| Summary: | 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].
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