Weak majorization inequalities and convex functions
By Jaspal Singh Aujla, Fernando Silva. 2003
152 citations recorded by the source corpus.
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.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/d294ffb1ad1460ff802b00c2370973b56c350bf6
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In Applied Mathematics
- Jean-Christophe Bourin
- Y. Chu
- Mitsuru Uchiyama
- G.‐D. Wang
- X.‐H. Zhang
- Jaspal Singh Aujla
- Tie-hong Zhao
- Miao-Kun Wang
Citations in the corpus
- Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory (2005, unknown): “(Proof: See [42].”
- A matrix subadditivity inequality for f(A + B) and f(A) + f(B) (2007, background): “[3] J. S. Aujla and F. C. Silva, Weak majorization inequalities and convex functions, Linear Alg.”
- The Schur multiplicative and harmonic convexities of the complete symmetric function (2011, background): “Schur convexity was introduced by Schur in 1923 [46] and it has many important applications in analytic inequalities [4], [12], [22], [43], theory of statistical experiments [48], graphs and matrices [15], combinatorial optimization [25], reliability [26], gamma functions [35], information-theoretic”
- Unitary orbits of Hermitian operators with convex or concave functions (2011, background): “1 is first noted in Brown–Kosaki’s paper [15]; with these assumptions, it is also obtained in Aujla–Silva’s paper [5].”
- Eigenvalue inequalities for convex and log-convex functions (2007, background): “In [3,7] we gave several convexity inequalities for eigenvalues.”