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Jensen’s operator inequality and its converses. (English) Zbl 1151.47025

Let \(f\) be an operator convex function on an interval \(J\) and let \({\mathfrak A}\) and \({\mathfrak B}\) be unital \(C^*\)-algebras. If \((\varphi_t)_{t\in T}\) is a unital field of positive linear mappings \(\varphi_t: {\mathfrak A} \to {\mathfrak B}\) defined on a locally compact Hausdorff space \(T\) with a bounded Radon measure \(\mu\), then the inequality
\[ f\left(\int_T \varphi_t(A_t)\,d\mu(t)\right) \leq \int_T\varphi_t(f(A_t))\,d\mu(t) \]
holds for every bounded continuous field \((A_t)_{t\in T}\) of selfadjoint elements of \({\mathfrak A}\) with spectra contained in \(J\).
This result is an extension of the Jensen operator inequality and contains as special cases some inequalities of C. Davis [Proc. Am. Math. Soc. 8, 42–44 (1957; Zbl 0080.10505)], F. Hansen and G.-K. Pedersen [Math. Ann. 258, 229–241 (1982; Zbl 0473.47011)] and B. Mond and J. E. Pečarić [Houston J. Math. 21, No. 4, 739–754 (1995; Zbl 0846.47015)]. Moreover, the authors prove different types of converse inequalities.

MSC:

47A63 Linear operator inequalities
39B62 Functional inequalities, including subadditivity, convexity, etc.
26A51 Convexity of real functions in one variable, generalizations
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