Basic Schwarz-Pick type inequalities

Let Ω ⊂ ℂ̄ and Π ⊂ ℂ̄ be two domains equipped by the Poincaré metric. We are concerned with the set A(ω, Π) = {f: ω → Π of functions locally holomorphic or meromorphic in Ω and, in general, multivalued. Let λΩ (z), z ∈ Ω, and λΠ(w), w ∈ Π, denote the density of the Poincaré metric at z∈ Ω and w ∈ Π, respectively. © 2009 Birkhäuser Verlag AG