes of univalent functions” (1992), marked a turn- ing point by introducing a unified approach to subclasses of univalent functions. This breakthrough has inspired extensive research, leading to the exploration of various subclasses. This thesis aims to enrich the study of the Ma-Minda starlike class S ∗(ϕ) by introducing, in Chapters 2 and 3, a new subclass of univalent starlike functions associated with a bean-shaped region. These chapters explore the geometric properties of the new subclass, deriving inclusion relations, sharp radii estimates, and implications of first and second order differential subordinations. Addressing theoretical gaps between S ∗(ϕ) and other classes such as the Uralegaddi class M(κ), the starlike classes associated with the Booth lemniscate BS ∗(β ), the Cissoid of Diocles S ∗ cs(β ), and the Limaçon of Pascal S T ∗ L(s), this thesis makes significant contributions to the study of starlike func- tions in GFT. In bridging these gaps, Chapter 4 generalizes the Janowski function and its associated classes, originally introduced by Janowski in “Some extremal problems for certain families of analytic functions” (1973). This chapter extends the scope to include oblique and non-Carathéodory domains, marking a transition from Ma-Minda to non-Ma-Minda functions. Moreover, several new open-door-type results are established in sector domains. Building on the study of non-Ma-Minda classes, Chapter 5 introduces a non-Ma-Minda starlike class F [A, B] associated with a newly defined function ψA,B. The function ψA,B maps the unit disk onto an elliptical domain in some cases and a strip domain in others. Since functions in the class F [A, B] may lack univalence, the chapter derives the radius of univalence, starlikeness of order β , and other critical properties. Finally, Chapter 6 proposes a unified framework to study Ma-Minda and non-Ma-Minda classes un- der a common definitions Fm(ψ) and C F m(ψ), enabling their comparative analysis. The chapter provides sharp lower bounds for specific ratios involving the real parts of functions from these classes and their nth partial sums, offering new insights into their structural properties.