On the theorem of Hayman and Wu (Q2758974)

Let \(f:\Omega\to D\) be a conformal map of a domain \(\Omega\) onto the unit disc \(D\). If \(L\) is a line or a circle, then the Hayman-Wu theorem [\textit{W. K. Hayman}, \textit{J.-M. Wu}, Comment. Math. Helv. 56, 366-403 (1981; Zbl 0473.30012)] asserts that there is a universal constant \(C\) such that the length \((f(L))\leq C\). Øyma [\textit{K. Øyma}, Proc. Am. Math. Soc. 115, No. 3, 687-689 (1992; Zbl 0752.30014)] provided a short proof for this result and he also showed that \(C\leq 4\pi\). He also conjectured that the best value of \(C\) is \(\pi^2\). The author shows that \(C< \pi^2\). The method is the Øyma method combined with a careful analysis of the geodesic curvature [\textit{J. Fernandez}, \textit{A. Granados}, St. Petersburg Math. J. 9, No. 3, 615-637 (1998; Zbl 0930.53013)].