Spectral theory
On a Pólya’s inequality for planar convex sets
Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 241-246.

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

λ 1 (Ω)T(Ω) |Ω|π 2 12·1+πr(Ω) |Ω| 2 ,

where λ 1 , T, r and |·| 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

λ 1 (Ω)T(Ω) |Ω|π 2 121+22(6+π 2 )-π 2 4+π 2 2 0.996613

which improves Polyá’s inequality 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)π 2 4·1 r(Ω)+π |Ω| 2 ,

recently proved in [8].

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.292
Ftouhi, Ilias 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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Ftouhi, Ilias. On a Pólya’s inequality for planar convex sets. Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 241-246. doi : 10.5802/crmath.292. http://www.numdam.org/articles/10.5802/crmath.292/

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