A modulus inequality for condensers and conformal invariants of smooth Jordan domains (Q1271300)

For a smooth Jordan domain \(\Omega\) in the plane it is shown that \(M(\Omega)< R(\Omega)+1\) except for a disk or a half plane. Here \(M(\Omega)\) and \(R(\Omega)\) are the quasiextremal distance constant and the quasiconformal reflection constant of \(\Omega\), respectively. The proof employs a special modulus inequality, whose proof is based on the discrete version for the conformal capacity [\textit{T. Bagby}, J. Math. Mech. 17, 315-329 (1967; Zbl 0163.35204)].