Near order and metric-like functions on the cone of positive definite matrices

In this article we introduce a new relation on the cone of positive definite matrices and we study its properties and its effect on operator monotonicity and convexity. Furthermore, we use this new relation to establish analogies between the weighted geometric means A♯tB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\sharp _t B$$\end{document} and the spectral weighted geometric means A♮tB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A\natural _t B$$\end{document} of positive definite matrices A and B, via the Thompson metric d∞(A,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_\infty (A,B)$$\end{document} and the semi-metric d(A,B)=2‖log(A-1♯B)‖.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d(A,B)=2\Vert \log (A^{-1}\sharp B)\Vert .$$\end{document}