Inequalities for the carathéodory and poincaré metrics in open unit balls

We generalize a known inequality relating the Euclidean and hyperbolic metrics in Poincaré's unit ball model of hyperbolic space. Our generalization applies to Schwarz-Pick metrics in the open unit ball of any complex Banach space. We study the case of equality, both in the general case and the case when the open unit ball is homogeneous. For open unit balls of Hilbert spaces, we explicitly determine the case of equality, and we prove a distortion theorem that quantifies the failure of equality when the inequality is strict. An analogous distortion theorem for real hyperbolic space follows readily.