Characterization of the Robin function by extremal problems.

In [`Robin function and conformal welding - A new proof of the existence', submitted] a new proof of the existence of the Robin function is given using conformal welding. This proof is essentially based on the simply connected case and a conformal mapping which maps the domain onto a complement of a disk and a system of radial spires. The aim of this note is to show that this proof also has a wider meaning, namely that this mapping is the unique solution of extremal problems, which minimizes the length or the diameter of a subset of the boundary curve. As a consequence we obtain sharp inequalities for the Robin capacity and the logarithmic capacity. [ABSTRACT FROM AUTHOR]