Jensen's inequality relative to matrix-valued measures
Jensen's inequality states that if \((X,\Sigma,\mu)\) is a measure space with \(\mu(X)=1\), \(\alpha<\beta\), and \(f:X\to\mathbb R\) is an integrable function with \(f(X)\subset(\alpha,\beta)\), then \[ \mathcal I\Big(\int_X f\;d\mu\Big) \leq \int_X (\mathcal I\circ f) d\mu \] for every convex function \(\mathcal I: (\alpha,\beta)\to\mathbb R\). In this paper, Jensen's inequality is extended to both scalar-valued and matrix-valued probability measures.
scientific article; zbMATH DE number 5083323