We study the Cauchy--Schwarz and some related inequalities in a semi-inner product module over a C^* -algebra { mathscr A} . The key idea is to consider a semi-inner product { mathscr A} -module as a semi-inner product { mathscr A} -module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related to the Gram matrix. The induced semi-inner products are also related to the the notion of covariance and variance. Furthermore, we obtain a sequence of nested inequalities that emerges from the Cauchy--Schwarz inequality. As a consequence, we derive some interesting operator-theoretical corollaries. In particular, we show that the sequence arising from our construction, when applied to a positive invertible element of a C^* -algebra, converges to its inverse.

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