On a Polya functional involving torsional rigidity and first Dirichlet Laplacian eigenvalue

An inequality proved by Pólya establishes a bound from above for the functional obtained as product between the torsional rigidity and the first Dirichlet eigenvalue for the laplacian in un domain with fixed measure. We will discuss the sharpness of the bound and the possibility to obtain upper and lower bounds in the class of convex domains. This is a joint work with M. vand den Berg, C. Nitsch and C. Trombetti.