Jensen’s operator inequality for functions of two variables

Hansen, Frank

doi:10.1090/s0002-9939-97-04003-3

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2000

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Cited by 7 publications

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Operator Inequalities Associated with Jensen’s Inequality

Hansen

2000

Survey on Classical Inequalities

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Abstract. We give a survey of various operator inequalities associated with Jensen's inequality and study the class of operator convex functions of several variables. Related questions are considered. Jensen's classical inequalityLet I be a real interval. A function f : I → R is said to be convex, iffor all t, s ∈ I and every λ ∈ [0, 1]. Notice that the definition, in order to be meaningful, requires that f can be evaluated in λt + (1 − λ)s, or equivalently that I is convex. But this is satisfied because the convex subsets of R are the intervals. If f satisfies (1.1) just for λ = 1/2, then f is said to be midpoint convex. It is easy to establish that a continuous and mid-point convex function is convex. The geometric interpretation of (1.1) is that the graph of f is below the chord and consequently above the extensions of the chord. This entails that a convex function defined on an open interval is continuous. Condition (1.1) can be reformulated asfor all mutually different numbers t, s, r ∈ I. If we for such numbers define the divided difference [ts] of f taken in the points t, s as

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