Spectral theory
On a Pólya’s inequality for planar convex sets
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 241-246.

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

 $\frac{{\lambda }_{1}\left(\Omega \right)T\left(\Omega \right)}{|\Omega |}\le \frac{{\pi }^{2}}{12}·{\left(1+\sqrt{\pi }\frac{r\left(\Omega \right)}{\sqrt{|\Omega |}}\right)}^{2},$

where ${\lambda }_{1}$, $T$, $r$ and $|\phantom{\rule{0.166667em}{0ex}}·\phantom{\rule{0.166667em}{0ex}}|$ 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

 $\frac{{\lambda }_{1}\left(\Omega \right)T\left(\Omega \right)}{|\Omega |}\le \frac{{\pi }^{2}}{12}{\left(1+\frac{2\sqrt{2\left(6+{\pi }^{2}\right)}-{\pi }^{2}}{4+{\pi }^{2}}\right)}^{2}\approx 0.996613\cdots$

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

The novel ingredient of the proof is the sharp inequality

 ${\lambda }_{1}\left(\Omega \right)\le \frac{{\pi }^{2}}{4}·{\left(\frac{1}{r\left(\Omega \right)}+\sqrt{\frac{\pi }{|\Omega |}}\right)}^{2},$

recently proved in [8].

Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.292

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
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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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