It seems to be a fairly common exercise to show that $$ 0\preceq A \preceq B \implies \sqrt A \preceq \sqrt B $$ for positive semi-definite matrices. On the other hand, in general, $$ 0\preceq A \preceq B \ \nRightarrow\ A^2 \preceq B^2. $$ (See e.g. https://math.stackexchange.com/a/510999/620957).
Along with being able to take inverses, this is more or less the usual simple results I've seen of this kind. But do they hint at a more general theorem behind the scenes? I'm imaging some result along the lines of
If $0\preceq A \preceq B$, and if $f$ is a [concave, increasing, positive, ...?] function, then $f(A)\preceq f(B)$.
I realize that this question is somewhat vague, so sorry about that, but is there a nice result to discovered here? My googling is being unsuccessful... Thanks in advance.