A characterization of matrix inequality \(A\geqslant B\geqslant C\) via Karcher mean (Q2790764)

Given three positive definite matrices \(A,B,C\), the main result of this paper shows that \(A \geqslant B \geqslant C\) iff the following two Karcher mean inequalities NEWLINE\[NEWLINE\begin{align*}{ & \Lambda (\omega ;{A^{ - {p_1}}},{B^{ - {p_2}}},{B^t}{{\natural}_s}{C^{{p_3}}}) \leqslant C \cr & \Lambda (\omega ;{C^{ - {p_1}}},{B^{ - {p_2}}},{B^t}{{\natural}_s}{A^{{p_3}}}) \geqslant A \cr}\end{align*}NEWLINE\]NEWLINE hold for \(t \in [0,1]\), \(s \geqslant 1\), \({p_1},{p_2} > 0\), \({p_3} > 1\), \(\hat \omega = (\tfrac{1}{{{p_1} + 1}},\tfrac{1}{{{p_2} + 1}},\tfrac{2}{{({p_3} - 1)s + t - 1}})\), \(\omega = ({w_1},{w_2},{w_3}) = \tfrac{{\hat \omega }}{{{{\left\| {\hat \omega } \right\|}_1}}}\).