Let (mathcal{X} , langle cdot,cdotrangle) be a semi-inner product module over a C^*-algebra mathcal{A} . For arbitrary n in mathbb{N}$ and x_1, cdots, x_n in mathcal{X} we study the so-called n times n Gram matrix [ langle x_i, x_j rangle] with entries in mathcal{A} , construct a non-decreasing sequence of positive matrices in M_n( mathcal{A}) which is majorized by [ langle x_i, x_j rangle] and apply it to obtain generalizations of covariance–variance inequality, an extension of the Ostrowski inequality and an improvement of the Kantorovich inequality involving operator means.

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