An improvement to a Berezin-Li-Yau type inequality

In this article we improve a lower bound for $\sum_{j=1}^k\beta_j$ (a Berezin-Li-Yau type inequality) that appeared in an earlier paper of Harrell and Yolcu. Here $\beta_j$ denotes the $j$th eigenvalue of the Klein Gordon Hamiltonian $H_{0,\Omega}=\vert p\vert$ when restricted to a bounded set $\Omega\subset {\mathbb{R}}^n$. $H_{0,\Omega}$ can also be described as the generator of the Cauchy stochastic process with a killing condition on $\partial \Omega$. To do this, we adapt the proof of Melas, who improved the estimate for the bound of $\sum_{j=1}^k\lambda_j$, where $\lambda_j$ denotes the $j$th eigenvalue of the Dirichlet Laplacian on a bounded domain in ${\mathbb{R}}^d$.