KYUNGPOOK Math. J. 55(2015), 429-438 http://dx.doi.org/10.5666/KMJ.2015.55.2.429 pISSN 1225-6951 eISSN 0454-8124 c ⃝ Kyungpook Mathematical Journal Coefficient Inequality for Transforms of Starlike and Convex Functions with Respect to Symmetric Points Deekonda Vamshee Krishna ∗ and Bollineni Venkateswarlu Department of Mathematics, GIT, GITAM University, Visakhapatnam 530-045, A.P., India e-mail : vamsheekrishna1972@gmail.com and bvlmaths@gmail.com Thoutreddy Ramreddy Department of Mathematics, Kakatiya University, Warangal 506-009, T.S., India e-mail : reddytr2@gmail.com Abstract. The objective of this paper is to obtain sharp upper bound for the second Hankel functional associated with the k th root transform [ f (z k ) ] 1 k of normalized analytic function f (z) when it belongs to the class of starlike and convex functions with respect to symmetric points, defined on the open unit disc in the complex plane, using Toeplitz determinants. 1. Introduction Let A denote the class of all functions f (z) of the form (1.1) f (z)= z + ∞ ∑ n=2 a n z n in the open unit disc E = {z : |z| < 1}. Let S be the subclass of A consisting of univalent functions. For a univalent function in the class A, it is well known that the n th coefficient is bounded by n. The geometric properties of these functions were determined by the study of their coefficient bounds. For example, the bound for the second coefficient of normalized univalent function readily yields the growth and distortion properties for univalent functions. The Hankel determinant of f for * Corresponding Author. Received January 2, 2014; revised June 4, 2014; accepted October 30, 2014. 2010 Mathematics Subject Classification: 30C45, 30C50. Key words and phrases: starlike and convex functions with respect to symmetric points, upper bound, second Hankel functional, positive real function, Toeplitz determinants. 429