Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 22 (2021), No. 1, pp. 375–382 DOI: 10.18514/MMN.2021.1807 , THIRD HANKEL DETERMINANT FOR CERTAIN SUBCLASS OF p-VALENT ANALYTIC FUNCTIONS D. VAMSHEE KRISHNA This paper is dedicated to Professor T. RAMREDDY on his 72 nd birthday. Received 15 October, 2015 Abstract. The objective of this paper is to obtain an upper bound to the third Hankel determinant for certain subclass of p-valent functions, using Toeplitz determinants. 2010 Mathematics Subject Classification: 30C45; 30C50 Keywords: p-valent analytic function, upper bound, Hankel determinant, positive real function, Toeplitz determinants 1. I NTRODUCTION Let A p denote the class of functions f of the form f (z)= z p + a p+1 z p+1 + ··· , (1.1) in the open unit disc E = {z : |z| < 1} with p ∈ N = {1, 2, 3, ...}. Let S be the subclass of A 1 = A, consisting of univalent functions. In 1985, Louis de Branges de Bourcia proved the Bieberbach conjecture, i.e., for a univalent function its n th - coefficient is bounded by n (see [3]). The bounds for the coefficients of these functions give information about their geometric properties. In particular, the growth and distortion properties of a normalized univalent function are determined by the bound of its second coefficient. The Hankel determinant of f for q ≥ 1 and n ≥ 1 (when p = 1) was defined by Pommerenke [10] as follows and has been extensively studied. H q (n)= a n a n+1 ··· a n+q−1 a n+1 a n+2 ··· a n+q . . . . . . . . . . . . a n+q−1 a n+q ··· a n+2q−2 . (1.2) One can easily observe that the Fekete-Szeg˝ o functional is H 2 (1)= a 3 − a 2 2 . Fekete and Szeg˝ o then further generalized the estimate |a 3 − μa 2 2 | with μ real and f ∈ S. Fur- ther, sharp upper bounds for the functional H 2 (2)= a 2 a 3 a 3 a 4 = a 2 a 4 − a 2 3 , the Hankel determinant in the case of q = 2 and n = 2, known as the second Hankel determinant © 2021 Miskolc University Press