Green energy and extremal decompositions

We give two precise estimates for the Green energy of a discrete charge, concentrated in an even number of points on the circle, with respect to the concentric ring. The lower estimate for the Green energy is attained for the points with a nonstandard symmetry. The well-known Pólya-Schur inequality for the logarithmic energy is a special case of this estimate. The proof is based on the application of dissymmetrization and an asymptotic formula for the conformal capacity of a generalized condenser in the case when some of its plates contract to given points. The upper bound is established for a charge that takes values of opposite signs. Its proof reduces to solving a problem on the so-called extremal decomposition of a circular ring with free poles on a circle.