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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2507.14291 |
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| _version_ | 1866918097688985600 |
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| author | Chuaqui, Martin |
| author_facet | Chuaqui, Martin |
| contents | 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14291 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Ahlfors-Weill reflection on convex domains and Nehari quasidisks Chuaqui, Martin Complex Variables 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 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. |
| title | The Ahlfors-Weill reflection on convex domains and Nehari quasidisks |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2507.14291 |