Weighted trace inequalities of monotonicity

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.