ESTIMATES OF SECOND AND THIRD HANKEL DETERMINANTS FOR BAZILEVIC FUNCTION OF ORDER GAMMA

JAMIU OLUSEGUN HAMZAT

JAMIU OLUSEGUN HAMZAT DEPARTMENT OF MATHEMATICS, UNIVERSITY OF LAGOS, AKOKA, LAGOS STATE, NIGERIA

Keywords: Analytic, univalent, starlike, convex, Bazilevic

Abstract

In the recent time, the study of Bazilevic functions became so popular that researchers, especially in Geometric Function Theory, have had to study different subclasses of Bazilevic functions in different directions. How- ever, their study seem to lack full stamina addressing relevant connections of Bazilevic functions to some properties such as coefficient bounds, sharp bounds of the Fekete-Szego functional as well as the Hankel determinants for functions belonging to some specific subclasses of Bazilevic functions. Consequently, in this article, with the aid of Salagean derivative operator, the author derived the Bazilevic class T_n^α(γ), of order γ, type α, via the convolution of the fractional analytic function g(z)^αand the normalized univalent function f (z) in the open unit disk. In the sequel, sharp bounds on the Taylor-Maclaurin coeffi- cients a_k(α) for functions belonging to the aforementioned class were obtained while the relationship of these bounds to the classical Fekete-Szego inequality H₂(1) and Hankel determinants H₂(2) and H₃(1) were established using a clear Mathematical approach. Some of the consequences of the results so obtained were discussed as corollaries.

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