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

We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves classical form of this inequality. As consequence new we can rederive known doubly weighted inequalities. Our motivation comes from theory Schrödinger operators and explain use inequalities in that context.