  We generalize a classical inequality between the eigenvalues of the
Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar
domains: in 1955, Payne proved that below the $k$-th eigenvalue of the
Dirichlet Laplacian there exist at least $k + 2$ eigenvalues of the Neumann
Laplacian, provided the domain is convex. It has, however, been conjectured
that this should hold for any domain. Here we show that the statement indeed
remains true for all simply connected planar Lipschitz domains. The proof
relies on a novel variational principle.
