Lower bounds for the half-plane capacity of compact sets and symmetrization

Given a bounded relatively closed subset $E$ of the upper half-plane $H=\{z:\operatorname{Im}z>0\}$, a new representation of the half-plane capacity of $E$ is obtained in terms of the inner radius of the connected component of the set $H\setminus E$ which goes off to infinity. For this capacity, new lower bounds in terms of the capacities of sets obtained by application of a series of geometric transformations of the set $E$, including the Steiner and circular symmetrizations, are established, and its behaviour under linear and radial averaging transformations of families of compact sets $\{E_k\}_{k=1}^n$ is examined.
Bibliography: 10 titles.