Comparison of hyperbolic metric and triangular ratio metric in a square
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917529614548992 |
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| author | Kushaeva, A. Nasyrov, S. |
| author_facet | Kushaeva, A. Nasyrov, S. |
| contents | Let $K$ be a square in the plane and $ρ_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in K$ to the value $\inf_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $þ(ρ_K(x,y)/2)$ to $s_K(x,y)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_04733 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Comparison of hyperbolic metric and triangular ratio metric in a square Kushaeva, A. Nasyrov, S. Complex Variables 30C20, 51M09, 51M16 Let $K$ be a square in the plane and $ρ_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in K$ to the value $\inf_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $þ(ρ_K(x,y)/2)$ to $s_K(x,y)$. |
| title | Comparison of hyperbolic metric and triangular ratio metric in a square |
| topic | Complex Variables 30C20, 51M09, 51M16 |
| url | https://arxiv.org/abs/2602.04733 |