Weak majorization inequalities and convex functions

Let f be a convex function defined on an interval I, 0 α 1 and A,B n×n complex Hermitian matrices with spectrum in I. We prove that the eigenvalues of f(αA+(1−α)B) are weakly majorized by the eigenvalues of αf(A)+(1−α)f(B). Further if f is log convex we prove that the eigenvalues of f(αA+(1−α)B) are weakly majorized by the eigenvalues of f(A)αf(B)1−α. As applications we obtain generalizations of the famous Golden–Thomson trace inequality, a representation theorem and a harmonic–geometric mean inequality. Some related inequalities are discussed.