Hansen, Frank; Pečarić, Josip; Perić, Ivan Jensen’s operator inequality and its converses. (English) Zbl 1151.47025 Math. Scand. 100, No. 1, 61-73 (2007). 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. Reviewer: Mohammad Sal Moslehian (Mashhad) Cited in 3 ReviewsCited in 34 Documents MSC: 47A63 Linear operator inequalities 39B62 Functional inequalities, including subadditivity, convexity, etc. 26A51 Convexity of real functions in one variable, generalizations Keywords:operator convex; Jensen’s inequality; selfadjoint operator; mean; reverse inequality Citations:Zbl 0080.10505; Zbl 0473.47011; Zbl 0846.47015 PDFBibTeX XMLCite \textit{F. Hansen} et al., Math. Scand. 100, No. 1, 61--73 (2007; Zbl 1151.47025) Full Text: DOI arXiv