Schwarz-Pick inequalities for hyperbolic domains in the extended plane

Let Ω and Π be two hyperbolic simply connected domains in the extended complex plane C̄ = C ∪ {∞}. We derive sharp upper bounds for the modulus of the nth derivative of a holomorphic, resp. meromorphic function f: Ω → Π at a point z0 εΩ. The bounds depend on the densities λΩ and λΠ of the Poincaré metrics and on the hyperbolic distances of the points z0 and f(z0) to the point ∞