Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_{\sigma}(z) := \sum_k{(z -\lambda_k)+^{\sigma}}. Here ${{\lambda }_k}_{k=1}^{\mathrm{\infty }}$ are the ordered eigenvalues of the Laplacian on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, and $x_{+}:=\max (0,x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on ${\lambda }_k$, on averages such as $\overline{{\lambda }_k}:=\frac{1}{k}{\sum }_{\mathrm{\ell }\le k}{\lambda }_{\mathrm{\ell }}$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k\ge j\frac{1+\frac{d}{2}}{1+\frac{d}{4}}$, {\bar{\lambda{k}}}/{\bar{\lambda_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}.