Just to avoid confusion, a function is called matrix monotone in an interval $[a, b]$ if $A - B \geq 0$ implies $f(A) - f(B) \geq 0$ for any Hermitian Matrices $A, B$ (we can restrict to finite dimensions) with spectrum in $[a, b]$. ($\geq 0$ means that the matrix is positive semi-definite)
I am currently interested in the function $f(t) = 1 - ( 1 - t^{1/2} )^2$ and the interval $[0, 1]$. The function is monotonically increasing and concave in this interval, so there is reason to hope that it is also operator monotone.
However, I do not know how to prove this and was wondering if there exists a general strategy to prove operator monotonicity of functions. Any ideas?