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 nx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}n complex matrices, then AB*2 <= min(B*BA*A,A*AB*B).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \left| \left| \text { }\left| AB<^>{*}\right| <^>{2}\right| \right| \right| \le \min (\left| \left| \left| B<^>{*}B\right| \right| \right| \left\| A<^>{*}A\right\| ,\left| \left| \left| A<^>{*}A\right| \right| \right| \left\| B<^>{*}B\right\| ). \end{aligned}$$\end{document}In particular, if center dot=center dot,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| \left| \left| \cdot \right| \right| \right| =\left\| \cdot \right\| ,$$\end{document} then AB*2 <= A*AB*B,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\| AB<^>{*}\right\| <^>{2}\le \left\| A<^>{*}A\right\| \left\| B<^>{*}B\right\| , \end{aligned}$$\end{document}which is known as the Cauchy-Schwarz inequality. Also, we prove that if A and B are nx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document}n complex matrices, then AB*2 <= wA*AB*B,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text { }\left\| AB<^>{*}\right\| <^>{2}\le w\left( A<^>{*}AB<^>{*}B\right) , \end{aligned}$$\end{document}which is a refinement of the above Cauchy-Schwarz inequality. Here center dot,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left| \left| \left| \cdot \right| \right| \right| ,$$\end{document}center dot,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\| \cdot \right\| ,$$\end{document} and w(center dot)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w(\cdot )$$\end{document} denote any unitarily invariant norm, the spectral norm, and the numerical radius of matrices, respectively.