Operator Inequalities Associated with Jensen’s Inequality

Abstract

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

On regularization algorithms in learning theory

“…Thus, condition (18) is not restrictive at all. In fact it is automatically satisfied for the best a priori choice of the regularization parameter (see Theorem 14) balancing the values of the terms in the estimates (19) and (20). Finally, the condition < 1 is considered only to simplify the results and can be replaced by < a for some positive constant a (and in particular for a = ) that would eventually appear in the bound.…”

“…In general, the smoothness expressed through source conditions is not stable with respect to perturbations in the involved operator T. In Learning Theory only the empirical covariance operator T x is available [19]) that each operator monotone function on (0, b) admits an analytic continuation in the corresponding strip of the upper half-plane with positive imaginary part. Important implications of the concept of operator monotonicity in the context of regularization can be seen from the following result (see [20,22]).…”

“…It simply indicates the range of regularization parameters, for which the error estimates (19) and (20) are non-trivial. For example, if does not satisfy (18) then right-hand side of (20) becomes larger than a fixed constant C 4 /(2 √ 2 2 ), which is not reasonable.…”

Matrix operator inequalities based on Mond, Pecaric, Jensen and Ando's

“…Beesack and Pĕcarić in [1] showed the Jensen's inequality for real valued convex functions. Hansen in [6] also gave operator inequalities associated with Jensen's inequality. In this article, we will give some new operator inequalities of matrix version based on Mond, Pecaric, Jensen and Ando's.…”

“…In early of study operator inequalities, there have been many research results, where useful references centered mainly literatures [1][2][3][4]. In nearly 20 years, some new research results of operator inequalities continuously appears, readers may refer to Mond and Pecaric [5], Beesack [1], Hansen [6] and Kubo and Ando [2] and so on. In [5], Mond Pecaric gave several matrix operator inequalities associated with positive linear maps by means of concavity and convexity theorems.…”