Upper bounds for courant-sharp neumann and robin eigenvalues

We consider the eigenvalues of the Laplacian, with a Neumann or Robin boundary condition, on an open, bounded, connected set in R-n with a C-2 boundary. We obtain upper bounds for the eigenvalues that have a corresponding eigenfunction that achieves equality in Courant's Nodal Domain theorem. In the case where the set is also assumed to be convex, we obtain explicit upper bounds in terms of some of the geometric quantities of the set.