The second Hankel determinant for starlike and convex functions of order alpha

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions f is an element of S given by f(z) = z + Sigma(infinity)(n=2) a(n)z(n) for D = {z is an element of C : vertical bar z vertical bar < 1} has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form H-2,H-2 (f) = a(2)a(4) - a(3)(2). A non-sharp bound for H-2,H-2 (f ) when f is an element of K(alpha), alpha is an element of [0, 1) consisting of convex functions of order alpha was found by Krishna and Ramreddy (Hankel determinant for starlike and convex functions of order alpha. Tbil Math J. 2012;5:65-76), and later improved by Thomas et al. (Univalent functions: a primer. Berlin: De Gruyter; 2018). In this paper, we give the sharp result. Moreover, we obtain sharp results for H-2,H-2 (f(-1)) for the inverse functions f(-1) when f is an element of K(alpha), and when f is an element of S* (alpha), the class of starlike functions of order alpha. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and f(-1) for the classes S* (alpha), K(alpha), S-beta* and K-beta, where S-beta*, and K-beta are, respectively, the classes of strongly starlike, and strongly convex functions of order beta.