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

Let $\Omega$ and $\Pi$ 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 $A(\Omega,\Pi)=\{f\colon\Omega\to\Pi\}$ of functions locally holomorphic in $\Omega$; for the domain $\Omega$ we consider the exterior of the unit disk and the upper half-plane. The obtained results generalize the well-known theorems of Szasz and Ruscheweyh about the exact estimates of derivatives of analytic functions defined on the disk $|z|<1$.