Improvements of the Hermite-Hadamard inequality for the simplex

Journal of Inequalities and Applications, Jan 2017

In this study, the simplex whose vertices are barycenters of the given simplex facets plays an essential role. The article provides an extension of the Hermite-Hadamard inequality from the simplex barycenter to any point of the inscribed simplex except its vertices. A two-sided refinement of the generalized inequality is obtained in completion of this work. MSC: 26B25, 52A40.

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Improvements of the Hermite-Hadamard inequality for the simplex

Pavic´ Journal of Inequalities and Applications Improvements of the Hermite-Hadamard inequality for the simplex Zlatko Pavic´ In this study, the simplex whose vertices are barycenters of the given simplex facets plays an essential role. The article provides an extension of the Hermite-Hadamard inequality from the simplex barycenter to any point of the inscribed simplex except its vertices. A two-sided refinement of the generalized inequality is obtained in completion of this work. MSC: 26B25; 52A40 2 Convex functions on the simplex The section is a review of the known results on the Hermite-Hadamard inequality for simplices, and it refers to its generic background. The main notification is concentrated in Lemma ., which is also the generalization of the Hermite-Hadamard inequality. © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. A · · · An+ = conv{A, . . . , An+}. The convex hull of n vertices is called the facet or (n – )-face of the given n-simplex. The analytic presentation of points of an n-simplex A = A · · · An+ in Rn arises from the n-volume by means of the Lebesgue measure or the Riemann integral. We will use the abbreviation vol instead of voln. Let A ∈ A be a point, and let Ai be the convex hull of the set containing the point A and vertices Aj for j = i, formally as Ai = conv{A, . . . , Ai–, A, Ai+, . . . , An+}. A = where we have the coefficients Let A, . . . , An+ ∈ Rn be points so that the points A – An+, . . . , An – An+ are linearly independent. The convex hull of the points Ai written in the form of A · · · An+ is called the n-simplex in Rn, and the points Ai are called the vertices. So, we use the denotation S = S x dμ(x) , . . . , S xn dμ(x) μ(S) μ(S) is called the μ-barycenter of the set S. In the above integrals, points x ∈ S are used as x = (x, . . . , xn). The μ-barycenter S belongs to the convex hull of S. When we use the Lebesgue measure, we say just barycenter. If S is closed and convex, then a μ-integrable continuous convex function f : S → R satisfies the inequality S x dμ(x) , . . . , S xn dμ(x) μ(S) μ(S) as a special case of Jensen’s inequality for multivariate convex functions; see the excellent McShane paper in []. If f is affine, then the equality is valid in (). A = A μx(Adμ) (x) , . . . , A μxn(Adμ) (x) = Then each convex function f : A → R satisfies the double inequality Proof We have three cases depending on the position of the μ-barycenter A within the simplex A. If A is an interior point of A, then we take a supporting hyperplane xn+ = h(x) at the graph point (A, f (A)), and the secant hyperplane xn+ = h(x) passing through the graph points (A, f (A)), . . . , (An+, f (An+)). Using the affinity of the functions h and h, we get We consider a convex function f defined on the n-simplex A = A · · · An+. The following lemma presents a basic inequality for a convex function on the simplex, and it refers to the connection of the simplex barycenter with simplex vertices. Lemma . Let μ be a positive measure on Rn. Let A = A · · · An+ be an n-simplex in Rn such that μ(A) > . Let A be the μ-barycenter of A, and let in=+ αiAi be its unique convex combination by means of = h(A) because h(Ai) = f (Ai). So, formula () works for the interior point A. If A is a relative interior point of a certain k-face where  ≤ k ≤ n – , then we can apply the previous procedure to the respective k-simplex. For example, if A · · · Ak+ is the observed k-face, then the coefficients α, . . . , αk+ are positive, and the coefficients αk+, . . . , αn+ are equal to zero. If A is a simplex vertex, suppose that A = A, then the trivial inequality f (A) ≤ f (A) ≤ f (A) represents formula (). More generally, if the μ-barycenter A lies in the interior of A, the inequality in formula () holds for all μ-integrable functions f : A → R that admit a supporting hyperplane at A, and satisfy the supporting-secant hyperplane inequality h(x) ≤ f (x) ≤ h(x) for every point x of the simplex A. A = Then each convex function f : A → R satisfies the double inequality A = nin=++Ai , and its use in formula () implies the Hermite-Hadamard inequality Corollary . Let A = A · · · An+ be an n-simplex in Rn, and let P, . . . , Pm ∈ A be points. Let A = jm= λjPj be a convex combination of the points Pj, and let in=+ αiAi be the unique convex combination of the vertices Ai such that Lemma . was obtained in [], Corollary , the case αi = /(n + ) was obtained in [], Theorem , and a similar result was obtained in [], Theorem .. By applying the Lebesgue measure or the Riemann integral in Lemma ., the condition in () gives the barycenter can be utilized in Lemma . to obtain the discrete inequality in formula (). Putting jm= λjPj instead of in=+ αiAi within the first term of formula (), we obtain the Jensen inequality extended to the right. Corollary . in the case αi = /(n + ) was obtained in [], Corollary . One of the most influential results of the theory of convex functions is the Jensen inequality (see [] and []), and among the most beautiful results is certainly the HermiteHadamard inequality (see [] and []). A significant generalization of the Jensen inequality for multivariate convex functions can be found in []. Improvements of the HermiteHadamard inequality for univariate convex functions were obtained in []. As for the Hermite-Hadamard inequality for multivariate convex functions, one may refer to [, , , –], and []. Figure 1 The inscribed simplex as the barycenter extension. 3 Main results Bi = and let B = B · · · Bn+ be the n-simplex of the vertices Bi. The simplices A and B in our three-dimensional space are tetrahedrons presented in Figure . Our aim is to extend the Hermite-Hadamard inequality to all points of the inscribed simplex B excepting its vertices. So, we focus on the non-peaked simplex B = B \ {B, . . . , Bn+}. follows. The last of the convex combinations A = Ai = Ci = confirms that the point A belongs to the simplex B. Let us assume that the point A belongs to the simplex B. Then we have the convex combination A = in=+ λiBi. Using equation () in the reverse direction, we get the convex combinations equality We need another subsimplex of A. Let A be a point belonging to the interior of A. In this case, the sets Ai defined by formula () are n-simplices. Let Ci stand for the barycenter of the simplex Ai by means of Lemma . Let A = A · · · An+ be an n-simplex in Rn, and let A = in=+ αiAi be a convex combination of the vertices Ai with coefficients αi satisfying αi > . The point A belongs to the non-peaked simplex C = C \ {C, . . . , Cn+} if and only if the coefficients αi satisfy the additional limitations αi ≤ /n. Proof Suppose that the coefficients αi satisfy  < αi ≤ /n. Let βi be the coefficients as in equation (). Using the trivial equality A = A/(n + ) + nA/(n + ), and the coefficient connections of equation (), we get A = i= i=j= n +  i= i=j= n +  indicating that the point A lies in the simplex C. To show that the convex combination in=+ βiCi does not represent any vertex, we will assume that some βi = . Then αi =  as opposed to the assumption that all αi are positive. The proof of the reverse implication goes exactly in the same way as in the proof of Lemma .. Each simplex C is homothetic to the simplex B. Namely, combining equations () and (), we can represent each vertex Ci by the convex combination Then it follows that Ci = n +  A + n n+  Bi. The point A used in the previous corollary lies in the interior of the simplex A because the coefficients αi are positive. In that case, the sets Ai are n-simplices, and they will be used in the main theorem that follows. Theorem . Let A = A · · · An+ be an n-simplex in Rn, let A = in=+ αiAi be a convex combination of the vertices Ai with coefficients αi satisfying  < αi ≤ /n, and let βi =  – nαi. Let Ai be the simplices defined by formula (). and each convex function f : A → R satisfies the double inequality Then each convex function f : A → R satisfies the double inequality Proof Using the convex combinations equality Jensen inequality to f ( in=+ βiCi), we get Summing the products of the Hermite-Hadamard inequalities for the function f on the simplices Ai and the coefficients βi, it follows that Repeating the procedure which was used for the derivation of formula (), we obtain the series of equalities of which the most important part is the double inequality in formula (). The inequality in formula () is a generalization and refinement of the HermiteHadamard inequality. Taking the coefficients αi = /(n + ), in which case βi = /(n + ), n +  i= n +  i= n +  i= n +  i= Bringing together all of the above, we obtain the multiple inequality can be called the μ-barycenter of the function g. It is about the following measure. Introducing the measure ν as we realize the five terms inequality where the second and fourth terms refine the Hermite-Hadamard inequality. The third term is generated from all of n +  simplices Ai. In the present case, these simplices have the same volume equal to vol(A)/(n + ). The inequality in formula () excepting the second term was obtained in [], Theorem . Similar inequalities concerning the standard n-simplex were obtained in [, ] and []. Special refinements of the left and right-hand side of the Hermite-Hadamard inequality were recently obtained in [] and []. 4 Generalization to the function barycenter If μ is a positive measure on Rn, if S ⊆ Rn is a measurable set, and if g : S → R is a nonnegative integrable function such that S g(x) dμ(x) > , then the integral mean point S = S xg(x) dμ(x) , . . . , S xng(x) dμ(x) S g(x) dμ(x) S g(x) dμ(x) S = Thus the μ-barycenter of the function g coincides with the ν-barycenter of its domain S . So, the barycenter S belongs to the convex hull of the set S . By using the unit function g(x) =  in formula (), it is reduced to formula (). Utilizing the function barycenter instead of the set barycenter, we have the following reformulation of Lemma .. Lemma . Let μ be a positive measure on Rn. Let A = A · · · An+ be an n-simplex in Rn, and let g : A → R be a nonnegative integrable function such that A g(x) dμ(x) > . Let A be the μ-barycenter of g, and let in=+ αiAi be its unique convex combination by means of A = AAxgg(x(x))ddμμ(x(x)) , . . . , AAxgng(x(x))ddμμ(x(x)) = Then each convex function f : A → R satisfies the double inequality The proof of Lemma . can be employed as the proof of Lemma . by using the measure ν in formula () or by utilizing the affinity of the hyperplanes h and h in the form of the equalities A xg(x) dμ(x) , . . . , A xng(x) dμ(x) A g(x) dμ(x) A g(x) dμ(x) Lemma . is an extension of the Fejér inequality (see []) to multivariable convex functions. As regards univariable convex functions, using the Lebesgue measure on R and a closed interval as -simplex in Lemma ., we get the following generalization of the Fejér inequality. Corollary . Let [a, b] be a closed interval in R, and let g : [a, b] → R be a nonnegative integrable function such that ab g(x) dx > . Let c be the barycenter of g, and let αa + βb be its unique convex combination by means of Then each convex function f : [a, b] → R satisfies the double inequality Fejér used a nonnegative integrable function g that is symmetric with respect to the midpoint c = (a + b)/. Such a function satisfies g(x) = g(c – x), and therefore (x – c)g(x) dx = . As a consequence it follows that Using the barycenters of the restrictions of g onto simplices Ai in formula (), we have the following generalization of Theorem .. let A = Then each convex function f : A → R satisfies the double inequality Proof The first step of the proof is to apply Lemma . to the functions f and gi on the simplex Ai in the way of in=+j= f (Aj) Ci = in=+j= f (Aj) containing the double inequality in formula (). with the barycenter Ci = (A + duces to the inequality in formula (). Competing interests The author declares that he has no competing interests. 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Zlatko Pavić. Improvements of the Hermite-Hadamard inequality for the simplex, Journal of Inequalities and Applications, 2017, 3, DOI: 10.1186/s13660-016-1273-z