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Let (A,m) be a commutative quasi-local ring with non-zero identity and M be an Artinian A-module with dimM =d.If I is an ideal of A with e(0:M I) ＜ o∞,then we show that for a minimal reduction J of I,(0:M JI) =(0:M I2) if and only if e(0:M In+1) =e(0:M J)(n+dd)--e((0:M J)/(0:M I))(n+d-1d-1) for all n ≥ 0.Moreover,we study the dual of Burch’s inequality.In particular,the Burch’s inequality becomes an equality if G(I,M) is co-Cohen-Macaulay.