Transformation of the quasiconformal reflection coefficient and of the Fredholm eigenvalue (Q2016442)

Let \(\gamma\) be a closed Jordan curve in the complex plane. The reflection coefficient \(q \in [0,1]\) of \(\gamma\) is the smallest value for which a \((1+q)/(1-q)\)-quasiconformal reflection exists and the reciprocal Fredholm eigenvalue \(\kappa \in [0,1]\) could be defined as \(\kappa = \sup(|D_G(u) - D_{G^*}|)/(D_G(u) + D_{G^*})\) where \(G\) and \(G^*\) are the interior and exterior domains of \(\gamma\), respectively, and \(D\) refers to the Direchlet integral of \(u\) over the marked domain. The supremum is taken over all functions continuous in \(\overline{\mathbb C}\) and harmonic in \(G \cup G^*\), see [\textit{S. Krushkal}, in: Analysis and Mathematical Physics, Trends in Mathematics, Birkhäuser. 349--368 (2009; Zbl 1297.30011)]. The fundamental Ahlfors inequality says that \(\kappa \leq q\) but for many curves there is equality. The author studies the behavior of these quantities under the square root transformation. He assumes that \(\gamma\) is the image of the unit circle under the hydrodynamically normalised univalent mapping \(f(z) = z + a_0 + a_1/z + ...\) of \(|z| > 1\) and \(f\) is analytic on \(|z| = 1\) with \(f'(z) \neq 0\). The square root transformation \(\sqrt{f(z^2)}\) produces from \(\gamma\) a Jordan curve \(\gamma^*\) centrally symmetric with respect to \(0\). Let \(q^*\) and \(\kappa^*\) denote the aforementioned quantities associated with \(\gamma^*\). The main results are: \(q^* \geq q\) and if \(q= \kappa\) and \(q^* = q\), then \(\kappa^* = \kappa\). The paper contains several interesting examples and a detailed study of the case \(\kappa^* = \kappa\).