Some inequalities involving perimeter and torsional rigidity

We consider shape functionals of the form $F_q(Omega)=P(Omega)T^q(Omega)$ on the class of open sets of prescribed Lebesgue measure. Here $q>0$ is fixed, $P(Omega)$ denotes the perimeter of $Omega$ and $T(Omega)$ is the torsional rigidity of $Omega$. The minimization and maximization of $F_q(Omega)$ is considered on various classes of admissible domains $Omega$: in the class ${cal A}_{all}$ of {it all domains}, in the class ${cal A}_{convex}$ of {it convex domains}, and in the class ${cal A}_{thin}$ of {it thin domains}.