Schur-convex function

Every function that is convex and symmetric (under permutations of the arguments) is also Schur-convex. Every Schur-convex function is symmetric, but not necessarily convex. If f is (strictly) Schur-convex and g is (strictly) monotonically increasing, then g\circ f is (strictly) Schur-convex. If g is a convex function defined on a real interval, then \sum_{i=1}^n g(x_i) is Schur-convex. Schur–Ostrowski criterion If f is symmetric and all first partial derivatives exist, then f is Schur-convex if and only if : (x_i - x_j)\left(\frac{\partial f}{\partial x_i} - \frac{\partial f}{\partial x_j}\right) \ge 0 for all x \in \mathbb{R}^d holds for all 1\le i,j\le d. == Examples ==

• f(x)=\min(x) is Schur-concave while f(x)=\max(x) is Schur-convex. This can be seen directly from the definition. • The Shannon entropy function \sum_{i=1}^d{P_i \cdot \log_2{\frac{1}{P_i}}} is Schur-concave. • The Rényi entropy function is also Schur-concave. • x \mapsto \sum_{i=1}^d{x_i^k},k \ge 1 is Schur-convex if k \geq 1, and Schur-concave if k \in (0, 1). • The function f(x) = \prod_{i=1}^d x_i is Schur-concave, when we assume all x_i > 0 . In the same way, all the elementary symmetric functions are Schur-concave, when x_i > 0 . • A natural interpretation of majorization is that if x \succ y then x is less spread out than y . So it is natural to ask if statistical measures of variability are Schur-convex. The variance and standard deviation are Schur-convex functions, while the median absolute deviation is not. • A probability example: If X_1, \dots, X_n are exchangeable random variables, then the function \text{E} \prod_{j=1}^n X_j^{a_j} is Schur-convex as a function of a=(a_1, \dots, a_n) , assuming that the expectations exist. • The Gini coefficient is strictly Schur convex. == References ==