As per my understanding, roughly stated, $f$ is an operator monotone function of order $n$ if for all $n\times n$ (Hermitian) matrices, $X,Y\ge0$ which satisfy $X\ge Y$, we have $f(X)\ge f(Y)$.
If $f$ is an operator monotone function of all orders $n$, then $f$ is said to be an operator monotone function.
Löwner gave a nice characterisation of operator monotone functions $f$ as $$ f(t) = \int _0 ^1 \frac{t}{\lambda + (1-\lambda)t}d\mu (\lambda) $$ and I was wondering if a similar characterisation exists for operator monotone functions of order $n$.
