Some inequalities for mean values in statistical mechanics

Some new upper bounds are proved for mean values of the form $\langle\operatorname{BB^+}\rangle$ in equilibrium statistical mechanics. As an illustration, a rigorous upper bound is obtained for the mean kinetic energy of a molecule of a quantum system of distinguishable particles and some majorizing estimates are considered in the method of approximating Hamiltonians of N. N. Bogolyubov (Jr.).