Extensions of the Schur majorisation inequalities (Q2158293)

For an \(n\)-by-\(n\) Hermitian matrix \(A\) with eigenvalues \[ \lambda_1\leq \dots \leq\lambda_n, \] and diagonal entries \[ a_{11}\leq \dots \leq a_{nn}, \] the Schur majorisation inequalities read: \[ \sum_{i=1}^r\lambda_i\leq \sum_{i=1}^r a_{ii}, \] for every \(1\leq r\leq n\). In this paper, the authors find inequalities stronger than the Schur majorisation. The techniques used are not easy but they yield good proofs and provide interesting relations between the eigenvalues and the diagonal entries of an Hermitian matrix.