Norm inequalities for positive semidefinite matrices and a question of Bourin III

Let A and B be n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n $$\end{document} positive semidefinite matrices. It is shown that B1-tA2t-1B1-t+A1-tB2t-1A1-t≤22t-1A+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left| \left| \left| B^{1-t}A^{2t-1}B^{1-t}+A^{1-t}B^{2t-1}A^{1-t}\right| \right| \right| }\le 2^{2t-1}{\left| \left| \left| A+B\right| \right| \right| }$$\end{document} fort∈[12,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\; t\in [\frac{1}{2},1]$$\end{document} and for every unitarily invariant norm. It is also shown that for t∈[0,1]andp≥1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\;t\in [0,1]\;\text {and} p \ge 1,$$\end{document}‖AtXB1-t+BtX∗A1-t‖p≤212-12p‖X‖∞‖A+B‖p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Vert A^tXB^{1-t}+B^tX^{*}A^{1-t}\Vert _p \le 2^{\frac{1}{2}-\frac{1}{2p}}\Vert X\Vert _{\infty }\Vert A+B\Vert _p$$\end{document} for every n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\times n $$\end{document} matrix X. This gives a partial answer to a question of Bourin. Some related weak majorization inequalities are given.