For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$ P_\mu(p, A, B) = A^{ 1/2}\left(\frac{ 1+(A^{ -1/2}BA^{ -1/2})^p}{ 2}\right )^{ 1/p} A^{ 1/2}\quad (p \ge 0). $$ In this paper, for $0\le p \le 1 \le q$, we show that if one of the following inequalities \begin{ align*} f(P_\mu(p, A, B)) \le f(P_\mu(1, A, B)) \le f(P_\mu(q, A, B))\nonumber \end{ align*} holds for any positive definite matrices $A$ and $B$, then the function $f$ is operator monotone on $(0, \infty).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers $1/2$ and $2$. As a consequence, we establish new charaterizations of operator monotone functions with the non-Kubo-Ando matrix power means.
Categories: math.FA math.OA