## 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.

### MSC:

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