A Proof of a Conjecture on a Pólya Functional

Let $λ_1(Ω)$ and $T(Ω)$ denote the first Dirichlet eigenvalue and torsional rigidity of a bounded convex domain $Ω\subset\mathbb{R}^2$, and let $M=\max_Ωu$, where $u$ is the torsion function. We prove the sharp inequalities $π^2/24<λ_1(Ω)T(Ω)/|Ω|<π^2/12$. This proves the two-dimensional case of Conj