Estimates for conformal capacity

Let a,b,c,d be distinct points on \overline \bf R ⁿ . By p we denote the minimal conformal capacity of all rings (E,F) with a,b ∈ E and c,d∈ F . For n=2 , we use explicit expressions of p in terms of complete elliptic integrals to prove a sharp inequality that connects p and the conformal capacity of Teichmüller's ring. We also show, by a concrete example, how we can use techniques involving polarization and hyperbolic geometry to prove estimates for the conformal capacity of rings. November 16, 1998. Date revised: March 2, 1999. Date accepted: May 7, 1999.