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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2508.01607 |
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| _version_ | 1866915423496175616 |
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| author | Maji, Bibekananda Naskar, Pritam Sahoo, Swadesh Kumar |
| author_facet | Maji, Bibekananda Naskar, Pritam Sahoo, Swadesh Kumar |
| contents | Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $ζ_D$, and a version of Gehring-Osgood's distance ratio metric $j_D'$ [{\tt J. Analyse Math.} {\bf 36} (1979), 50--74], denoted by $ζ_D'$, are introduced to better understand how quasihyperbolic geometry interacts with bounded uniform domains in $\mathbb{R}^n$. We show that the metric $m_D$, introduced in [{\tt arXiv:2505.10964v2}], is the inner metric of $ζ_D$ and explore their relations to several well-known hyperbolic-type metrics. The paper includes ball inclusion properties of these metrics associated with the metric $m_D$ and other hyperbolic-type metrics. The distortion properties of them are also considered under several important classes of mappings. Furthermore, as an application, we demonstrate that uniform domains can be characterized in terms of metrics $ζ_D$ and $m_D$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01607 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Modified Distance Ratio Metrics via Domain Diameter and their geometric implications Maji, Bibekananda Naskar, Pritam Sahoo, Swadesh Kumar Metric Geometry Primary: 30F45, 30L15, 51K05, Secondary: 30C65, 30L10, 51M10 Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $ζ_D$, and a version of Gehring-Osgood's distance ratio metric $j_D'$ [{\tt J. Analyse Math.} {\bf 36} (1979), 50--74], denoted by $ζ_D'$, are introduced to better understand how quasihyperbolic geometry interacts with bounded uniform domains in $\mathbb{R}^n$. We show that the metric $m_D$, introduced in [{\tt arXiv:2505.10964v2}], is the inner metric of $ζ_D$ and explore their relations to several well-known hyperbolic-type metrics. The paper includes ball inclusion properties of these metrics associated with the metric $m_D$ and other hyperbolic-type metrics. The distortion properties of them are also considered under several important classes of mappings. Furthermore, as an application, we demonstrate that uniform domains can be characterized in terms of metrics $ζ_D$ and $m_D$. |
| title | Modified Distance Ratio Metrics via Domain Diameter and their geometric implications |
| topic | Metric Geometry Primary: 30F45, 30L15, 51K05, Secondary: 30C65, 30L10, 51M10 |
| url | https://arxiv.org/abs/2508.01607 |