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\begin{document}$$\Omega $$\end{document} be an open set in Euclidean space with finite Lebesgue measure |Ω|\documentclass[12pt]{minimal}
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\begin{document}$$\vert \Omega \vert $$\end{document}. We obtain some properties of the set function F:Ω↦R+\documentclass[12pt]{minimal}
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\begin{document}$$F:\Omega \mapsto {\mathbb {R}}^+$$\end{document} defined by F(Ω)=T(Ω)λ1(Ω)|Ω|,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F(\Omega )=\frac{T(\Omega )\lambda _1(\Omega )}{\vert \Omega \vert } , \end{aligned}$$\end{document}where T(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$T(\Omega )$$\end{document} and λ1(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _1(\Omega )$$\end{document} are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound F(Ω)≤1,\documentclass[12pt]{minimal}
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\begin{document}$$F(\Omega )\le 1,$$\end{document} and show that F(Ω)≤1-νmT(Ω)|Ω|-1-2m,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F(\Omega )\le 1- \nu _m T(\Omega )|\Omega |^{-1-\frac{2}{m}}, \end{aligned}$$\end{document}where νm\documentclass[12pt]{minimal}
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\begin{document}$$\nu _m$$\end{document} depends only on m. For any m=2,3,…\documentclass[12pt]{minimal}
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\begin{document}$$m=2,3,\ldots $$\end{document} and ϵ∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon \in (0,1)$$\end{document} we construct an open set Ωϵ⊂Rm\documentclass[12pt]{minimal}
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\begin{document}$$\Omega _{\epsilon }\subset {\mathbb {R}}^m$$\end{document} such that F(Ωϵ)≥1-ϵ\documentclass[12pt]{minimal}
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\begin{document}$$F(\Omega _{\epsilon })\ge 1-\epsilon $$\end{document}.
