In last years, there has been some effort to get quantitative versions of the most classical spectral inequalities, namely the Faber-Krahn inequality for the first eigenvalue of the Dirichlet Laplacian and its generalization to the case of the p-Laplacian. In this talk, we will present two new results, which involve the second eigenvalue of the Dirichlet Laplacian and the second (that is, the first non-trivial) eigenvalue of the Neumann Laplacian. We will also discuss the sharpness of these two estimates. The results are in collaboration with Lorenzo Brasco (Napoli).