Isoperimetric inequalities for capacities in the plane (Q1190808)

Let \(d=d_ e(K)\) denote the exterior deficiency of a connected compact set \(K\) in the complex plane with strictly positive area and connected complement \((d_ e(K)=r_ e(K)/r_ 0(G)-1\), \(r_ e(K)\) exterior radius of \(K\), \(r_ 0(K)\) radius of a disk \(K_ 0\) having the same area as \(K\)). It is shown that, for every \(c>1/2\), \[ \hbox{Cap} K\geq (1+c(d^ 2/\ln(1/d)))\hbox{Cap} K_ 0 \] if \(d\) is sufficiently small. Examples of starlike sets are given where the converse inequality holds for every \(c>1\).