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 \(\times \) $\times$ n complex matrices, then \(\begin{aligned} \left| \left| \left| \text { }\left| AB^{*}\right| ^{2}\right| \right| \right| \le \min (\left| \left| \left| B^{*}B\right| \right| \right| \left\| A^{*}A\right\| ,\left| \left| \left| A^{*}A\right| \right| \right| \left\| B^{*}B\right\| ). \end{aligned}\) $$\begin{array}{c}\hfill \left|\left|\left|,{\left|A,B^{\ast }\right|}^2\right|\right|\right|\le min(\left|\left|\left|B^{\ast },B\right|\right|\right|∥A^{\ast },A∥,\left|\left|\left|A^{\ast },A\right|\right|\right|∥B^{\ast },B∥).\end{array}$$ In particular, if \(\left| \left| \left| \cdot \right| \right| \right| =\left\| \cdot \right\| ,\) $\left|\left|\left|·\right|\right|\right|=∥·∥,$ then \(\begin{aligned} \left\| AB^{*}\right\| ^{2}\le \left\| A^{*}A\right\| \left\| B^{*}B\right\| , \end{aligned}\) $$\begin{array}{c}\hfill {∥A,B^{\ast }∥}^2\le ∥A^{\ast },A∥∥B^{\ast },B∥,\end{array}$$ which is known as the Cauchy–Schwarz inequality. Also, we prove that if A and B are n \(\times \) $\times$ n complex matrices,  then \(\begin{aligned} \text { }\left\| AB^{*}\right\| ^{2}\le w\left( A^{*}AB^{*}B\right) , \end{aligned}\) $$\begin{array}{c}\hfill {∥A,B^{\ast }∥}^2\le w\left(A^{\ast },A,B^{\ast },B\right),\end{array}$$ which is a refinement of the above Cauchy–Schwarz inequality. Here \( \left| \left| \left| \cdot \right| \right| \right| ,\) $\left|\left|\left|·\right|\right|\right|,$ \(\left\| \cdot \right\| ,\) $∥·∥,$ and \(w(\cdot )\) $w(·)$ denote any unitarily invariant norm, the spectral norm, and the numerical radius of matrices, respectively.