On domain monotonicity of Neumann eigenvalues of convex domains

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.