The Ahlfors-Weill reflection on convex domains and Nehari quasidisks

2025/07/18 by Chuaqui, Martin
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.14291

Abstract

The estimate \RR\a₂f\gt;-\frac12 derived for convex mappings in \citeFMR, is interpreted here in terms of the Ahlfors-Weill reflection to show that for such domains \Om, the mediatrix of the segment [w, \mR_w] joining a point w∈\Om and its reflection \mR_w lies always outside \Om. In particular, the midpoint of the segment is also outside \Om. We determine the extremal cases when such a midpoint can lie of the boundary ∂\Om. The normalization \fd=(f)/(1+a₂f) to a Möbius equivalent mapping with vanishing second coefficient leads to important distinctions between bounded an unbounded domains. We finally derive a geometric characterization of Nehari quasidisks in terms of the distance to the boundary of the Ahlfors-Weill reflection.

Related

• Ahlfors-Weill Extensions for a Class of Minimal Surfaces
• Ahlfors reflection theorem for p-morphisms
• Distortion of quasiconformal mappings with identity boundary values
• Uniformly perfect domains and convex hulls
• Quasiconformal reflection with respect to the boundary of an isosceles trapezoid