Abstract: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.
| Comments: | 7 pages, 1 figure |
| Subjects: | Complex Variables (math.CV) |
| MSC classes: | 51M09, 51M16, 30C20 |
| Cite as: | arXiv:2605.25779 [math.CV] |
| (or arXiv:2605.25779v1 [math.CV] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25779 arXiv-issued DOI via DataCite (pending registration) |
From: Semen Nasyrov Raphailovich [view email]
[v1]
Mon, 25 May 2026 12:27:50 UTC (34 KB)
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