Extremal problems on the class of convex functions of order −1/2

Lawrence Zalcman’s conjecture states that if f ( z ) = z + ∑ n = 2 ∞ a n z n is analytic and univalent in the unit disk | z | < 1 , then | a n 2 - a 2 n - 1 | ≤ ( n - 1 ) 2 , for each n ≥ 2 , with equality only for the Koebe function k ( z ) = z / ( 1 - z ) 2 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 Re 1 + z f ′ ′ ( z ) f ′ ( z ) > - 1 2 for | z | < 1 . Functions satisfying this condition are known to be convex in some direction (and hence close-to-convex and univalent) in | z |