Definition: Let A and B be self-adjoint matrices, with the partial order $A\ge B$ if A-B is positive semidefinite. If A is self-adjoint with spectrum in the interval [a,b] and f:[a,b]->R is a real-function, define f(A) using the spectral theorem. The function f is MATRIX MONOTONE if $A\ge B$ implies $f(A)\ge f(B)$ for all A,B with spectra in the domain [a,b] of f.

Loewner's theorem: A function f:[a,b]->R is matrix monotone iff it has an analytic extension to the upper and lower half-planes so that the each of these half-planes is mapped into itself.

Definition: Let $A$ and $B$ be self-adjoint matrices, with the partial order $A\ge B$ if $A-B$ is positive semidefinite. If $A$ is self-adjoint with spectrum in the interval $[a,b]$ and $f\colon [a,b] \to \mathbb{R}$ is a real-function, define $f(A)$ using the spectral theorem. The function $f$ is called matrix monotone if $A\ge B$ implies $f(A)\ge f(B)$ for all $A,B$ with spectra in the domain $[a,b]$ of $f$.

Loewner's theorem: A function $f\colon [a,b] \to \mathbb{R}$ is matrix monotone iff it has an analytic extension to the upper and lower half-planes so that the each of these half-planes is mapped into itself.