Abstract
There are a lot of papers dealing with applications of the so-called cyclic refinement of the discrete Jensen’s inequality. A significant generalization of the cyclic refinement, based on combinatorial considerations, has recently been discovered by the author. In the present paper we give the integral versions of these results. On the one hand, a new method to refine the integral Jensen’s inequality is developed. On the other hand, the result contains some recent refinements of the integral Jensen’s inequality as elementary cases. Finally some applications to the Fejér inequality (especially the Hermite–Hadamard inequality), quasi-arithmetic means, and f-divergences are presented.
Highlights
The significance of convex functions is rightly due to Jensen’s inequality
We say that the numbersi∈I represent a discrete probability distribution if pi ≥ 0 (i ∈ I) and i∈I pi = 1
Theorem 1 (a) Let C be a convex subset of a real vector space V, and let f : C → R be a convex function
Summary
The significance of convex functions is rightly due to Jensen’s inequality. A real function f defined on an interval C ⊂ R is called convex if it satisfies f αt1 + (1 – α)t2 ≤ αf (t1) + (1 – α)f (t2). Theorem 1 (discrete Jensen’s inequalities, see [16] and [17]) (a) Let C be a convex subset of a real vector space V , and let f : C → R be a convex function. Theorem 2 (integral Jensen’s inequality, see [16]) Let φ be an integrable function on a probability space (X, A, μ) taking values in an interval C ⊂ R. Let (pi)i∈I and (λj)j∈J represent positive probability distributions. Lemma 6 Let φ be an integrable function on a probability space (X, A, μ) taking values in an interval C ⊂ R. (pi)i∈I represents a positive probability distribution, and by the integral Jensen’s inequality, vi ∈ C (i ∈ I).
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