In this paper, we establish a new refinement of the left-hand side of
Hermite-Hadamard inequality for convex functions of several variables defined
on simplices

Nowicka, Monika

Witkowski, Alfred

Journal of Inequalities and Applications

English

arXiv

Monika Nowicka

Alfred Witkowski

Crossref

Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 DOI 10.1186/s13660-015-0904-0RESEARCH Open AccessA reﬁnement of the left-hand side ofHermite-Hadamard inequality for simplicesMonika Nowicka* and Alfred Witkowski*Correspondence:monika.nowicka@utp.edu.plInstitute of Mathematics andPhysics, UTP University of Scienceand Technology, Al. Prof.S. Kaliskiego 7, Bydgoszcz, 85-796,PolandAbstractIn this paper, we establish a new reﬁnement of the left-hand side ofHermite-Hadamard inequality for convex functions of several variables deﬁned onsimplices.MSC: Primary 26D15Keywords: Hermite-Hadamard inequality; convex function; barycentric coordinates1 Introduction, deﬁnitions, and notationsThe classical Hermite-Hadamard inequality [] states that if a function f : [a,b] → R isconvex, thenf(a + b)≤ b – a∫ baf (t) dt ≤ f (a) + f (b) .This inequality has been discussed bymanymathematicians.We refer to [–] and the ref-erences therein. In the last few decades, several generalizations of the Hermite-Hadamardinequality have been established and studied. One of them [–] says that if  ⊂ Rn is asimplex with barycenter b and vertices x, . . . ,xn and f :  →R is convex, thenf (b)≤ Vol()∫f (x) dx ≤ f (x) + · · · + f (xn)n +  . (.)Wąsowicz and Witkowski in [] and Mitroi and Spiridon in [] investigated the relation-ship between the left- and right-hand sides of (.).An interesting reﬁnement of both inequalities in (.) was obtained by Raïssouli andDragomir in []. In this paper we use their method to obtain another reﬁnement of theleft-hand side of Hermite-Hadamard inequality on simplices.Before we formulate the main theorem of this paper, we ﬁrst give some deﬁnitions andnotations. For a ﬁxed natural number n≥  letN = {, , . . . ,n}. Suppose x, . . . ,xn ∈Rn aresuch that the vectors #     »xxi, i = , . . . ,n are linearly independent. The set = conv{xi : i ∈N}is called a simplex. Such a simplex is an n-dimensional object andwe shall call it sometimesan n-simplex if we would like to emphasize its dimension. The pointb = n + (x + · · · + xn)© 2015 Nowicka and Witkowski. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna-tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commonslicense, and indicate if changes were made.Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 Page 2 of 6is called the barycenter of . For any subset K of N of cardinality k ≤ n we deﬁne an(n – k)-simplex [K ] as follows. For each j ∈N \K letx[K ]j =n + ∑i∈Kxi +n +  – kn +  xj (.)and[K ] = conv{x[K ]j : j ∈N \K}. (.)Obviously [∅] = and [K ] = b if cardN \K = .The integration over a k-dimensional simplex will be always with respect to thek-dimensional Lebesgue measure denoted by dx and the k-dimensional volume will bedenoted by Vol. There will be no ambiguity, as the dimension will be obvious from thecontext.The purpose of this paper is to prove that if f :  → R is convex and K ⊂ L N , thenthe average value of f on [L] does not exceed its average on [K ].By H(a,λ) : Rn → Rn we denote the homothety with center a and scale λ, given by theformulaH(a,λ)(x) = a + λ(x – a).2 Reﬁnement of the left-hand sideThis is the main result of our paper.Theorem. Let n ∈N and x, . . . ,xn ∈Rn be such that the vectors #     »xxi, i = , . . . ,n are lin-early independent. If f : conv{x, . . . ,xn} → R is a convex function and K ⊂ L {, . . . ,n},thenVol[L]∫[L]f (x) dx ≤ Vol[K ]∫[K ]f (x) dx,where [K ] := conv{ n+∑i∈K xi + n+–kn+ xj : j ∈ {, . . . ,n} \K} and k is the cardinality of K .Given the remark stated after equation (.) it is clear that Theorem . reﬁnes the left-hand side of (.).Let us begin with two observations, which will make clear the nature of simplices [K ].The ﬁrst observation follows immediately from (.).Observation . All simplices [K ] have a common barycenter.Observation . If K ⊂ L  N and cardL = cardK + , then [L] arises from [K ] in thefollowing way:let l ∈ L \K and let [K ]l be the face of [K ] opposite to x[K ]l .Then[L] =H(x[K ]l ,n – cardKn +  – cardK)([K ]l).Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 Page 3 of 6Proof Assume, without loss of generality, that K = {, . . . ,k} and L = {} ∪K . Let k < s≤ n.By (.) the vertices of [L] arex[L]s =n + k∑i=xi +n – kn +  xs.Thenx[L]s =n + k∑i=xi +n + x +n – kn +  xs= n + k∑i=xi +n +  – kn +  x +n – kn +  xs –n – kn +  x= n + k∑i=xi +n +  – kn +  x +n – kn +  – k(n +  – kn +  xs –n +  – kn +  x)= x[K ] +n – kn +  – k(x[K ]s – x[K ] ) =H(x[K ] ,n – kn +  – k)(x[K ]s ). Let us brief on the approach proposed by Dragomir and Raïssouli in []. They con-structed the sequence of subsimplices of  as follows.Let b be the barycenter of . One can divide  into n +  subsimplicesDi = conv{x, . . . ,xi–,b,xi+, . . . ,xn}, i = , , . . . ,n.It is important to note that all these simplices have the same volume.Denote by D the set of simplices created this way. The set Dp+ is constructed by ap-plying the above procedure to all simplices inDp. Dragomir and Raïssouli proved that fora convex function f : →R one hasf (b)≤ cardDp∑δ∈Dpf (bδ)≤ cardDp+∑δ∈Dp+f (bδ)andlimp→∞cardDp∑δ∈Dpf (bδ) =Vol∫f (x) dx, (.)where bδ denotes the barycenter of δ.We shall use the above procedure to prove the main result of this paper.Proof of Theorem . Obviously it is enough to prove the result in the case cardK +  =cardL. As above we may assume K = {, . . . ,k} and L = {} ∪ K . Let  = [K ] denote theface of [K ] opposite to x[K ] . For simplicity denote by H the homothety with center x[K ]and scale n–kn+–k . Then, by Observation . we see that [L] =H().Let us apply the Dragomir-Raïssouli process to . Thus we obtain a sequence of sets ofsubsimplices of  denoted by Dp.Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 Page 4 of 6Fix p ≥ . For every σ ∈ Dp let σ = conv(σ ∪ {x[K ] }). Clearly the simplices σ form apartition of [K ] into simplices of the same height thus Volσ = Vol[K ]/ cardDp.Now we apply the left-hand side of the Hermite-Hadamard inequality to all simplicesσ to obtaincardDp∑σ∈Dpf (bσ )≤∑σ∈DpVolσ∫σf (x) dx = Vol[K ]∫[K ]f (x) dx. (.)Since [L] is the image of  by H , the setsH(Dp) ={H(σ ) : σ ∈Dp}form the Dragomir-Raïssouli sequence for [L]. Moreover, comme par miracle [], thebarycenters of σ and that of H(σ ) coincide, i.e.bσ = bH(σ ). (.)From (.) and (.) we concludecardDp∑σ∈Dpf (bH(σ ))≤ Vol[K ]∫[K ]f (x) dx, (.)and applying (.)limp→∞cardDp∑σ∈Dpf (bH(σ )) =Vol[L]∫[L]f (x) dx. (.)Now the assertion follows immediately from (.) and (.). From Theorem . we obtain the following corollary.Corollary . Let K,K, . . . ,Kn be a sequence of subsets of N such thatK ⊂ K ⊂ · · · ⊂ Kn and cardKi = i, i = , , . . . ,n.If f : →R is convex, thenf (b) = Vol([Kn])∫[Kn]f (x) dx ≤ Vol([Kn–])∫[Kn–]f (x) dx≤ · · · ≤ Vol([K])∫[K]f (x) dx≤ Vol([K])∫[K]f (x) dx= Vol()∫f (x) dx(note that Vol([Ki]) denotes (n – i)-dimensional volume and∫[Ki] · · · dx denotes integra-tion with respect to (n – i)-dimensional Lebesgue measure).Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 Page 5 of 6Applying Theorem . to all possible proper subsets of N of the same cardinality andsumming the obtained inequalities, we obtain the following corollary.Corollary . If f :  →R is a convex function, thenVol∫f (x) dx≥ (n+k) ∑KNcardK=kVol[K ]∫[K ]f (x) dx.From Theorem . we can derive the following corollary.Corollary . Let f :  →R be a convex function and let k < l ≤ n. Then(n+l) ∑LcardL=lVol[L]∫[L]f (x) dx≤ (n+k) ∑KcardK=kVol[K ]∫[K ]f (x) dx.Proof Clearly it is suﬃcient to prove the corollary only in the case l = k + . Fix K ={, . . . ,k}. We have n +  – k oversets of K of cardinality k + . Applying Theorem . toK and all such oversets and summing the obtained inequalities, we deduce∑L⊃KcardL=k+Vol[L]∫[L]f (x) dx ≤ (n +  – k) Vol[K ]∫[K ]f (x) dx.Summing this over all possible K , we obtain(k + )∑LcardL=k+Vol[L]∫[L]f (x) dx≤ (n +  – k)∑KcardK=kVol[K ]∫[K ]f (x) dx,since every L has k + subsets of cardinality k. We complete the proof by multiplying bothsides by k!(n–k)!(n+)! . Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAW came up with an idea, MN extended it and performed all necessary calculations. All authors read and approved theﬁnal manuscript.Received: 10 September 2015 Accepted: 21 November 2015References1. Hadamard, J: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann.J. Math. Pures Appl. 58, 171-215 (1893)2. Dragomir, SS, Pearce, CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIAMonographs. Victoria University, Melbourne (2000). http://ajmaa.org/RGMIA/monographs.php3. Niculescu, CP, Persson, LE: Old and new on the Hermite-Hadamard inequality. Real Anal. Exch. 29, 663-686(2003/2004)4. Niculescu, CP, Persson, LE: Convex Functions and Their Applications: A Contemporary Approach. Springer, New York(2006)5. Neuman, E: Inequalities involving multivariate convex functions II. Proc. Am. Math. Soc. 109(4), 965-974 (1990)6. Niculescu, CP: The Hermite-Hadamard inequality for convex functions of a vector variable. Math. Inequal. Appl. 5(4),619-623 (2002)7. Bessenyei, M: The Hermite-Hadamard inequality on simplices. Am. Math. Mon. 115(4), 339-345 (2008)8. Wa˛sowicz, S, Witkowski, A: On some inequality of Hermite-Hadamard type. Opusc. Math. 32(3), 591-600 (2012)Nowicka and Witkowski Journal of Inequalities and Applications  (2015) 2015:373 Page 6 of 69. Mitroi, FC, Spiridon, CI: Reﬁnements of Hermite-Hadamard inequality on simplices. Math. Rep. (Bucur.) 15(65), 69-78(2013)10. Raïssouli, M, Dragomir, SS: Reﬁning recursively the Hermite-Hadamard inequality on a simplex. Bull. Aust. Math. Soc.92(1), 57-67 (2015)11. Prévert, J: Histoires et d’autres histoires. Gallimard, Paris (1963)

A refinement of the left-hand side of Hermite-Hadamard inequality for simplices

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A refinement of the left-hand side of Hermite-Hadamard inequality for simplices

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