J ournal of Classical Analysis Volume 2, Number 2 (2013), 167–181 doi:10.7153/jca-02-14 COEFFICIENT ESTIMATES FOR SOME FAMILIES OF BI–BAZILEVI ˇ C FUNCTIONS OF THE MA–MINDA TYPE INVOLVING THE HOHLOV OPERATOR H. M. SRIVASTAVA, G. MURUGUSUNDARAMOORTHY AND K. VIJAYA Abstract. In this paper, we introduce and investigate a new subclass of the function class Σ of bi- univalent functions of the Bazilevi˘ c type defined in the open unit disk, which are associated with the Hohlov operator and satisfy some subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients |a 2 | and |a 3 | for functions in the new subclass introduced here. Several (known or new) consequences of the results are also pointed out. 1. Introduction, definitions and preliminaries Let A denote the class of functions f (z) of the form: f (z)= z + ∞ ∑ n=2 a n z n , (1.1) which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1}. Further, by S we shall denote the class of all functions f (z) in A which are univalent in U and indeed normalized by f (0)= f ′ (0) − 1 = 0. Some of the important and well-investigated subclasses of the univalent function class S include (for example) the class S ∗ (α )(0 ≦ α < 1) of starlike functions of order α in U and the class K (α )(0 ≦ α < 1) of convex functions of order α in U. It is well known that every function f ∈ S has an inverse f −1 , defined by f −1 ( f (z) ) = z (z ∈ U) and f ( f −1 (w) ) = w  |w| < r 0 ( f ); r 0 ( f ) ≧ 1 4  , Mathematics subject classification (2010): Primary 30C45; Secondary 30C50. Keywords and phrases: Analytic functions, univalent functions, bi-univalent functions, Bazilevi˘ c type functions, bi-starlike and bi-convex functions, Hohlov operator, Gaussian hypergeometric function, Taylor- Maclaurin coefficients, Dziok-Srivastava operator, general fractional calculus operator, Srivastava-Wright operator, Carlson-Shaffer operator, Ruscheweyh derivative operator, Bernardi-Libera-Livingston operator. c  , Zagreb Paper JCA-02-14 167