Two-Point Distortion Theorems for Spherically Convex Functions

Abstract

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 ∈ D sharp lower bounds on dP(f(a), f(b)), the spherical distance between f(a) and f(b), in terms of dD(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 twopoint distortion theorems are equivalent to sharp two-point comparison theorems between hyperbolic and spherical geometry on a spherically convex region Ω on P. Each of these two-point comparison theorems characterize spherically convex regions.