  In this paper we study optimal lower and upper bounds for functionals
involving the first Dirichlet eigenvalue $\lambda_{F}(p,\Omega)$ of the
anisotropic $p$-Laplacian, $1<p<+\infty$. Our aim is to enhance how, by means
of the $\mathcal P$-function method, it is possible to get several sharp
estimates for $\lambda_{F}(p,\Omega)$ in terms of several geometric quantities
associated to the domain. The $\mathcal P$-function method is based on a
maximum principle for a suitable function involving the eigenfunction and its
gradient.
