Generalization of theorems of Szász and Ruscheweyh on exact bounds for derivatives of analytic functions

Let Ω and Π be two domains in the extended complex plane equipped by the Poincaré metric. In this paper we obtain analogs of Schwarz-Pick type inequalities in the class A(Ω, gH) = {f: Ω → Π} of functions locally holomorphic in Ω; for the domain Ω we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szász and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk |z| < 1.