Weighted Monotonicity Inequalities for Traces on Operator Algebras

We study inequalities of the form
$$ \tau(w(A)^{1/2}f(A)w(A)^{1/2})\le\tau(w(A)^{1/2}f(B)w(A)^{1/2}),\qquad A\le B, $$
where $\tau$ is a trace on a von Neumann algebra or a $C^*$-algebra, $A$ and $B$ are self-adjoint elements of the algebra in question, $f$ and $w$ are real-valued functions, and the “weight” function $w$ is nonnegative.