Schur-convex function

Summary

A Schur-convex function is a function of $n$ real variables that preserves the order of majorization. If $x$ is more spread out than $y$ and has the same sum, then $f(x) \ge f(y)$ for a Schur-convex $f$. Every symmetric convex function is Schur-convex, and many classical inequalities, such as Karamata's inequality, are statements that some function is Schur-convex. The definition gives the precise condition.