Treffer: Univalent harmonic mappings and a conjecture of J. C. C. Nitsche

Let \(f\) be a univalent harmonic mapping of \(A(r) = \{z: r < |z|< 1\}\) onto \(A(R) =\{z: R < |z|< 1\}\). J. C. C. Nitsche has shown that for each \(r\) there is an upper bound \(\kappa(r) < 1\) for \(R\). Nitsche conjectured \(\kappa(r) \leq 2r/(1+r^2)\). Apparently, no actual upper bound for \(\kappa(r)\) is known. In this report, the author sketches the proof of the theorem that \(\kappa(r) < s\) where \(B(s)\) is Grötzsch's ring domain conformally equivalent to \(A(r)\). (This is the unit disk less the line segment \(\{x: 0 \leq x \leq s\}\).) The details are in a preprint which should appear soon.