Inequalities of singular values and unitarily invariant norms for sums and products of matrices

In this work, we investigate inequalities of singular values and unitarily invariant norms for sums and products of matrices. First, we prove that s2(XY & lowast;)< wlogs((X & lowast;X)q(Y & lowast;Y)(X & lowast;X)1-q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s<^>{2}\big (XY<^>{*}\big )\prec _{w\log }s\big ((X<^>{*}X)<^>{q}(Y<^>{*}Y)(X<^>{*}X)<^>{1-q}\big )$$\end{document}, where X,Y is an element of Mn(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X,\ Y\in M_{n}(C)$$\end{document} and 0<q<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q<1$$\end{document}. Based on this result, we present some inequalities between sum of the t-geometric mean and sum of the product of matrices. Those obtained results are the generalization of the present results. In the end, we present a singular values version of Audenaert's inequality [1].