抄録
Let D∉OAB be a plane domain and let CD(z) be its analytic capacity at z∈D. Let \mathscr{K}D(z) be the curvature of the Carathéodory metric CD(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∉OAB. Several other properties related to the Carathéodory metric are proven.
