Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians

We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kroger inequalities in the limit as the eigenvalues tend to infinity. We accomplish this in the framework of the Riesz mean R-1(z) of the eigenvalues by applying the averaged variational principle with families of test functions that have been corrected for boundary behaviour.