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).