  Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal
generated by a set $\alpha$ of homogeneous elements, and B = A/I. In this note,
some inequalities between Hilbert series of algebras $A,B$ and the number of
elements of the set $\alpha$ are announced. As in the Golod--Shafarevich
inequality as in our case the equality in every estimate is exact iff the set
$\alpha$ is strongly free: so we obtain some new characterizations of such
sets. As a consequence it is proved that over a field of zero characteristic
for the class of finitely defined graded algebras there is no algorithm to
answer the following question: for an algebra $A$ and a rational number $R$, is
the convergence radius of the Hilbert series of $A$ equal to $R$?
