The highest smoothness of the Green function implies the highest density of a set (Q1880451)

Let \(E\) be a subset of the interval \([0,1]\) and \(\Omega\) be the complement of \(E\) in the extended plane. The author observes that the density of \(E\) at zero is somehow related to the smoothness of the Green function \(g_\Omega\) of \(\Omega\) at the origin. Among other results, he shows that if the Green function \(g_\Omega\) satisfies a Lipschitz condition \(g_\Omega(z)\leq c|z|^{1/2}\), \(z\in C\) at the origin, then the density of \(E\) in a small neighborhood of zero, measured in terms of logarithmic capacity, is arbitrarily close to the density of \([0,1]\) in that neighborhood.