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Infinite sets of inequalities which generalize all the known inequalitiesthat can be used in the majorization step of the Approximating Hamiltonianmethod are derived. They provide upper bounds on the difference between thequadratic fluctuations of intensive observables of a $N$-particle system andthe corresponding Bogoliubov-Duhamel inner product. The novel feature is that,under sufficiently mild conditions, the upper bounds have the same form andorder of magnitude with respect to $N$ for all the quantities derived by afinite number of commutations of an original intensive observable with theHamiltonian. The results are illustrated on two types of exactly solvable modelsystems: one with bounded separable attraction and the other containinginteraction of a boson field with matter.

Generalized inequalities for the Bogoliubov-Duhamel inner product with applications in the Approximating Hamiltonian Method