Hyperbolically convex functions and the generalized Fekete–Szegö functional

We describe extremal functions for the generalized Fekete–Szegö functional over the class of hyperbolically convex functions. We apply the Julia variational formula to reduce the problem to mappings onto hyperbolic polygons having no more than two proper sides. In general, this is the best result possible. We show that both one-sided and two-sided maps can be extremal for different sub-classes.