  Let W be an associative PI-algebra over a field F of characteristic zero.
Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote
the codimension growth of W and of the identity component W_e, respectively.
The following inequality had been conjectured by Bahturin and Zaicev:
exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is
affine (i.e. finitely generated). Here we prove the conjecture in general.
