On generalizations of Schur's inequality

TL;DR

This paper explores broad generalizations of Schur's inequality, extending it to multiple terms, diverse functions, and algebraic structures like vectors and matrices, enhancing its theoretical scope.

Contribution

It introduces new generalized forms of Schur's inequality applicable to various algebraic structures and functions, expanding the inequality's theoretical framework.

Findings

Extended Schur's inequality to multiple terms

Applied inequality to vectors and Hermitian matrices

Provided new bounds and conditions for generalizations

Abstract

Schur's inequality for the sum of products of the differences of real numbers states that for $x,y,z,t\geq 0$, $x^t(x-y)(x-z) + y^t(y-z)(y-x) + z^t(z-x)(z-y) \geq 0$. In this paper we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as vectors and Hermitian matrices.