  Classically, Jensen's Inequality asserts that if $X$ is a compact convex set,
and $f:K\to \mathbb{R}$ is a convex function, then for any probability measure
$\mu$ on $K$, that $f(\text{bar}(\mu))\le \int f\;d\mu$, where
$\text{bar}(\mu)$ is the barycenter of $\mu$. Recently, Davidson and Kennedy
proved a noncommutative ("nc") version of Jensen's inequality that applies to
nc convex functions, which take matrix values, with probability measures
replaced by ucp maps. In the classical case, if $f$ is only a separately convex
function, then $f$ still satisfies the Jensen inequality for any probability
measure which is a product measure. We prove a noncommutative Jensen inequality
for functions which are separately nc convex in each variable. The inequality
holds for a large class of ucp maps which satisfy a noncommutative analogue of
Fubini's theorem. This class of ucp maps includes any free product of ucp maps
built from Boca's theorem, or any ucp map which is conditionally free in the
free-probabilistic sense of M{\l}otkowski. As an application to free
probability, we obtain some operator inequalities for conditionally free ucp
maps applied to free semicircular families.
