  Recently, new thermodynamic inequalities have been obtained, which set bounds
on the quadratic fluctuations of intensive observables of statistical
mechanical systems in terms of the Bogoliubov - Duhamel inner product and some
thermal average values. It was shown that several well-known inequalities in
equilibrium statistical mechanics emerge as special cases of these results. On
the basis of the spectral representation, lower and upper bounds on the
one-sided fidelity susceptibility were derived in analogous terms. Here, these
results are reviewed and presented in a unified manner. In addition, the
spectral representation of the symmetric two-sided fidelity susceptibility is
derived, and it is shown to coincide with the one-sided case. Therefore, both
definitions imply the same lower and upper bounds on the fidelity
susceptibility.
