Let *G* be a finite *p *-group of order pn, Green proved that M(G), its Schur multiplier is of order at most . Later Berkovich showed that the equality holds if and only if *G * is elementary abelian of order pn. In the present paper, we prove that if *G* is a non-abelian*p *-group of order pn with derived subgroup of order pk, then . In particular, , and the equality holds in this last bound if and only ifG=H×Z, where *H * is extra special of order p^{3} and exponent *p*, and *Z* is an elementary abelian *p*-group.