  We show, using the Kobayashi and Caratheodory metrics on special holomorphic
disks in the universal Teichmuller space, that a wide class of holomorphic
functionals on the space of univalent functions in the disk is maximized by the
Koebe function or by its root transforms; their extremality is forced by
hyperbolic features. As consequences, this implies the proofs of the famous
Zalcman and Bieberbach conjectures.
