  Inspired by a recent result of Funano's, we provide a sharp quantitative
comparison result between the first nontrivial eigenvalues of the Neumann
Laplacian on bounded convex domains $\Omega_{1} \subset \Omega_{2}$ in any
dimension $d$ greater than or equal to two, recovering domain monotonicity up
to an explicit multiplicative factor. We provide upper and lower bounds for
such multiplicative factors for higher-order eigenvalues, and study their
behaviour with respect to the dimension and order. We further consider
different scenarios where convexity is no longer imposed. In a final section we
formulate some related open problems.
