On a Pólya’s inequality for planar convex sets
In this short note, we prove that for every bounded, planar and convex set $\Omega $, one has \[ \frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}\le \frac{\pi ^2}{12}\cdot \left(1+\sqrt{\pi }\frac{r(\Omega )}{\sqrt{|\Omega |}}\right)^2, \] where $\lambda _1$, $T$, $r$ and $|{\,\cdot \,}|$ are the fir...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-03-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.292/ |
| _version_ | 1797651492933468160 |
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| author | Ftouhi, Ilias |
| author_facet | Ftouhi, Ilias |
| author_sort | Ftouhi, Ilias |
| collection | DOAJ |
| description | In this short note, we prove that for every bounded, planar and convex set $\Omega $, one has
\[ \frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}\le \frac{\pi ^2}{12}\cdot \left(1+\sqrt{\pi }\frac{r(\Omega )}{\sqrt{|\Omega |}}\right)^2, \]
where $\lambda _1$, $T$, $r$ and $|{\,\cdot \,}|$ are the first Dirichlet eigenvalue, the torsion, the inradius and the volume. The inequality is sharp as the equality asymptotically holds for any family of thin collapsing rectangles.As a byproduct, we obtain the following bound for planar convex sets
\[ \frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}\le \frac{\pi ^2}{12}\left(1+\frac{2\sqrt{2(6+\pi ^2)}-\pi ^2}{4+\pi ^2}\right)^2\approx 0.996613\dots \]
which improves Polyá’s inequality $\frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}<1$ and is slightly better than the one provided in [3].The novel ingredient of the proof is the sharp inequality
\[ \lambda _1(\Omega )\le \frac{\pi ^2}{4}\cdot \left(\frac{1}{r(\Omega )}+\sqrt{\frac{\pi }{|\Omega |}}\right)^2, \]
recently proved in [8]. |
| first_indexed | 2024-03-11T16:16:35Z |
| format | Article |
| id | doaj.art-bfeeb284e218418d91190a6d53f0293f |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:35Z |
| publishDate | 2022-03-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-bfeeb284e218418d91190a6d53f0293f2023-10-24T14:19:47ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-03-01360G324124610.5802/crmath.29210.5802/crmath.292On a Pólya’s inequality for planar convex setsFtouhi, Ilias0Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair of Applied Analysis (Alexander von Humboldt Professorship), Cauerstr. 11, 91058 Erlangen, GermanyIn this short note, we prove that for every bounded, planar and convex set $\Omega $, one has \[ \frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}\le \frac{\pi ^2}{12}\cdot \left(1+\sqrt{\pi }\frac{r(\Omega )}{\sqrt{|\Omega |}}\right)^2, \] where $\lambda _1$, $T$, $r$ and $|{\,\cdot \,}|$ are the first Dirichlet eigenvalue, the torsion, the inradius and the volume. The inequality is sharp as the equality asymptotically holds for any family of thin collapsing rectangles.As a byproduct, we obtain the following bound for planar convex sets \[ \frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}\le \frac{\pi ^2}{12}\left(1+\frac{2\sqrt{2(6+\pi ^2)}-\pi ^2}{4+\pi ^2}\right)^2\approx 0.996613\dots \] which improves Polyá’s inequality $\frac{\lambda _1(\Omega )T(\Omega )}{|\Omega |}<1$ and is slightly better than the one provided in [3].The novel ingredient of the proof is the sharp inequality \[ \lambda _1(\Omega )\le \frac{\pi ^2}{4}\cdot \left(\frac{1}{r(\Omega )}+\sqrt{\frac{\pi }{|\Omega |}}\right)^2, \] recently proved in [8].https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.292/ |
| spellingShingle | Ftouhi, Ilias On a Pólya’s inequality for planar convex sets Comptes Rendus. Mathématique |
| title | On a Pólya’s inequality for planar convex sets |
| title_full | On a Pólya’s inequality for planar convex sets |
| title_fullStr | On a Pólya’s inequality for planar convex sets |
| title_full_unstemmed | On a Pólya’s inequality for planar convex sets |
| title_short | On a Pólya’s inequality for planar convex sets |
| title_sort | on a polya s inequality for planar convex sets |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.292/ |
| work_keys_str_mv | AT ftouhiilias onapolyasinequalityforplanarconvexsets |