  The purpose of this note is to supply an upper and a lower bound (which are
in general sharp) for the $h$-vector of a level algebra which is relatively
compressed with respect to any arbitrary level algebra $A$. The useful concept
of relatively compressed algebra was recently introduced in $[MMN]$ by Migliore
{\it et al.} (whose investigations mainly focused on the particular case of $A$
a complete intersection). The key idea of this note is the simple observation
that the level algebras which are relatively compressed with respect to $A$
coincide (after an obvious isomorphism) with the generic level quotients of
suitable truncations of $A$. Therefore, we are able to apply to relatively
compressed algebras the main result of our recent work $[Za3]$.

Commutative Algebra