On a Pólya’s inequality for planar convex sets

Ilias Ftouhi

doi:10.5802/crmath.292

Théorie spectrale

On a Pólya’s inequality for planar convex sets

Ilias Ftouhi ¹

¹ Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair of Applied Analysis (Alexander von Humboldt Professorship), Cauerstr. 11, 91058 Erlangen, Germany

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 241-246.

Résumé

In this short note, we prove that for every bounded, planar and convex set $Ω$ , one has

\frac{λ_{1} (Ω) T (Ω)}{| Ω |} \leq \frac{π^{2}}{12} \cdot {(1 + \sqrt{π} \frac{r (Ω)}{\sqrt{| Ω |}})}^{2},

where $λ_{1}$ , $T$ , $r$ 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{λ_{1} (Ω) T (Ω)}{| Ω |} \leq \frac{π^{2}}{12} {(1 + \frac{2 \sqrt{2 (6 + π^{2})} - π^{2}}{4 + π^{2}})}^{2} \approx 0.996613 \dots

which improves Polyá’s inequality $\frac{λ_{1} (Ω) T (Ω)}{| Ω |} < 1$ and is slightly better than the one provided in [3].

The novel ingredient of the proof is the sharp inequality

λ_{1} (Ω) \leq \frac{π^{2}}{4} \cdot {(\frac{1}{r (Ω)} + \sqrt{\frac{π}{| Ω |}})}^{2},

recently proved in [8].

Reçu le : 2021-08-28
Révisé le : 2021-10-26
Accepté le : 2021-10-26
Publié le : 2022-03-31

DOI : 10.5802/crmath.292

Affiliations des auteurs :

Ilias Ftouhi ¹

¹ Friedrich-Alexander-Universität Erlangen-Nürnberg, Department of Mathematics, Chair of Applied Analysis (Alexander von Humboldt Professorship), Cauerstr. 11, 91058 Erlangen, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Ilias Ftouhi},
     title = {On a {P\'olya{\textquoteright}s} inequality for planar convex sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {241--246},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.292},
     language = {en},
}

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Ilias Ftouhi. On a Pólya’s inequality for planar convex sets. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 241-246. doi : 10.5802/crmath.292. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.292/

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