  We revisit an inequality due to Onsager, which states that the (quantum)
liquid structure factor has an upper bound of the form (const.) x |k|, for not
too large modulus of the wave vector k. This inequality implies the validity of
the Landau criterion in the theory of superfluidity with a definite, nonzero
critical velocity. We prove an auxiliary proposition for general Bose systems,
together with which we arrive at a rigorous proof of the inequality for one of
the very few soluble examples of an interacting Bose fluid, Girardeau's model.
The latter proof demonstrates the importance of the thermodynamic limit of the
structure factor, which must be taken initially at k different from 0. It also
substantiates very well the heuristic density functional arguments, which are
also shown to hold exactly in the limit of large wave-lengths. We also briefly
discuss which features of the proof may be present in higher dimensions, as
well as some open problems related to superfluidity of trapped gases.
