Remarks on a Question of Bourin for Positive Semidefinite Matrices

Let A and B be positive semidefinite matrices. For t∈34,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \left[ \frac{3}{4},1\right] $$\end{document} and for every unitarily invariant norm, it is shown that AtB1-t+BtA1-t≤22t-34A+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left| \left| \left| A^{t}B^{1-t}+B^{t}A^{1-t} \right| \right| \right| }\le 2^{2\left( t-\frac{3}{4}\right) }{\left| \left| \left| A+B \right| \right| \right| } \end{aligned}$$\end{document}and for t∈0,14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \left[ 0,\frac{1}{4}\right] $$\end{document}, AtB1-t+BtA1-t≤2214-tA+B.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left| \left| \left| A^{t}B^{1-t}+B^{t}A^{1-t} \right| \right| \right| }\le 2^{2\left( \frac{1}{4}-t\right) }{\left| \left| \left| A+B \right| \right| \right| }. \end{aligned}$$\end{document}These norm inequalities are sharper than an earlier norm inequality due to Alakhrass and closely related to an open question of Bourin. In fact, they lead to an affirmative solution of Bourin’s question for t=14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=\frac{1}{4}$$\end{document} and 34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{4}$$\end{document}, which is a result due to Hayajneh and Kittaneh (Int J Math 32 (2150043):7, 2021).