Inequality with Applications in Statistical Mechanics

We prove for Hermitian matrices (or more generally for completely continuous self‐adjoint linear operators in Hilbert space)A and B that Tr (eA+B ) ≤ Tr (eAeB ). The inequality is shown to be sharper than the convexity property (0 ≤ α ≤ 1) Tr (e αA+(1−α)B ) ≤ [Tr (eA )]α[Tr (eB )]1−α, and its possible use for obtaining upper bounds for the partition function is discussed briefly.

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