It is a common fact from Matrix Analysis (see e.g. Bhatia 1997, "Matrix Analysis") that the map $t\mapsto t^r$ is matrix and in fact operator monotone on $t\in[0,\infty)$ if and only if $r\in[0,1]$. I.e. $A\leq B \implies A^r\leq B^r$ for the above values of $r$. Somewhat related to this I was wondering if the following map is matrix or operator monotone: $$ A\mapsto \Lambda(A^r)^{\frac{1}{r}}, \hspace{2cm} \text{for } r\in\mathbb{R} $$ where $\Lambda:B(\mathscr{H})\to B(\mathscr{H})$ is some operator monotone map. In fact we may assume it to be a completely positive map. $B(\mathscr{H})$ should be the bounded operators over some seperable or finite Hilbert space $\mathscr{H}$.
If $\Lambda$ was not there this would be obvious, but with it I seem to have trouble proving the monotonicity. Or finding counterexamples for that matter.