The Carathéodory metric in plane domains

Abstract

Let D∉O_AB be a plane domain and let C_D(z) be its analytic capacity at z∈D. Let \mathscr{K}_D(z) be the curvature of the Carathéodory metric C_D(z)|dz|. We show that \mathscr{K}_D(z)<−4 the Ahlfors function of D with respect to z has a zero other than z. For finite D, \mathscr{K}_D(z){≤}−4 and equality holds if and only if D is simply connected. As a corollary we obtain a result proved first by Suita, namely, that \mathscr{K}_D(z){≤}−4 if D∉O_AB. Several other properties related to the Carathéodory metric are proven.