A Unitarily Invariant Norm Inequality for Positive Semidefinite Matrices and a Question of Bourin

In this article, we obtain a new unitarily invariant norm inequality for positive semidefinite matrices. In fact, we prove that if A and B are positive semidefinite matrices and 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}, then B1-tA2t-1B1-t+A1-tB2t-1A1-t≤24t-34A+B.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| \left| \left| B^{1-t}A^{2t-1}B^{1-t}+A^{1-t}B^{2t-1}A^{1-t} \right| \right| \right| \le 2^{4\left( t-\frac{3}{4}\right) }\left| \left| \left| A+B \right| \right| \right| . \end{aligned}$$\end{document}The significance of this result is that it is sharper than an earlier norm inequality and closely related to an open question of Bourin. In particular, this inequality gives a way to settle 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 [9].