Polarization and Circular Truncation of a Domain

Abstract

The paper considers the difference between the reduced module m(D) of a simply connected domain D with respect to the point z = 0 and the reduced module m(Dr) of its radial circular truncation, where Dr is the connected component of the set D ∩ {|z| < r} containing the point z = 0. It is proved that this difference does not decrease under polarization and circular symmetrization.