Generalization of a matrix inequality of Ky Fan (Q1892530)

The authors give some new and interesting inequalities for matrices. One of the main results is: Let \(f : I \subset \mathbb{R} \to \mathbb{R}\) be a convex function and \(A_j\), \(j = 1, \ldots, k\), are Hermitian matrices with eigenvalues in \(I\), \(x_j \in \mathbb{C}^n\), \(j = 1, \ldots, k\), with \(\sum^k_{j = 1} (x_j, x_j) = 1\). Then \[ f \left( \sum^n_{j = 1} (A_jx_j,x_j) \right) \leq \sum^n_{j = 1} \bigl( f(A_j) x_j,x_j \bigr). \]