Third Hankel determinant of bounded analytic functions

Abstract

Let B(α) denote the class of functions f(z) which are analytic with an open unit disk U = {z : z ε < 1} and satisfy Re{ αz2f″(z)g(z)+zf′(z)g(z) }>0 for some 0 ≤ α < 1, where g(z) is a normalized starlike function. This paper focused on attaining the sharp bound of the third Hankel determinant, H3(1) by the help of some related lemmas, application of triangle inequalities and maximization of the function. The results obtained leads to development of Hankel determinant and more generally to the geometric function theory.