On the order of Schur multiplier of non-abelian p-groups

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 if G=H×Z, where H is extra special of order p3 and exponent p, and Z is an elementary abelian p-group.