Jensen’s Inequality for Separately Convex Noncommutative Functions

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