On a Pólya’s inequality for planar convex sets

Ftouhi, Ilias

Comptes Rendus. Mathématique, Tome 360 (2022) no. G3, pp. 241-246.

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

$$\frac{{\lambda }_1\left(\Omega \right)T\left(\Omega \right)}{\left|\Omega \right|}\le \frac{{\pi }^2}{12}·{\left(1+\sqrt{\pi }\frac{r\left(\Omega \right)}{\sqrt{\left|\Omega \right|}}\right)}^2,$$

where ${\lambda }_1$, $T$, $r$ and $\left|·\right|$ 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

$$\frac{{\lambda }_1\left(\Omega \right)T\left(\Omega \right)}{\left|\Omega \right|}\le \frac{{\pi }^2}{12}{\left(1+\frac{2\sqrt{2(6+{\pi }^2)}-{\pi }^2}{4+{\pi }^2}\right)}^2\approx 0.996613\cdots$$

which improves Polyá’s inequality $\frac{{\lambda }_1\left(\Omega \right)T\left(\Omega \right)}{\left|\Omega \right|}<1$ and is slightly better than the one provided in [3].

The novel ingredient of the proof is the sharp inequality

$${\lambda }_1\left(\Omega \right)\le \frac{{\pi }^2}{4}·{\left(\frac{1}{r\left(\Omega \right)}+\sqrt{\frac{\pi }{\left|\Omega \right|}}\right)}^2,$$

recently proved in [8].

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