Lower Bounds for the Helmholtz Function

A mathematical theorem is established for traces of products of bounded Hermitian and definite operators, a and b: $\mathrm{Tr}{\left(\mathrm{ab}\right)}^{2^{p+1\mathrm{}}}<~\mathrm{Tr}{\left({\mathrm{a}}^2{\mathrm{b}}^2\right)}^{2^p}$ for $p$ a non-negative integer. This theorem is applied to the equilibrium partition function by exploiting an infinite-product representation of the exponential function of the sum of two operators. As a result, a set of inequalities is established which yields a set of upper bounds for the partition function. This result is invariant to the particle statistics of the system. A general argument yields the result that the classical Helmholtz free-energy function serves as a lower bound to the corresponding quantum result.