  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.
