Universal lower bounds for Dirichlet eigenvalues

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain and let ${\lambda }_1,{\lambda }_2,\dots$ denote the sequence of eigenvalues of the Laplacian subject to Dirichlet boundary conditions. We consider inequalities for ${\lambda }_n$ that are independent of the domain $\mathrm{\Omega }$. A well–known such inequality follows from the Berezin–Li–Yau approach. The purpose of this paper is to point out a certain degree of flexibility in the Li–Yau approach. We use it to prove a new type of two-point inequality which are strictly stronger than what is implied by Berezin-Li-Yau itself. For example, when $d=2$, one has $2{\lambda }_n+{\lambda }_{2n}\ge 10\pi n\mathrm{/}\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{.}$