Lawrence Zalcman's conjecture states that if is analytic and univalent in the unit disk , then for each , with equality only for the Koebe function and its rotations. This conjecture remains open although it has been verified for a few geometric subclasses of the class of univalent analytic functions. In this paper, we consider this problem for the family of normalized functions f analytic and univalent in the unit disk |z| < 1 satisfying the condition Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in |z| < 1. A few other ...