The highest smoothness of the Green function implies the highest density of a set

Abstract

We investigate local properties of the Green function of the complement of a compact set Eυ[0,1] with respect to the extended complex plane. We demonstrate, that if the Green function satisfies the 1/2-Hölder condition locally at the origin, then the density of E at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]..