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.
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