A generalization of the Polia–Szego and Makai inequalities for torsional rigidity

We prove generalizations of the classical inequalities of Polia — Szegо and Makai about torsional rigidity of convex domains. The main idea of the proof is to apply an exact isoperimetric inequality of for Euclidean moments of a domain. This inequality has a wide class of extremal regions and is of independent interest.