On certain convex sets in the space of locally schlicht functions

Abstract:

Let $H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $|z| < 1$ as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If $M \subset H$ there is a closed nonempty convex set in the $\alpha \beta$-plane such that for $(\alpha ,\beta )$ in this set ${\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C$ (in K) whenever f, $g \in M$. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.