Distortion of the hyperbolic Robin capacity under conformal mapping and extremal configurations

This paper is connected with recent results of Duren and Pfaltzgraff (J. Anal. Math., 78, 205–218 (1999)). We consider the problem on the distortion of the hyperbolic Robin capacity $\delta_h(A,\Omega)$ of the boundary set $A\subset\partial\Omega$ under a conformal mapping of a domain $\Omega\subset U$ into the unit disk $U$. It is shown that, for sets consisting of a finite number of boundary arcs or complete boundary components, the inequality
\begin{equation} \operatorname{cap}_hf(A)\ge\delta_h(A,\Omega) \tag{1} \end{equation}
is sharp in the class of conformal mappings $f\colon\Omega\to U$ such that $f(\partial U)=\partial U$. Here $\operatorname{cap}_hf(A)$ is the hyperbolic capacity of a compact set $f(A)\subset U$. We give some examples demonstrating properties of functions which realize the case of equality in relation (1).