Jensen's Operator Inequality

Jensen's operator inequality and Jensen's trace inequality forreal functions defined on an interval are established in whatmight be called their definitive versions. This is accomplishedby the introduction of genuine non-commutative convex combinationsof operators, as opposed to the contractions considered in earlierversions of the theory by the authors, and by Brown and Kosaki.As a consequence, one no longer needs to impose conditions onthe interval of definition. It is shown how this relates tothe pinching inequality of Davis, and how Jensen's trace inequalitygeneralizes to C*-algebras. 2000 Mathematics Subject Classification47A56 (primary), 46L10, 47C15, 26A51 (secondary).