Verification of Brannan and Clunie's conjecture for certain subclasses of bi-univalent functions

Let sigma denote the class of bi-univalent functions f, that is both f(z) = z+a(2)z(2)+... and its inverse f(-1) are analytic and univalent on the unit disk. We consider the classes of strongly bi-close-to-convex functions of order a and of bi-close-to-convex functions of order beta, which turn out to be subclasses of sigma. We obtain upper bounds for vertical bar a(2)vertical bar and vertical bar a(3)vertical bar for those classes. Moreover, we verify Brannan and Clunie's conjecture vertical bar a(2)vertical bar <= root 2 for some of our classes.. In addition, we obtain the Fekete-Szego relation for these classes.