Weak majorization inequalities and convex functions

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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)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. © 2003 Elsevier Science Inc. All rights reserved.

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Aujla, J. S., & Silva, F. C. (2003). Weak majorization inequalities and convex functions. Linear Algebra and Its Applications, 369(SUPP.), 217–233. https://doi.org/10.1016/S0024-3795(02)00720-6

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