Operator monotone function

Summary

A real function $f$ on an interval is operator monotone if it keeps the order between Hermitian matrices: from $A \le B$ it follows that $f(A) \le f(B)$. Here $A \le B$ is the Loewner order, and $f(A)$ is defined through the eigenvalues of $A$. Every operator monotone function is a monotone function, but the converse fails, so operator monotonicity is a much stronger condition. Löwner's theorem describes these functions through complex analysis.