On a Pólya's inequality for planar convex sets

In this short note, we prove that for every bounded, planar and convex set Ω, one has (equation presented) where λ1, T , r and j j 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 (equation presented) which improves Polyá's inequality λ1(Ω)T (Ω) jΩj Ç 1 and is slightly better than the one provided in [3]. The novel ingredient of the proof is the sharp inequality (equation presented) recently proved in [8].