  Let $\phi$ be a linear map from the $n\times n$ matrices ${\mathcal M}_n$ to
the $m\times m$ matrices ${\mathcal M}_m$. It is known that $\phi$ is
$2$-positive if and only if for all $K\in {\mathcal M}_n$ and all strictly
positive $X\in {\mathcal M}_n$, $\phi(K^*X^{-1}K) \geq
\phi(K)^*\phi(X)^{-1}\phi(K)$. This inequality is not generally true if $\phi$
is merely a Schwarz map. We show that the corresponding tracial inequality
${\rm Tr}[\phi(K^*X^{-1}K)] \geq {\rm Tr}[\phi(K)^*\phi(X)^{-1}\phi(K)]$ holds
for a wider class of positive maps that is specified here. We also comment on
the connections of this inequality with various monotonicity that have found
wide use in mathematical physics, and apply it, and a close relative, to obtain
some new, definitive results.
