Distortion of the triangular ratio metric under Moebius transforms
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913161517465600 |
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| author | Nasyrov, S. |
| author_facet | Nasyrov, S. |
| contents | Let $\mathbb{U}$ be the unit disk in the complex plane. Denote by $s_\mathbb{U}(x,y)$ the triangular ratio metric in $\mathbb{U}$; for $x\neq y$ the value of $s_\mathbb{U}(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in \mathbb{U}$ to the value $\inf_{z\in \partial \mathbb{U}}(|x-z|+|z-y|)$. In the monograph by P.~Hariri, R.~Klén, and M.~Vuorinen "Conformally invariant metrics and quasiconformal mappings" (2020) the following problem was stated: for every Moebius automorphism of the unit disk, $w=f(z)=\frac{z+a}{1+za}$, $0\le a<1$, and every points $z_1$, $z_2\in \mathbb{U}$ the sharp inequality $s_\mathbb{U}(f(z_1),f(z_2))\le (1+a)s_\mathbb{U}(z_1,z_2)$ holds. We prove that the conjecture is valid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_25779 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distortion of the triangular ratio metric under Moebius transforms Nasyrov, S. Complex Variables 51M09, 51M16, 30C20 Let $\mathbb{U}$ be the unit disk in the complex plane. Denote by $s_\mathbb{U}(x,y)$ the triangular ratio metric in $\mathbb{U}$; for $x\neq y$ the value of $s_\mathbb{U}(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in \mathbb{U}$ to the value $\inf_{z\in \partial \mathbb{U}}(|x-z|+|z-y|)$. In the monograph by P.~Hariri, R.~Klén, and M.~Vuorinen "Conformally invariant metrics and quasiconformal mappings" (2020) the following problem was stated: for every Moebius automorphism of the unit disk, $w=f(z)=\frac{z+a}{1+za}$, $0\le a<1$, and every points $z_1$, $z_2\in \mathbb{U}$ the sharp inequality $s_\mathbb{U}(f(z_1),f(z_2))\le (1+a)s_\mathbb{U}(z_1,z_2)$ holds. We prove that the conjecture is valid. |
| title | Distortion of the triangular ratio metric under Moebius transforms |
| topic | Complex Variables 51M09, 51M16, 30C20 |
| url | https://arxiv.org/abs/2605.25779 |