On functionals involving the torsional rigidity related to some classes of nonlinear operators

In this paper we study optimal estimates for two functionals involving the anisotropic $p$-torsional rigidity $T_p(\Omega)$, $1<p<+\infty$. More precisely, we study $\Phi(\Omega)=\frac{T_p(\Omega)}{|\Omega|M(\Omega)}$ and $\Psi(\Omega)=\frac{T_p(\Omega)}{|\Omega|[R_{F}(\Omega)]^{\frac{p}{p-1}}}$, where $M(\Omega)$ is the maximum of the torsion function $u_{\Omega}$ and $R_F(\Omega)$ is the anisotropic inradius of $\Omega$.