Schurconvex

Schurconvex refers to a property of functions in mathematics, particularly in the study of inequalities and optimization. A function is Schurconvex if it is symmetric and its value increases when any of its variables are "majorized" by others. Majorization is a concept that compares two vectors. A vector $x$ majorizes a vector $y$ if the sum of the largest $k$ elements of $x$ is greater than or equal to the sum of the largest $k$ elements of $y$ for all $k$, and the total sums of the elements in both vectors are equal.

More formally, a function $f: \mathbb{R}^n \to \mathbb{R}$ is Schurconvex if for any two vectors $x, y \in \mathbb{R}^n$, if $x$ majorizes $y$, then $f(x) \ge f(y)$. This property is closely related to majorization theory, a field developed by Issai Schur. Schurconvex functions are often encountered in areas such as probability theory, information theory, and economics, where they are used to establish inequalities and analyze the behavior of various quantities.

A key characteristic of Schurconvex functions is that they are constant on the set of vectors that are permutations of each other. This means that the order of the arguments does not affect the function's output. Examples of Schurconvex functions include the sum of squares, the Gini mean difference, and certain entropy measures. Understanding Schurconvexity is crucial for proving certain types of inequalities, especially those involving symmetric functions.