Mediterr. J. Math. DOI 10.1007/s00009-016-0733-5 c  Springer International Publishing 2016 Upper Bound of Second Hankel Determinant for Bi-Bazilevi ˘ c Functions S¸ahsene Altınkaya and Sibel Yal¸ cın Abstract. A function is said to be bi-Bazilevi˘ c in the open unit disk U if both the function and its inverse are Bazilevi˘ c there. Making use of the Hankel determinant, in this work, we obtain coefficient expansions for Bi-Bazilevi˘ c functions. Mathematics Subject Classification. Primary 30C45; Secondary 30C50. Keywords. Analytic functions, Bi-Bazilevi˘ c functions, Hankel determi- nant. 1. Introduction Let A denote the class of functions f which are analytic in the open unit disk U = {z : |z| < 1} within the form f (z)= z + ∞  n=2 a n z n . (1) Let S be the subclass of A consisting of the form (1) which are also univalent in U. The Koebe one-quarter theorem [10] states that the image of U under every function f from S contains a disk of radius 1 4 . Thus, every such univalent function has an inverse f −1 which satisfies f −1 (f (z)) = z (z ∈ U ) and f ( f −1 (w) ) = w  |w| <r 0 (f ) ,r 0 (f ) ≥ 1 4  , where f −1 (w)= w − a 2 w 2 + ( 2a 2 2 − a 3 ) w 3 − ( 5a 3 2 − 5a 2 a 3 + a 4 ) w 4 + ··· . A function f ∈ A is said to be bi-univalent in U if both f and f −1 are univalent in U. Let Σ denote the class of bi-univalent functions defined in the