COEFFICIENT ESTIMATES AND SUBORDINATION THEOREMS FOR CLASSES OF ANALYTIC FUNCTIONS

Abstract

Geometric Function Theory is a branch of mathematics distinguished by its intriguing connection between geometry and analysis. This thesis presents several important results on coefficient estimates for classes of analytic functions. The upper bounds for the second Hankel determinant and symmetric Toeplitz determinants are investigated for functions belonging to two subclasses defined in the open unit disk, namely subclasses of analytic and univalent functions, such as starlike and convex functions. In addition, two classes of order α are introduced, specifically the classes of analytic starlike and convex functions of order α, where 0 ≤ α < 1. Coefficient estimates for these classes are obtained via symmetric Toeplitz determinants, from which several special cases are identified. Furthermore, estimates of the initial Maclaurin coefficients are derived, and Fekete-Szego ̋ type inequalities are established for three new classes of analytic and bi-univalent functions defined by Gegenbauer polynomials. Certain special cases and their corresponding consequences are also discussed. This thesis also addresses differential subordination, differential superordination, and quasi-subordination results for analytic functions. Three subclasses of analytic and bi-univalent functions are defined in the open unit disk, and upper bounds for the second Hankel determinant are obtained for functions in these subclasses. Finally, a suitable class of admissible functions is considered using the Dziok–Srivastava linear operator, where subordination and superordination results are investigated. The findings in this thesis on coefficient estimates of analytic functions are expected to contribute to and stimulate further research in this area.