  We propose an inequality between the longitudinally polarized density and the
transversity of a quark in a nucleon. This inequality, whose validity is
limited to very small scales, is based on considerations about Lorentz
transformations and on commonly accepted models. Therefore it may be used as a
consistecy check with other models. It turns out to agree with most model
preedictions. Moreover it allows to establish, thanks to the positivity
constraint, another inequality between the longitudinally polarized and the
unpolarized valence quark density. We show that this latter inequality may be
extended to any Q^2, consistently with commonly used factorization schemes and
with some nonperturbative evolution models. This inequality finds nontrivial
applications to the valence "d"-quark densities and is compared with data
analyses, with model predictions and with parametrizations of quark densities.
