Abstract
In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.
Highlights
Different variants of Jensen’s inequality and other inequalities have their origin in the notion of convexity
Theorem 1.1. (a) Let C be a convex subset of a real vector space V, and let f : C → R be a convex function
(H2) For each j = 1, . . . , k let πj be a permutation of the set {1, . . . , n}. (H3) Let C be a convex subset of a real vector space V, and f : C → R be a convex function. (C1) Let the set J denote either {1, . . . , k} for some k ≥ 2 or N+
Summary
Different variants of Jensen’s inequality and other inequalities have their origin in the notion of convexity. (discrete Jensen’s inequalities, see [11] and [13]) (a) Let C be a convex subset of a real vector space V , and let f : C → R be a convex function. AEM (b) Let C be a closed convex subset of a real Banach space V , and let f : C → R be a convex function. Again, to the best of my knowledge, there are no refinements of the discrete Jensen’s inequality (1.2) in such generality. C is a convex subset of a real vector space V , f : C → R is a convex function, and v1, . An essential generalization of Theorem 1.2 is given, on the other hand, refinements of (1.2) are developed without assuming that V is a special Banach space. We give some applications concerning information theory, the norm function, Holder’s inequality and the inequality of arithmetic and geometric means
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