majorizes

In mathematics, particularly in the fields of optimization, functional analysis, and economics, the concept of majorization plays a crucial role in comparing vectors or sequences in terms of their relative ordering. Introduced by Brunk (1955) and later formalized by Hardy, Littlewood, and Pólya, majorization provides a way to assess how one sequence "dominates" another in a partial ordering sense.

Given two real sequences \( x = (x_1, x_2, \dots, x_n) \) and \( y = (y_1, y_2, \dots, y_n) \) arranged in non-increasing order, \( x \) is said to *majorize* \( y \) (denoted \( x \succ y \)) if the following conditions hold:

1. The sum of the first \( k \) components of \( x \) is at least as large as the sum of the first \( k \) components of \( y \) for all \( k \) from 1 to \( n-1 \).

2. The total sums of the sequences are equal, i.e., \( \sum_{i=1}^n x_i = \sum_{i=1}^n y_i \).

This relationship is transitive and reflexive, forming a partial order. Majorization is closely tied to Schur-convex functions, which preserve the ordering under majorization. A function \( f \) is Schur-convex if \( x \succ y \) implies \( f(x) \geq f(y) \). Examples of Schur-convex functions include the sum of squares and the entropy function.

Majorization finds applications in various domains, such as optimization problems where constraints are expressed in terms of majorization (e.g., in the Karamata inequality), economic theory (e.g., in the analysis of resource allocation), and statistics (e.g., in the study of order statistics). The concept also extends to infinite sequences and continuous functions, broadening its utility in advanced mathematical analysis.