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