Operator monotone functions

Question

By definition, I know that a function $f$ is operator monotone if $A - B \geq 0 \Rightarrow f(A) - f(B) \geq 0$. For instance, we have $A^2 \leq B^2 \Rightarrow A \leq B$ because the root function is operator monotone. However, does this also imply that $A^{1/2} \leq 2A^{1/2}$?