Two consequences of Davies's Hardy inequality

Davies' version of the Hardy inequality gives a lower bound for the Dirichlet integral of a function vanishing on the boundary of a domain in terms of the integral of the squared function with a weight containing the averaged distance to the boundary. This inequality is applied to easily derive two classical results of spectral theory, E. Lieb's inequality for the first eigenvalue of the Dirichlet Laplacian and G. Rozenblum's estimate for the spectral counting function of the Laplacian in an unbounded domain in terms of the number of disjoint balls of preset size whose intersection with the domain is large enough.