Weak majorization inequalities and convex functions

✍ Scribed by Jaspal Singh Aujla; Fernando C. Silva

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 143 KB
• Volume: 369
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(02)00720-6

✦ Synopsis

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 + (1α)B) are weakly majorized by the eigenvalues of αf (A) + (1α)f (B). Further if f is log convex we prove that the eigenvalues of f (αA + (1α)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.