Estimates of the hyperbolic radius gradient and Schwarz–Pick inequalities for the eccentric annulus

Let $\Omega$ and $\Pi$ be hyperbolic domains in the complex plane $\mathbb C$. By $A(\Omega,\Pi)$ we shall designate the class of functions $f$ which are holomorphic or meromorphic in $\Omega$ and such that $f(\Omega)\subset\Pi$. Estimates of the higher derivatives $|f^{(n)}(z)|$ of the analytic functions from the class $A(\Omega,\Pi)$ with the punishing factor $C_n(\Omega,\Pi)$ is one of the main problems of geometric theory of functions. These estimates are commonly referred to as Schwarz–Pick inequalities. Many results concerning this problem have been obtained for simply connected domains. Therefore, the research interest in such problems for finitely connected domains is natural. As known, the constant $C_2(\Omega,\Pi)$ for any pairs of hyperbolic domains depends only on the hyperbolic radius gradient of the corresponding domains. The main result of this paper is estimates of the hyperbolic radius gradient and the punishing factor in the Schwarz–Pick inequality for the eccentric annulus. We also consider the extreme case – the randomly punctured circle.