For any bounded and convex set Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document} (N≥2\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2$$\end{document}), with smooth boundary ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document}, and any real number p>1,\documentclass[12pt]{minimal}
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\begin{document}$$p>1,$$\end{document} we denote by up\documentclass[12pt]{minimal}
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\begin{document}$$u_{p}$$\end{document} the p-torsion function on Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, that is the solution of the torsional creep problemΔpu=-1\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{p}u=-1$$\end{document} in Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, u=0\documentclass[12pt]{minimal}
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\begin{document}$$u=0$$\end{document} on ∂Ω\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Omega $$\end{document}, where Δpu:=div(∇up-2∇u)\documentclass[12pt]{minimal}
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\begin{document}$$\Delta _{p}u:=div( \left| \nabla u\right| ^{p-2}\nabla u) $$\end{document} is the p-Laplace operator. Our aim is to investigate the monotonicity with respect to p for the p-torsional rigidity on Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}, defined as TpΩ:=∫Ωupdx\documentclass[12pt]{minimal}
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\begin{document}$$T_{p}\left( \Omega \right) :=\int _{\Omega }u_{p}dx$$\end{document}. More precisely, we show that there exist two constants D1∈12,e-1N+1\documentclass[12pt]{minimal}
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\begin{document}$$D_1\in \left[ \frac{1}{2},e^{\frac{-1}{N+1}}\right] $$\end{document} and D2∈1,N\documentclass[12pt]{minimal}
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\begin{document}$$D_2\in \left[ 1,N\right] $$\end{document} such that for each bounded and convex set Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document} with |∂Ω||Ω|≤D1\documentclass[12pt]{minimal}
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\begin{document}$$\frac{|\partial \Omega |}{|\Omega |}\le D_1$$\end{document} the function p→Tp(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is decreasing on 1,∞\documentclass[12pt]{minimal}
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\begin{document}$$\left( 1,\infty \right) $$\end{document}, while for each bounded and convex set Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document}, with |∂Ω||Ω|≥D2\documentclass[12pt]{minimal}
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\begin{document}$$\frac{|\partial \Omega |}{|\Omega |}\ge D_2$$\end{document}, the function p→Tp(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is increasing on 1,∞\documentclass[12pt]{minimal}
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\begin{document}$$\left( 1,\infty \right) $$\end{document}. Moreover, for each real number s∈(D1,D2)\documentclass[12pt]{minimal}
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\begin{document}$$s\in (D_1,D_2)$$\end{document} there exists a bounded and convex set Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document}, with |∂Ω||Ω|=s\documentclass[12pt]{minimal}
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\begin{document}$$\frac{|\partial \Omega |}{|\Omega |}=s$$\end{document}, such that the function p→Tp(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is not monotone on (1,∞)\documentclass[12pt]{minimal}
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\begin{document}$$(1,\infty )$$\end{document}.
