Gen. Math. Notes, Vol. 21, No. 1, March 2014, pp. xx-xx ISSN 2219-7184; Copyright © ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in Fekete-Szegö Inequality for a New Class and Its Certain Subclasses of Analytic Functions Gurmeet Singh Khalsa College, Patiala E-mail: meetgur111@gmail.com (Received: / Accepted:) Abstract We introduce some classes of analytic functions, its subclasses and obtain sharp upper bounds of the functional | 3 μ 2 2 | for the analytic function  = + =2 ,  <1 belonging to these classes and subclasses. Keywords: Univalent functions, Starlike functions, Close to convex functions and bounded functions. 1 Introduction Let denote the class of functions of the form  = + =2 (1.1) which are analytic in the unit disc ={:|| < 1|}. Let be the class of functions of the form (1.1), which are analytic univalent in . In 1916, Bieber Bach ( [7], [8] ) proved that  2 ൑ 2 for the functions (). In 1923, Löwner [5] proved that  3 ൑ 3 for the functions ().