A Sharp Bound On The Schwarzian Derivatives Of Hyperbolically Convex Functions

We complete the determination of how far convex maps can distort discs in each of the three classical geometries. The euclidean case was settled by Nehari in 1976, and the spherical case by Mej\'ia and Pommerenke in 2000. We find sharp bounds on the Schwarzian derivatives of hyperbolically convex functions and thus complete the hyperbolic case. This problem was first posed by Ma and Minda in a series of papers in the 1980's. Mej\'ia and Pommerenke then produced partial results and a conjecture as to the extremal function. Their function maps onto a domain bounded by two proper geodesic sides, a "hyperbolic strip." Applying a generalization of the Julia variation and a critical Step Down Lemma, we show that there is an extremal function mapping onto a domain with at most two geodesic sides. We then verify using special function theory that among the remaining candidates, Mej\'ia and Pommerenke's two-sided function is in fact extremal. This correlates nicely with the euclidean and spher