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

Abstract

In recent years, the study of Hankel determinants for various subclasses of normalised univalent functions given by for has produced many interesting results. The main focus of interest has been estimating the second Hankel determinant of the form . A non-sharp bound for when , consisting of convex functions of order α 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 for the inverse functions when , and when , the class of starlike functions of order α. Thus, the results in this paper complete the set of problems for the second Hankel determinants of f and for the classes , , and , where and are, respectively, the classes of strongly starlike, and strongly convex functions of order β.