On generalizations of Schur's inequality on sums of products of differences of real numbers

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