SUBORDINATION, INEQUALITIES AND RADIUS CONSTANTS OF CERTAIN ANALYTIC FUNCTIONS

Abstract

The Bieberbach conjecture, undoubtedly the most famous coefficient problem in univalent function theory, played a significant role in the development of the field. Various subclasses of the class of normalized analytic univalent functions, denoted by S , were introduced, and determining the sharp estimate of n th Taylor coefficients for functions in these subclasses of S is still a challenging and in triguing problem. For the classes S ∗ (ϕ) and C (ϕ) introduced by Ma and Minda in ‘A unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis, Tianjin, Conf Proc Lecture Notes Anal., I Int Press, Cambridge, MA. 157-169 (1992)’, the estimate of |an| for n = 2,3,4 were known. In Chapter 2, we obtain the sharp estimate of |a5| for these classes. The derived estimates coincide with some already known bounds for other subclasses of starlike and convex functions, while also providing new cases. In continuation of coefficient problems, Chapter 3 gives the sharp bounds of second and third-order Hermitian-Toeplitz determinants for the same class es along with the class of close-to-convex functions. The established bounds directly extend to various subclasses as well, which show the applicability and significance of the results. This thesis also dealt in the radius problems along with the coefficient problems for a class of semigroup generator denoted by Aβ in Chapter 4. Using the sharp estimate of n th coefficient for functions in the class Aβ , we estab lish the sharp Bohr radius, Bohr-Rogosinski radius and radius of starlikeness of order α. Additionally, Hankel determinants, Toeplitz and Hermitian Toeplitz determinants, Zalcman functional and bounds of successive coefficient difference also investigated for the same class. In the subsequent chapter, we introduce and study the notion of Toeplitz determinants in the case of higher dimensions. The sharp bounds of second and third-order Toeplitz determinants constructed over the Taylor coefficients of biholomorphic mappings are established. Chapters 5 and 6 give the sharp estimates of Toeplitz de terminants for the subclasses of starlike mappings and quasi-convex mappings, respectively, defined on the unit ball in a complex Banach space and on the unit poly disk in C n . These derived bounds extend the already known bounds for univalent functions to higher dimensions.