Weak majorization inequalities and convex functions

Abstract

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) α f( 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.