Optimal bounds for Neumann eigenvalues in terms of the diameter

In this paper, we obtain optimal upper bounds for all the Neumann eigenvalues in two situations (that are closely related). First we consider a one-dimensional Sturm-Liouville eigenvalue problem where the density is a function $h(x)$ whose some power is concave. We prove existence of a maximizer for $\mu_k(h)$ and we completely characterize it. Then we consider the Neumann eigenvalues (for the Laplacian) of a domain $\Omega\subset \mathbb{R}^d$ of given diameter and we assume that its profile function (defined as the $d-1$ dimensional measure of the slices orthogonal to a diameter) has also some power that is concave. This includes the case of convex domains in $\mathbb{R}^d$, containing and generalizing previous results by P. Kr\"oger. On the other hand, in the last section, we give examples of domains for which the upper bound fails to be true, showing that, in general, $\sup D^2(\Omega)\mu_k(\Omega)= +\infty$.