New refinements of the discrete Jensen’s inequality generated by finite or infinite permutations

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.


Introduction
Different variants of Jensen's inequality and other inequalities have their origin in the notion of convexity. A real function f defined on a convex subset C of a real vector space is called convex if it satisfies for all v 1 , v 2 ∈ C and all α ∈ [0, 1].
The set of positive integers will be denoted by N + . The following versions of Jensen's inequality are well known. Theorem 1.1. (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. If p 1 , . . . , p n are nonnegative numbers with n i=1 p i = 1, and v 1 , . . . , v n ∈ C, then (1.1) (b) Let C be a closed convex subset of a real Banach space V , and let f : C → R be a convex function. If p 1 , p 2 , . . . are nonnegative numbers with To give refinements of the discrete Jensen's inequality (1.1) is an extensively investigated theme with numerous methods and results (see e.g. the book [8] and references therein), and applications (see e.g. [5] and [6]). Then again, to the best of my knowledge, there are no refinements of the discrete Jensen's inequality (1.2) in such generality. There are some refinements of (1.2) when C is an interval of R: either one estimates formulas in (1.2) in a suitable way (see [12]) or one can obtain results from refinements of integral Jensen's inequality (see [7]).
The following refinement of (1.1) can be found in [9] (see also [1] It is easy to think that the previous result cannot be generalized for infinite sums, but we can observe that the middle term C dis in (1.3) can be rewritten in the following form where π j (j = 1, . . . , k) is the (j − 1)-cyclic permutation of the set {1, . . . , n} to the right (all elements are moved to the right j − 1 times with elements overflowing from the right being inserted to the left).
In this paper we show that formulas like (1. itself. On the one hand, 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. Finally, we give some applications concerning information theory, the norm function, Hölder's inequality and the inequality of arithmetic and geometric means.

Main results
The positive part f + and the negative part f − of a real valued function f are defined in the usual way. Let the set I denote either {1, . . . , n} for some n ≥ 1 or N + . We say that the numbers (p i ) i∈I represent a (positive) discrete probability distribution if (p i > 0) p i ≥ 0 (i ∈ I) and i∈I p i = 1. A permutation π of I refers to a bijection from I onto itself.
We need the following hypotheses which are partitioned into two classes: (H 1 ) Let k, n ≥ 2 be integers, and let p 1 , . . . , p n and λ 1 , . . . , λ k represent positive probability distributions.
. . and (λ j ) j∈J represent positive probability distributions. (C 2 ) For each j ∈ J let π j be a permutation of the set N + . (C 3 ) Let C be a closed convex subset of a real Banach space (V, · ), and f : C → R be a convex function.
Proof. (a) By using Theorem 1.1 (a) and the fact that π j is a permutation of the set {1, . . . , n}, The left hand side inequality can be proved similarly. Since the discrete Jensen's inequality implies that The proof is divided into four parts. I. We first prove that the series Vol. 94 (2020) New refinements of the discrete Jensen's inequality 1113 is a rearrangement of the absolutely convergent series ∞ i=1 p i v i , and hence it is also absolutely convergent and (ii) Assume J = N + . The property of absolute convergence of (2.6) implies that Therefore, as it is well known, the order of summation can be interchanged in the double sum and hence the series is also absolutely convergent.
(2.9) We have seen in part I that the series (2.4) is convergent and its sum is Thus by (2.8) or by (2.9), the second inequality in (2.2) will be proved if we succeed in showing that the series is convergent. It is known that the positive part of f is also convex. The convergence of (2.4) implies the convergence of the series It now follows that we can copy the proofs of (2.8) and (2.9) with f + instead of f . Taking account of the nonnegativity of f + , we obtain Vol. 94 (2020) New refinements of the discrete Jensen's inequality 1115 From this it follows that the series (2.10) will be convergent (absolutely) if and only if III. In this step we show that the series (2.12) is convergent and the first inequality in (2.2) holds assuming f is bounded below, that is the series (2.12) is convergent. According to (2.11) and (2.12) the series (2.10) is absolutely convergent. Since the series (2.3) and (2.10) are absolutely convergent and (2.5) holds, we can apply Theorem 1.1 (b), and obtain the first inequality in (2.2).
IV. At this point we abandon the lower boundedness hypothesis on f .
Let the function f n : C → R be defined by Then f n (n ∈ N + ) is convex and bounded below, and f n ≥ f (n ∈ N + ). From this and from the results of part III, we get that for each n ∈ N + the series is absolutely convergent and .
Since the sequence (f n ) n∈N+ is decreasing and lim n→∞ f n = f pointwise, the previous two assertions imply that B. Levi's theorem can be applied, and it gives that the series (2.10) is absolutely convergent and the first inequality in (2.2) holds.
The proof is complete.

Applications
We begin with some inequalities corresponding to information theory.
The following notion was introduced by Csiszár in [2] and [3]. tional is Based on this concept, we have introduced a new functional in [10], and this functional can be further generalized: Definition 3.2. Let C be a convex subset of a real vector space V , and f : C → R be a convex function. If w := (w 1 , . . . , w n ) ∈ V n and q := (q 1 , . . . , then define Proof. By applying Theorem 2.1 (a) with Vol. 94 (2020) New refinements of the discrete Jensen's inequality 1117 The proof is complete.
From Proposition 3.3 we can obtain the following refinement of this inequality: