The Berezin and Gårding inequalities

Let phi be a convex function on C, let 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 is an element of Lambda(sigma). Under certain natural assumptions about properties of pseudodifferential operators, we prove that Sigma(lambda is an element of Lambda sigma), m(lambda)phi(lambda)