On a conjecture of a P\'olya functional for triangles and rectangles

We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by P\'olya and Szeg\H{o} in 1951 who showed that it is bounded above by 1 for all domains. It has been conjectured that within the class of bounded convex planar domains the functional is bounded below by $\pi^{2}/24$ and above by $\pi^{2}/12$ and that these bounds are sharp. Remarkably, the conjecture remains open even within the class of triangles. The purpose of this paper is to prove the conjecture in this case. The conjecture is also proved for rectangles where a stronger monotonicity property is verified. Finally, the upper bound also holds for tangential quadrilateral.