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Let ϕ be a convex function on ℂ, let ℒ(σ) be a pseudodifferential operator with symbol σ, let Λσ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λσ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that <img src="/fulltext-image.asp?format=htmlnonpaginated&src=FJ7M0317635X2P58_html\10688_2005_Article_51_TeX2GIFIE1.gif" border="0" alt="$$\sum {_{\lambda  \in \Lambda _\sigma  } m(\lambda )} \varphi (\lambda ) \leqslant \operatorname{Tr} L(\varphi (\sigma )) + R$$" />, where R is an error term of the same order as the remainder term in the Garding inequality.

The Berezin and Garding Inequalities