Upper bounds for the free energy. A generalisation of the Bogolubov inequality and the Feynman inequality

Abstract

A set of lower bounds is obtained for (e-A) from which a corresponding set of upper bounds is derived for the free energy. If the expectation values are expressed in the operator formalism the Bogolubov inequality (for Hermitian operators) is obtained as the first upper bound. When the expectation value is written as a path integral the Feynman inequality (for real actions) results as the first upper bound. It is shown that the second upper bound is identical to the one derived by Zeile (1978) for the path integral formalism while in the operator formalism this second upper bound is the one given by Dorre et al. (1979).