We generalize to the p-Laplacian
Δp a spectral inequality proved by M.-T.
Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower bound
on the first Dirichlet eigenvalue of Δp of a
set in terms of its p-torsional rigidity. The result is valid in every
space dimension, for every
1 < p < ∞ and for every open
set with finite measure. Moreover, it holds by replacing the first eigenvalue with more
general optimal Poincaré-Sobolev constants. The method of proof is based on a
generalization of the rearrangement technique introduced by Kohler−Jobin.