The Ahlfors-Weill reflection on convex domains and Nehari quasidisks

TL;DR

This paper explores geometric properties of convex domains and Nehari quasidisks using the Ahlfors-Weill reflection, revealing how certain reflections relate to domain boundaries and extremal configurations.

Contribution

It interprets a key estimate for convex mappings via Ahlfors-Weill reflection, characterizes Nehari quasidisks geometrically, and distinguishes bounded from unbounded domains.

Findings

The mediatrix of a segment joining a point and its reflection lies outside the domain.

Midpoints of such segments are outside the domain, with extremal cases on the boundary.

A geometric characterization of Nehari quasidisks in terms of reflection distance.

Abstract

The estimate $$\RR\{a_2f\}>-\frac12$$ derived for convex mappings in \cite{FMR}, 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 $w\in\Om$ 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 $\partial\Om$. The normalization $\fd=\frac{f}{1+a_2f}$ to a M\"obius 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.