  Let $\Omega\subset\mathbb{R}^n$, $n\geq 2$, be a bounded, open and convex set
and let $f$ be a positive and non-increasing function depending only on the
distance from the boundary of $\Omega$. We consider the $p-$torsional rigidity
associated to $\Omega$ for the Poisson problem with Dirichlet boundary
conditions, denoted by $T_{f,p}(\Omega)$. Firstly, we prove a P\'olya type
lower bound for $T_{f,p}(\Omega)$ in any dimension; then, we consider the
planar case and we provide two quantitative estimates in the case $f\equiv 1 $.
