AN EXACT ESTIMATE OF THE THIRD HANKEL DETERMINANTS FOR FUNCTIONS INVERSE TO CONVEX FUNCTIONS.

Invesigation of bounds for Hankel determinat of analytic univalent functions is prominentintrest of many researcher from early twenth century to study geometric properties. Manyauthors obtained non sharp upper bound of third Hankel determinat for different subclasses ofanalytic univalent functions until Kwon et al. [5] obtained exact estimation of the fourth coefficeientof Carath'eodory class. Recently authors made use of an exact estimation of the fourthcoefficient, well known second and third coefficient of Carath'eodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions.Let w = f(z) = z + a₂z² + · · · be analytic in the unit disk D = {z ∊ C: |z| < 1}, and Sbe the subclass of normalized univalent functions with f(0) = 0, and f'(0) = 1. Let z = f-1be the inverse function of f, given by f^-1(w) = w + t₂w² + · · · for some |w| < r_o(f). LetS^c ⊂ S be the subset of convex functions in D. In this paper, we estimate the best possibleupper bound for the third Hankel determinant for the inverse function z = f^-1 when f ∊ S^c.Let S^c be the class of convex functions. We prove the following statements (Theorem): Iff ∊ S^c, then - H_3,1(f^-1)| = 1 36 and the inequality is attained for p₀(z) = (1 + z³)/(1 - z³). [ABSTRACT FROM AUTHOR]