The inverse problem of geometric and golden means of positive definite matrices

In this paper we prove that the inverse mean problem of geometric and golden means of positive definite matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{ \begin{aligned} & A = X\# Y\\ & B = \frac{1} {2}(X+X\# (4Y - 3X)) \end{aligned} \right. $$\end{document} is solvable (resp. uniquely solvable) if and only if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A \leq{\sqrt {3}B} \leq 2A({\text{resp}}. A \leq {\sqrt {3}B} \leq{\sqrt {3}A}) $$\end{document}.