matrix monotone

A real function $f$ on a real interval $I$ is said to be matrix monotone of $n$, if

$$A{\le }_{L}B\Rightarrow f(A)\le f(B)$$(1)

for all Hermitian $n\times n$ matrices $A,B$ with spectra contained in $I$. Here ${\le }_L$ denotes the Loewner order, and the notation $f(A)$ is explained in the entry functional calculus for Hermitian matrices.