Abstract
In this paper, we study some complementary inequalities to Jensen’s inequality for self-adjoint operators, unital positive linear mappings, and real valued twice differentiable functions. New improved complementary inequalities are presented by using an improvement of the Mond-Pečarić method. These results are applied to obtain some inequalities with quasi-arithmetic means.
Highlights
Hausdorff space T equipped with a bounded Radon measure μ, andt∈T is a unital field of positive linear mappings φt : A → B from A to another unital C∗-algebra B
Wheret∈T is a bounded continuous field of self-adjoint operators in a C∗-algebra B(H) with spectra in [m, M] for some scalars m < M,t∈T is a unital field of positive linear mappings φt : B(H) → B(K), and φ ∈ C([m, M]) is a strictly monotone function
We obtain a generalization of known inequalities for a wider class of twice differentiable functions
Summary
Applying Lemma A, we obtain the following inequalities for twice differentiable functions. Applying Lemma 2 to a strictly convex function f , we improve inequalities (2) and (5). Remark 4 Applying Theorem 4 to strictly positive operators and the functions f (z) = zp and g(z) = zq, p, q ∈ R, we obtain an improvement of inequalities given in [7, Corollary 7].
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