Explicit relations for the reduced module and harmonic measure
Using conformal mappings of simply connected domains onto the unit disk or its exterior, the values of some reduced moduli of families of curves separating connected sets are calculated in an elementary way. These reduced moduli are introduced to formulate the theorems, but the specific methods of ``extremal lengths'' don't seem (from this short communication) to have been applied here. -- In the particular case of the harmonic measure \(\omega (z, \alpha)\) of a boundary arc \(\alpha\) at an interior point \(z\) in a simply connected domain \(G\), the curves considered here are those separating \(z\) from \(\alpha\), including the closed curves surrounding \(z\). The elementary relation obtained in this case between \(\omega\) and the reduced modulus \(\mu\) can be interpreted (and proved directly) as the relation between \(\omega\) and the (conformal invariant) ratio of two ``conformal radii'' \(R_ z\) and \(\widetilde R_ z\), namely \(\sin^ 2 (\pi \omega/2) = \widetilde R_ z/R_ z\). The authors compare their elementary relation between \(\omega\) and \(\mu\) with the known relation (in terms of elliptic integrals) between \(\omega\) and the extremal length of the sub-family of curves with end-points on \(\partial G \backslash \alpha\), separating \(z\) from \(\alpha\). For this last relation they quote a paper of the reviewer (1955); but this idea stays already in the Oklahoma College lecture notes ``Conformal mapping'' of L. V. Ahlfors (Summer 1951) (page 91), of which also the reviewer had unfortunately no knowledge at that time.
scientific article; zbMATH DE number 635754