Two-point distortion theorems for spherically convex functions

One-parameter families of sharp two-point distortion theorems are established for spherically convex functions f, that is, meromorphic univalent functions f defined on the unit disk D such that f(D) is a spherically convex subset of the Riemann sphere P. These theorems provide for a, b is an element of D sharp lower bounds on dp(f(a), f(b)), the spherical distance between f(a) and f(b), in terms of d(D) (a, b), the hyperbolic distance between a and b, and the quantities (1 - \a\(2))f(#)(a),(1- \b\(2))f(#)(b), where f(#) = \f'\/(1 + \f\(2)) is the spherical derivative. The weakest lower bound obtained is an invariant form of a known growth theorem for spherically convex functions. Each of the two-point distortion theorems is necessary and sufficient for spherical convexity. These two-point distortion theorems are equivalent to sharp two-point comparison theorems between hyperbolic and spherical geometry on a spherically convex region Omega on P. Each of these two-point comparison theorems characterize spherically convex regions.