Weak majorization inequalities and convex functions (Q1399932)

Let \(f\) be a convex function defined on an interval \(I\), \(0\leq \alpha \leq 1\), and \(A\), \(B\) be \(n\times n\) complex Hermitian matrices with spectrum in \(I\). The authors prove that the eigenvalues of \(f(\alpha A+(1-\alpha)B)\) are weakly majorized by the eigenvalues of \(\alpha f(A)+(1-\alpha)f(B)\). Furthermore, if \(f\) is log convex they prove that the eigenvalues of \(f(\alpha A+(1-\alpha)B)\) are weakly majorized by the eigenvalues of \(f(A)^\alpha f(B)^{1-\alpha}\). Some related inequalities are also discussed.