Classical Families of Univalent Functions in the Hornich Space

In this paper the simple structure between some convex sets in the Banach space H introduced by Hornich is used to determine the extreme points of the families K(a) of convex functions of order ~ and V(k) of functions with bounded boundary rotation k ~. For close-to-convex functions of order {1,/3 ~ ]0, 1[, a partial result is given. The results for K(~) and V(k) agree with those that hold for the closed convex hulls of the same families with respect to the usual linear structure and the topology of locally uniform convergence. However, in this case, for kE]2,4[ the question of determining the extreme points of ~d V(k) is still open.