Universal lower bounds for Dirichlet eigenvalues

Let $\Omega \subset \mathbb{R}^d$ 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 $\Omega$. 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 $2 \lambda_n + \lambda_{2n} \geq 10 \pi n/|\Omega|.$