# Weighted trace inequalities of monotonicity

## (*Lobachevskii Journal of Mathematics, Volume XXVI*)

We study the inequality
Tr*(w(A)f(A)) ≤* Tr*(w(A)f(B))*,
where *w : R → R*^{+} is a ``weight
function'' and *A,B* are Hermitian matrices with *A ≤ B*, and find
corresponding characterizations of the trace.