Geodesic
Weighted trace inequalities of monotonicity
Lobachevskii journal of mathematics, Tome 26 (2007), pp. 63-67.

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We study the inequality $\mathrm{Tr}(w(A)f(A))\le\mathrm{Tr}(w(A)f(B))$, where $w\colon\mathbb R\to\mathbb R^+$ is a “weight function” and $A$, $B$ are Hermitian matrices with $A\le B$, and find corresponding characterizations of the trace.
Keywords: full matrix algebra, trace inequality, monotonicity. 
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Dinh Trung Hoa; O. E. Tikhonov. Weighted trace inequalities of monotonicity. Lobachevskii journal of mathematics, Tome 26 (2007), pp. 63-67. http://geodesic.mathdoc.fr/item/LJM_2007_26_a6/

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