A bound for reflections across Jordan curves. (English) Zbl 1061.30041

Let \(f\) be a conformal map of the upper half-plane \(U\! =\{z:\text{Re}\, z\!>\! 0\}\) onto the interior domain \(D=D(L)\) of a given quasicircle \(L\). How can this map be characterized? L. Ahlfors conjectured that it can be characterized by analytic properties of the invariant (logarithmic derivative) \(b_f=f''/f'\). The goal of this paper is to prove the following theorem which concerns the above conjecture and relies on the properties of the logarithmic derivative. Let a function \(f\) map conformally the upper half-plane \(U\) into \(\mathbb C\), and let the equation \[ w''(\zeta)=tb_f(\zeta)w'(\zeta),\quad \zeta\in U,\tag{1} \] have univalent solutions on \(U\) for all \(t\in [0,t_0]\), \(t_0>1\). Then the image \(f(U)\) is a quasidisk, and the reflection coefficient of its boundary \(L=f(\widehat{\mathbb R})=\partial f(U)\) satisfies \[ q_L\leqslant {1}/{t_0}.\tag{2} \] The bound given by (2) cannot be improved in the general case. The equality in (2) is attained by any quasicircle which contains two \(C^{1+\epsilon}\) smooth subarcs (\(\epsilon>0\)) with the interior intersection angle \(\alpha\pi\), where \(\alpha=1-1/t_0\) (under the univalence assumption for the logarithmic derivative \(b_f\)). In this case, \(q_L = {1}/{t_0}.\) The exact bound for the reflection coefficient \(q_L\) follows from (2) by choosing a maximal value to admitting the indicated univalence property for all \(t\in[0,t_0]\). The corresponding solution \(w_{t_0}\) of (1) for this value is also univalent on \(U\) (by the properties of holomorphic functions), but the domain \(w_{t_0}(U)\) is not a quasidisk.


30F60 Teichmüller theory for Riemann surfaces
30C62 Quasiconformal mappings in the complex plane
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