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

Schur's inequality 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 short note we study a generalization of this inequality to more terms, more general functions of the variables and algebraic structures such as Hermitian matrices.