島根大学総合理工学研究科
ISSN:1342-7121
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Use this link to cite this item : https://ir.lib.shimane-u.ac.jp/52526
Memoirs of the Graduate School of Science and Engineering, Shimane University. Series B, Mathematics 54
2021-01-30 発行
REVERSE JENSEN INTEGRAL INEQUALITIES FOR CONVEX FUNCTIONS AND POSITIVE LINEAR MAPS IN C∗-ALGEBRAS
DRAGOMIR, S. S.
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Let A and B be unital C∗ -algebras. In this paper we obtain several operator inequalities providing upper bounds for the difference
∫Tϕt (f (xt)) dμ (t) -f(∫Tϕt (xt) dμ (t)),
where f : I ! R is a convex function defined on an interval I , (ϕt)t∈T is a unital field of positive linear mappings ϕt : A ! B defined on a locally compact Hausdorff space T with a bounded Radon measure μ and (xt)t∈T is a bounded continuous field of selfadjoint elements in A with spectra contained in I. Several Hermite-Hadamard type inequalities are given. Some examples for convex and operator convex functions are also provided.
∫Tϕt (f (xt)) dμ (t) -f(∫Tϕt (xt) dμ (t)),
where f : I ! R is a convex function defined on an interval I , (ϕt)t∈T is a unital field of positive linear mappings ϕt : A ! B defined on a locally compact Hausdorff space T with a bounded Radon measure μ and (xt)t∈T is a bounded continuous field of selfadjoint elements in A with spectra contained in I. Several Hermite-Hadamard type inequalities are given. Some examples for convex and operator convex functions are also provided.