APPLICATIONS OF HYPERBOLIC CONVEXITY TO EUCLIDEAN AND SPHERICAL CONVEXITY DAVID MINDA Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, USA 1. Introduction In [8] a reflection principle for the hyperbolic metric was established. One application of this reflection principle was a sufficient condition for hyperbolic convexity. In particular, if a hyperbolic plane region ~ is starlike with respect to some point a in the closure of f2 and A is any disk with center a, then f2 A A is hyperbolically convex as a subset of fL In particular, if s is a euclidean convex region in the complex plane C, then f2 f~ A is hyperbolically convex for any disk A with center a in el(f2). This latter result is sharp: if f2 is a half-plane and A is a disk which does not have its center in el(f2), then ~ f~ A is not hyperbolically convex. These results on hyperbolic convexity lead to applications for regions that are convex in either euclidean or spherical geometry. For example, if f~ is a convex region in C and y is a hyperbolic geodesic in f2, then the center of any circle of curvature for y cannot lie in ~. Other results of this type are known. J~rgensen [5] noted that for a general hyperbolic region in C the circle of curvature must intersect the boundary. Osgood [10] observed that a result of Netanyahu [9] refined this for simply connected regions to the distance between z0 and the center of the circle of curvature for y at z0 must be at least ]8a(zo), where ~(z0) denotes the euclidean distance from z0 to af2. Our result on the center of the circle of curvature leads to several sharp distortion theorems for the hyperbolic metric which refine known facts for simply connected regions. For instance, I V~(z)l < l/8n(z) for a convex region ~, where 2~ denotes the density of the hyperbolic metric on f2, and equality holds if and only if f~ is a half-plane. This has a consequence in univalent function theory. If fEK, the class of normalized convex univalent functions in the unit disk, then ~/~D)(0) If"(0) [ < 1 with equality if and only if f (z) = z/( 1 - e~~ for some 0 E R. For the full class S of normalized univalent functions Netanyahu [9] demonstrated that the analogous sharp upper bound is ~ and that the Koebe function is not extremal. Finally, we consider regions on the Riemann sphere P which are convex in spherical geometry and obtain analogs of all of our results about euclidean convex regions. These lead to an application for the class Ks(a) of all univalent functions 90