On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue

Let $\mathrm{\Omega }$ be an open set in Euclidean space with finite Lebesgue measure $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }$. We obtain some properties of the set function $F:\Omega\mapsto \R^+$definedby$$F(\mathrm{\Omega })=\frac{T(\mathrm{\Omega }){\lambda }_1(\mathrm{\Omega })}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }},$$where$T(\Omega)$and$\lambda_1(\Omega)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical P'olya bound $F(\mathrm{\Omega })\le 1,$ and show that $$F(\mathrm{\Omega })\le 1-{\nu }_{m}T(\mathrm{\Omega })\mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{-1-\frac{2}{m}},$$ where ${\nu }_m$ depends only on $m$. For any $m=2,3,\dots$ and $ϵ\in (0,1)$ we construct an open set ${\mathrm{\Omega }}_{ϵ}\subset {\mathbb{R}}^m$ such that $F({\mathrm{\Omega }}_{ϵ})\ge 1-ϵ$.