Abstract
The purpose of this paper is to present some new versions of Hermite-Hadamard type inequalities for operator convex functions. We give refinements of Hermite-Hadamard type inequalities for convex functions of self-adjoint operators in a Hilbert space analogous to well-known inequalities of the same type. The results presented in this paper are more general than known results given by several authors. MSC:26D15, 47A63.
Highlights
Let f be a real-valued function defined on I ∈ R
Zabandan gave a refinement of the Hermite-Hadamard inequality for convex functions in [ ]
The first using the definition of operator convex functions and the second using the Hermite-Hadamard inequality for real-valued functions
Summary
Let f be a real-valued function defined on I ∈ R. Let f : [a, b] → R be a convex function and a, b ∈ R, with a < b, the inequality f a + b ≤ b f (x) dx ≤ f (a) + f (b) , a, b ∈ R, Is known in the literature as the Hermite-Hadamard inequality for convex functions, see [ ]. A real-valued continuous function f on an interval I is said to be operator convex (operator concave) if f ( – λ)A + λB ≤ (≥)( – λ)f (A) + λf (B)
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