Norm inequalities for product of matrices

In this paper, we prove some new norm inequalities for product of matrices. Among other results, we prove that if A and B are n \(\times \) n complex matrices, then

$$\begin{aligned} \left| \left| \left| \text { }\left| AB^{*}\right| ^{2}\right| \right| \right| \le \min (\left| \left| \left| B^{*}B\right| \right| \right| \left\| A^{*}A\right\| ,\left| \left| \left| A^{*}A\right| \right| \right| \left\| B^{*}B\right\| ). \end{aligned}$$

In particular, if \(\left| \left| \left| \cdot \right| \right| \right| =\left\| \cdot \right\| ,\) then

$$\begin{aligned} \left\| AB^{*}\right\| ^{2}\le \left\| A^{*}A\right\| \left\| B^{*}B\right\| , \end{aligned}$$

which is known as the Cauchy–Schwarz inequality. Also, we prove that if A and B are n \(\times \) n complex matrices,  then

$$\begin{aligned} \text { }\left\| AB^{*}\right\| ^{2}\le w\left( A^{*}AB^{*}B\right) , \end{aligned}$$

which is a refinement of the above Cauchy–Schwarz inequality. Here \( \left| \left| \left| \cdot \right| \right| \right| ,\) \(\left\| \cdot \right\| ,\) and \(w(\cdot )\) denote any unitarily invariant norm, the spectral norm, and the numerical radius of matrices, respectively.