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.
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