Abstract
In this paper we establish several Jensen-type operator inequalities for a class of superquadratic functions and self-adjoint operators. Our results are given in the so-called external form. As an application, we give improvements of the H?lder-McCarthy inequality and the classical discrete and integral Jensen inequality in the corresponding external forms. In addition, the established Jensen-type inequalities are compared with the previously known results and we show that our results provide more accurate estimates in some general settings.
Highlights
AND PRELIMINARIESThe famous Jensen inequality in its basic form asserts that if f : J ⊆ R → R is a convex function, the inequality f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) holds for all x, y ∈ J and λ ∈ [0, 1]
The Jensen-type operator inequalities are accompanied by an operator convex function
Some Jensen-type operator inequalities are accompanied by a usual convex function
Summary
AND PRELIMINARIESThe famous Jensen inequality in its basic form asserts that if f : J ⊆ R → R is a convex function, the inequality f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) holds for all x, y ∈ J and λ ∈ [0, 1]. Inner product, positive operator, superquadratic function, external form. [9, Theorem 2.1] If f : [0, ∞) → R is a continuous superquadratic function, the inequality
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