The Berezin and Gårding Inequalities

Let $\varphi$ be a convex function on $\mathbb{C}$, let $\mathcal{L}(\sigma)$ be a pseudodifferential operator with symbol $\sigma$, let $\Lambda_\sigma$ be the set of its eigenvalues, and let $m(\lambda)$ be the multiplicity of an eigenvalue $\lambda\in\Lambda_\sigma$. Under certain natural assumptions about properties of pseudodifferential operators, we prove that $\sum_{\lambda\in\Lambda_\sigma}m(\lambda)\varphi(\lambda)\le\operatorname{Re}\operatorname{Tr}\mathcal{L}(\varphi(\sigma))+R$, where $R$ is an error term of the same order as the remainder term in the Gårding inequality.