COEFFICIENT, RADIUS AND SUBORDINATION PROBLEMS FOR SOME ANALYTIC FUNCTIONS

Abstract

Univalent Function Theory is a fascinating branch of complex analysis that investigates func- tions possessing unique geometric properties. Due to its deep-rooted connections with ge- ometry, it forms a central component of Geometric Function Theory. One of the field’s most celebrated milestones is Bieberbach’s conjecture, posed in 1916, which remained unsolved until L. De Branges conclusively proved it in 1985 through methods involving special functions. The extensive efforts to solve this conjecture over the decades significantly advanced the the- ory, leading to the identification of key subclasses of analytic functions, such as S ∗ (starlike functions) and C (convex functions). Additionally, innovative techniques like differential subor- dination have further enriched the field’s methodological toolbox. This thesis is dedicated to addressing coefficient and radius problems, as well as sufficient conditions for second and third-order differential subordination implications for functions be- longing to certain well-established subclasses of starlike and convex functions. Moreover, a new subclass of S ∗(ψ) is introduced and systematically studied. The main contributions of the thesis are as follows: Chapter 2 introduces a novel subclass of starlike functions associated with a strip domain. The chapter investigates various geometric properties, derives sharp bounds for the initial coefficients (up to the fifth order), and explores second and third-order Hankel determinants. Chapter 3 builds upon the study of coefficient problems, where a conjecture regarding the sharp bound of the third-order Hankel determinant is successfully proven for functions in the class S ∗(1 + zez), associated with a cardioid-shaped domain. Furthermore, best improved bounds are established for the third-order Hankel determinant for functions in the classes S ∗(ez) and C (ez). Chapter 4 focuses on functions belonging to the classes S ∗(√1 + tanh z) and C (√1 + tanh z), associated with a bean-shaped domain. The chapter determines sharp bounds for the first five coefficients and the second and third-order Hankel determinants. Additionally, bounds for the sixth and seventh coefficients are derived to investigate potential estimates for the xi fourth-order Hankel determinant for functions belonging to the classes S ∗(1 + zez), S ∗(ez), S ∗(√1 + tanh z) and C (ez). Chapters 5 and 6 shift the focus to differential subordination and its implications. These chapters broaden the existing framework by establishing conditions for second and third- order differential subordination involving various real parameters. The results are derived using admissibility conditions associated with Ma-Minda functions, including ez, 1 + sin z, and 1 + sinh−1 z. The findings highlight the versatility and broad applicability of differential subordi- nation across multiple areas.