Inequalities between torsional rigidity and principal eigenvalue of the $p$-Laplacian

We consider the torsional rigidity and the principal eigenvalue related to the $p$-Laplace operator. The goal is to find upper and lower bounds to products of suitable powers of the quantities above in various classes of domains. The limit cases $p=1$ and $p=\mathrm{\infty }$ are also analyzed, which amount to consider the Cheeger constant of a domain and functionals involving the distance function from the boundary.