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Improvements of the Hermite-Hadamard inequality for the simplex

Journal of Inequalities and Applications20172017:3

https://doi.org/10.1186/s13660-016-1273-z

  • Received: 29 August 2016
  • Accepted: 6 December 2016
  • Published:

Abstract

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.

Keywords

  • convex combination
  • simplex
  • the Hermite-Hadamard inequality

MSC

  • 26B25
  • 52A40

1 Introduction

A concise approach to the concept of affinity and convexity is as follows. Let \(\mathbb{X}\) be a linear space over the field \(\mathbb{R}\). Let \(P_{1},\ldots,P_{m}\in\mathbb{X}\) be points, and let \(\lambda_{1},\ldots ,\lambda_{m}\in\mathbb{R}\) be coefficients. A linear combination
$$ \sum_{j=1}^{m} \lambda_{j}P_{j} $$
(1)
is affine if \(\sum_{j=1}^{m}\lambda_{j}=1\). An affine combination is convex if all coefficients \(\lambda_{j}\) are nonnegative.

Let \(\mathcal{S}\subseteq\mathbb{X}\) be a set. The set containing all affine combinations of points of \(\mathcal{S}\) is called the affine hull of the set \(\mathcal{S}\), and it is denoted with \(\operatorname{aff}\mathcal{S}\). A set \(\mathcal{S}\) is affine if \(\mathcal{S}=\operatorname{aff}\mathcal{S}\). Using the adjective convex instead of affine, and the prefix conv instead of aff, we obtain the characterization of the convex set.

A convex function \(f:\operatorname{conv}\mathcal{S}\to\mathbb{R}\) satisfies the Jensen inequality
$$ f \Biggl(\sum_{j=1}^{m} \lambda_{j}P_{j} \Biggr) \leq\sum _{j=1}^{m}\lambda_{j}f(P_{j}) $$
(2)
for all convex combinations of points \(P_{j}\in\mathcal{S}\). An affine function \(f:\operatorname{aff}\mathcal{S}\to\mathbb{R}\) satisfies the equality in equation (2) for all affine combinations of points \(P_{j}\in\mathcal{S}\).

Throughout the paper, we use the n-dimensional space \(\mathbb {X}=\mathbb{R}^{n}\) over the field \(\mathbb{R}\).

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 2.1, which is also the generalization of the Hermite-Hadamard inequality.

Let \(A_{1},\ldots,A_{n+1}\in\mathbb{R}^{n}\) be points so that the points \(A_{1}-A_{n+1},\ldots,A_{n}-A_{n+1}\) are linearly independent. The convex hull of the points \(A_{i}\) written in the form of \(A_{1} \cdots A_{n+1}\) is called the n-simplex in \(\mathbb{R}^{n}\), and the points \(A_{i}\) are called the vertices. So, we use the denotation
$$ A_{1}\cdots A_{n+1}=\operatorname{conv} \{A_{1},\ldots,A_{n+1}\}. $$
(3)
The convex hull of n vertices is called the facet or \((n-1)\)-face of the given n-simplex.

The analytic presentation of points of an n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\) in \(\mathbb{R}^{n}\) arises from the n-volume by means of the Lebesgue measure or the Riemann integral. We will use the abbreviation vol instead of \(\operatorname{vol}_{n}\).

Let \(A\in\mathcal{A}\) be a point, and let \(\mathcal{A}_{i}\) be the convex hull of the set containing the point A and vertices \(A_{j}\) for \(j\neq i\), formally as
$$ \mathcal{A}_{i}=\operatorname{conv}\{A_{1}, \ldots,A_{i-1},A,A_{i+1},\ldots ,A_{n+1}\}. $$
(4)
Each \(\mathcal{A}_{i}\) is a facet or n-subsimplex of \(\mathcal{A}\), so \(\operatorname{vol}(\mathcal{A}_{i})=0\) or \(0<\operatorname{vol}(\mathcal {A}_{i})\leq\operatorname{vol}(\mathcal{A})\), respectively. The sets \(\mathcal{A}_{i}\) satisfy \(\mathcal{A}=\bigcup_{i=1}^{n+1}\mathcal{A}_{i}\) and \(\operatorname{vol}(\mathcal{A}_{i}\cap\mathcal{A}_{j})=0\) for \(i\neq j\), and so it follows that \(\operatorname{vol}(\mathcal{A})=\sum_{i=1}^{n+1}\operatorname{vol}(\mathcal{A}_{i})\).
The point A can be uniquely represented as the convex combination of the vertices \(A_{i}\) by means of
$$ A=\sum_{i=1}^{n+1} \alpha_{i}A_{i}, $$
(5)
where we have the coefficients
$$ \alpha_{i}=\frac{\operatorname{vol}(\mathcal{A}_{i})}{\operatorname{vol}(\mathcal{A})}. $$
(6)
If the point A belongs to the interior of the n-simplex \(\mathcal {A}\), then all sets \(\mathcal{A}_{i}\) are n-simplices, and consequently all coefficients \(\alpha_{i}\) are positive. Furthermore, the reverse implications are valid.
If μ is a positive measure on \(\mathbb{R}^{n}\), and if \(\mathcal {S}\subseteq\mathbb{R}^{n}\) is a measurable set such that \(\mu(\mathcal{S})>0\), then the integral mean point
$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\mu(x)}{\mu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\mu(x)}{\mu(\mathcal {S})} \biggr) $$
(7)
is called the μ-barycenter of the set \(\mathcal{S}\). In the above integrals, points \(x\in\mathcal{S}\) are used as \(x=(x_{1},\ldots ,x_{n})\). The μ-barycenter S belongs to the convex hull of \(\mathcal{S}\). When we use the Lebesgue measure, we say just barycenter. If \(\mathcal{S}\) is closed and convex, then a μ-integrable continuous convex function \(f:\mathcal{S}\to\mathbb{R}\) satisfies the inequality
$$ f \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\mu(x)}{\mu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\mu(x)}{\mu(\mathcal {S})} \biggr) \leq \frac{\int_{\mathcal{S}}f(x) \,d\mu(x)}{\mu(\mathcal{S})} $$
(8)
as a special case of Jensen’s inequality for multivariate convex functions; see the excellent McShane paper in [1]. If f is affine, then the equality is valid in (8).

We consider a convex function f defined on the n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\). 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 2.1

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\) such that \(\mu(\mathcal{A})>0\). Let A be the μ-barycenter of \(\mathcal {A}\), and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be its unique convex combination by means of
$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1} \,d\mu(x)}{\mu(\mathcal {A})},\ldots, \frac{\int_{\mathcal{A}}x_{n} \,d\mu(x)}{\mu(\mathcal {A})} \biggr) = \sum_{i=1}^{n+1}\alpha_{i}A_{i}. $$
(9)
Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)\leq \frac{\int_{\mathcal{A}}f(x) \,d\mu(x)}{\mu(\mathcal{A})} \leq\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(10)

Proof

We have three cases depending on the position of the μ-barycenter A within the simplex \(\mathcal{A}\).

If A is an interior point of \(\mathcal{A}\), then we take a supporting hyperplane \(x_{n+1}=h_{1}(x)\) at the graph point \((A,f(A))\), and the secant hyperplane \(x_{n+1}=h_{2}(x)\) passing through the graph points \((A_{1},f(A_{1})),\ldots,(A_{n+1},f(A_{n+1}))\). Using the affinity of the functions \(h_{1}\) and \(h_{2}\), we get
$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) =& h_{1}(A) = \frac{\int_{\mathcal{A}}h_{1}(x) \,d\mu(x)}{\mu(\mathcal{A})} \\ \leq& \frac{\int_{\mathcal{A}}f(x) \,d\mu(x)}{\mu(\mathcal{A})} \\ \leq& \frac{\int_{\mathcal{A}}h_{2}(x) \,d\mu(x)}{\mu(\mathcal{A})} = h_{2}(A) \\ =& \sum_{i=1}^{n+1}\alpha_{i}h_{2}(A_{i}) = \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}) \end{aligned}$$
(11)
because \(h_{2}(A_{i})=f(A_{i})\). So, formula (10) works for the interior point A.

If A is a relative interior point of a certain k-face where \(1 \leq k \leq n-1\), then we can apply the previous procedure to the respective k-simplex. For example, if \(A_{1}\cdots A_{k+1}\) is the observed k-face, then the coefficients \(\alpha_{1},\ldots,\alpha_{k+1}\) are positive, and the coefficients \(\alpha_{k+2},\ldots,\alpha_{n+1}\) are equal to zero.

If A is a simplex vertex, suppose that \(A=A_{1}\), then the trivial inequality \(f(A_{1})\leq f(A_{1})\leq f(A_{1})\) represents formula (10). □

More generally, if the μ-barycenter A lies in the interior of \(\mathcal{A}\), the inequality in formula (10) holds for all μ-integrable functions \(f:\mathcal{A}\to\mathbb{R}\) that admit a supporting hyperplane at A, and satisfy the supporting-secant hyperplane inequality
$$ h_{1}(x) \leq f(x) \leq h_{2}(x) $$
(12)
for every point x of the simplex \(\mathcal{A}\).

Lemma 2.1 was obtained in [2], Corollary 1, the case \(\alpha_{i}=1/(n+1)\) was obtained in [3], Theorem 2, and a similar result was obtained in [4], Theorem 2.4.

By applying the Lebesgue measure or the Riemann integral in Lemma 2.1, the condition in (9) gives the barycenter
$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1} \,dx}{\operatorname{vol}(\mathcal {A})},\ldots, \frac{\int_{\mathcal{A}}x_{n} \,dx}{\operatorname{vol}(\mathcal{A})} \biggr) = \frac{\sum_{i=1}^{n+1}A_{i}}{n+1}, $$
(13)
and its use in formula (10) implies the Hermite-Hadamard inequality
$$ f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr)\leq \frac{\int_{\mathcal{A}}f(x) \,dx}{\operatorname{vol}(\mathcal{A})} \leq \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1}. $$
(14)
The above inequality was introduced by Neuman in [5]. An approach to this inequality can be found in [6].

The discrete version of Lemma 2.1 contributes to the Jensen inequality on the simplex.

Corollary 2.2

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(P_{1},\ldots,P_{m}\in\mathcal{A}\) be points. Let \(A=\sum_{j=1}^{m}\lambda_{j}P_{j}\) be a convex combination of the points \(P_{j}\), and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be the unique convex combination of the vertices \(A_{i}\) such that
$$ A=\sum_{j=1}^{m} \lambda_{j}P_{j}=\sum_{i=1}^{n+1} \alpha_{i}A_{i}. $$
(15)
Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq\sum _{j=1}^{m}\lambda_{j}f(P_{j}) \leq\sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(16)

Proof

The discrete measure μ concentrated at the points \(P_{j}\) by the rule
$$ \mu \bigl(\{P_{j}\} \bigr)=\lambda_{j} $$
(17)
can be utilized in Lemma 2.1 to obtain the discrete inequality in formula (16). □

Putting \(\sum_{j=1}^{m}\lambda_{j}P_{j}\) instead of \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) within the first term of formula (16), we obtain the Jensen inequality extended to the right.

Corollary 2.2 in the case \(\alpha_{i}=1/(n+1)\) was obtained in [3], Corollary 4.

One of the most influential results of the theory of convex functions is the Jensen inequality (see [7] and [8]), and among the most beautiful results is certainly the Hermite-Hadamard inequality (see [9] and [10]). A significant generalization of the Jensen inequality for multivariate convex functions can be found in [1]. Improvements of the Hermite-Hadamard inequality for univariate convex functions were obtained in [11]. As for the Hermite-Hadamard inequality for multivariate convex functions, one may refer to [2, 4, 5, 1216], and [17].

3 Main results

Throughout the section, we will use an n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\) in the space \(\mathbb{R}^{n}\), and its two n-subsimplices which will be denoted with \(\mathcal{B}\) and \(\mathcal{C}\).

Let \(B_{i}\) stand for the barycenter of the facet of \(\mathcal{A}\) not containing the vertex \(A_{i}\) by
$$ B_{i}=\frac{\sum_{i \neq j=1}^{n+1}A_{j}}{n}, $$
(18)
and let \(\mathcal{B}=B_{1}\cdots B_{n+1}\) be the n-simplex of the vertices \(B_{i}\).
The simplices \(\mathcal{A}\) and \(\mathcal{B}\) in our three-dimensional space are tetrahedrons presented in Figure 1. Our aim is to extend the Hermite-Hadamard inequality to all points of the inscribed simplex \(\mathcal{B}\) excepting its vertices. So, we focus on the non-peaked simplex \(\mathcal{B}'=\mathcal{B}\setminus\{B_{1},\ldots,B_{n+1}\}\).
Figure 1
Figure 1

The inscribed simplex as the barycenter extension.

Lemma 3.1

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\).

The point A belongs to the n-simplex \(\mathcal{B}=B_{1}\cdots B_{n+1}\) if and only if the coefficients \(\alpha_{i}\) satisfy \(\alpha _{i}\leq1/n\).

The point A belongs to the non-peaked simplex \(\mathcal{B}'=\mathcal {B}\setminus\{B_{1},\ldots,B_{n+1}\}\) if and only if the coefficients \(\alpha_{i}\) satisfy \(0<\alpha_{i}\leq1/n\).

Proof

The first statement, relating to the simplex \(\mathcal{B}\), will be covered as usual by proving two directions.

Let us assume that the coefficients \(\alpha_{i}\) satisfy the limitations \(\alpha_{i}\leq1/n\). Then the coefficients
$$ \beta_{i}=1-n\alpha_{i} $$
(19)
are nonnegative, and their sum is equal to 1. Since \(\beta_{i}=1-\sum_{i \neq j=1}^{n+1}\beta_{j}\), the reverse connection
$$ \alpha_{i}=\frac{\sum_{i \neq j=1}^{n+1}\beta_{j}}{n} $$
(20)
follows. The last of the convex combinations
$$\begin{aligned} \begin{aligned}[b] A &= \sum_{i=1}^{n+1} \alpha_{i}A_{i} \\ &= \sum_{i=1}^{n+1}\frac{\sum_{i \neq j=1}^{n+1}\beta_{j}}{n}A_{i} =\sum_{i=1}^{n+1}\beta_{i} \frac{\sum_{i \neq j=1}^{n+1}A_{j}}{n} \\ &= \sum_{i=1}^{n+1}\beta_{i}B_{i} \end{aligned} \end{aligned}$$
(21)
confirms that the point A belongs to the simplex \(\mathcal{B}\).
Let us assume that the point A belongs to the simplex \(\mathcal{B}\). Then we have the convex combination \(A=\sum_{i=1}^{n+1}\lambda_{i}B_{i}\). Using equation (21) in the reverse direction, we get the convex combinations equality
$$ \sum_{i=1}^{n+1} \lambda_{i}B_{i} =\sum_{i=1}^{n+1} \alpha_{i}A_{i} $$
(22)
with the coefficient connections \(\alpha_{i}=\sum_{i\neq j=1}^{n+1}\lambda_{j}/n\) from which we may conclude that \(\alpha_{i}\leq1/n\).

The second statement, relating to the non-peaked simplex \(\mathcal {B}'\), follows from the first statement and the convex combinations in formula (18) which uniquely represent the facet barycenters \(B_{i}\). □

We need another subsimplex of \(\mathcal{A}\). Let A be a point belonging to the interior of \(\mathcal{A}\). In this case, the sets \(\mathcal{A}_{i}\) defined by formula (4) are n-simplices. Let \(C_{i}\) stand for the barycenter of the simplex \(\mathcal{A}_{i}\) by means of
$$ C_{i}=\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1}, $$
(23)
and let \(\mathcal{C}=C_{1}\cdots C_{n+1}\) be the n-simplex of the vertices \(C_{i}\).

Lemma 3.2

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(\alpha_{i}>0\).

The point A belongs to the non-peaked simplex \(\mathcal{C}'=\mathcal {C}\setminus\{C_{1},\ldots,C_{n+1}\}\) if and only if the coefficients \(\alpha_{i}\) satisfy the additional limitations \(\alpha_{i}\leq1/n\).

Proof

Suppose that the coefficients \(\alpha_{i}\) satisfy \(0<\alpha_{i}\leq 1/n\). Let \(\beta_{i}\) be the coefficients as in equation (19). Using the trivial equality \(A=A/(n+1)+nA/(n+1)\), and the coefficient connections of equation (20), we get
$$\begin{aligned} A =& \sum_{i=1}^{n+1} \alpha_{i}A_{i} =\frac{1}{n+1}A +\frac{n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}A_{i} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A}{n+1} +\sum_{i=1}^{n+1}\sum _{i \neq j=1}^{n+1}\beta_{j} \frac{A_{i}}{n+1} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A}{n+1} +\sum_{i=1}^{n+1} \beta_{i}\sum_{i \neq j=1}^{n+1} \frac{A_{j}}{n+1} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} =\sum_{i=1}^{n+1} \beta_{i}C_{i} \end{aligned}$$
(24)
indicating that the point A lies in the simplex \(\mathcal{C}\). To show that the convex combination \(\sum_{i=1}^{n+1}\beta_{i}C_{i}\) does not represent any vertex, we will assume that some \(\beta_{i_{0}}=1\). Then \(\alpha_{i_{0}}=0\) as opposed to the assumption that all \(\alpha _{i}\) are positive.

The proof of the reverse implication goes exactly in the same way as in the proof of Lemma 3.1. □

Each simplex \(\mathcal{C}\) is homothetic to the simplex \(\mathcal{B}\). Namely, combining equations (23) and (18), we can represent each vertex \(C_{i}\) by the convex combination
$$ C_{i}=\frac{1}{n+1}A+\frac{n}{n+1}B_{i}. $$
(25)
Then it follows that
$$ C_{i}-A=\frac{n}{n+1}(B_{i}-A), $$
and using free vectors, we have the equality \(\overrightarrow {AC_{i}}=(n/(n+1))\overrightarrow{AB_{i}}\). So, the simplices \(\mathcal{C}\) and \(\mathcal{B}\) are similar respecting the homothety with the center at A and the coefficient \(n/(n+1)\).

If \(A\in\mathcal{B}'\), then \(\mathcal{C}\subset\mathcal{B}'\) by the convex combinations in formula (25). Combining Lemma 3.1 and Lemma 3.2, and applying Corollary 2.2 to the simplex inclusions \(\mathcal{C}\subset \mathcal{B}\) and \(\mathcal{B}\subset\mathcal{A}\), we get the Jensen type inequality as follows.

Corollary 3.3

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(0<\alpha_{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\).

Then it follows that
$$ \sum_{i=1}^{n+1} \beta_{i}C_{i} = \sum_{i=1}^{n+1} \beta_{i}B_{i} = \sum_{i=1}^{n+1} \alpha_{i}A_{i}, $$
(26)
and each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ \sum_{i=1}^{n+1} \beta_{i}f(C_{i}) \leq\sum_{i=1}^{n+1} \beta_{i}f(B_{i}) \leq\sum_{i=1}^{n+1} \alpha_{i}f(A_{i}). $$
(27)

The point A used in the previous corollary lies in the interior of the simplex \(\mathcal{A}\) because the coefficients \(\alpha_{i}\) are positive. In that case, the sets \(\mathcal{A}_{i}\) are n-simplices, and they will be used in the main theorem that follows.

Theorem 3.4

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(0<\alpha_{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\). Let \(\mathcal{A}_{i}\) be the simplices defined by formula (4).

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x) \,dx}{ \operatorname{vol}(\mathcal{A}_{i})} \leq \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(28)

Proof

Using the convex combinations equality \(\sum_{i=1}^{n+1}\alpha _{i}A_{i}=\sum_{i=1}^{n+1}\beta_{i}C_{i}\), and applying the Jensen inequality to \(f (\sum_{i=1}^{n+1}\beta _{i}C_{i} )\), we get
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i}f(C_{i}) = \sum_{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr). $$
Summing the products of the Hermite-Hadamard inequalities for the function f on the simplices \(\mathcal{A}_{i}\) and the coefficients \(\beta_{i}\), it follows that
$$ \sum_{i=1}^{n+1} \beta_{i}f \biggl(\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x) \,dx}{ \operatorname{vol}(\mathcal{A}_{i})} \leq \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1}. $$
Repeating the procedure which was used for the derivation of formula (24), we obtain the series of equalities
$$\begin{aligned} \sum_{i=1}^{n+1} \beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} =& \frac{1}{n+1}f(A) + \frac{n}{n+1}\sum_{i=1}^{n+1} \beta_{i} \frac{\sum_{i \neq j=1}^{n+1}f(A_{j})}{n} \\ =& \frac{1}{n+1}f(A) +\frac{n}{n+1}\sum_{i=1}^{n+1} \frac{\sum_{i \neq j=1}^{n+ 1}\beta_{j}}{n}f(A_{i}) \\ =& \frac{1}{n+1}f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)+\frac {n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). \end{aligned}$$
Finally, applying the Jensen inequality to \(f (\sum_{i=1}^{n+1}\alpha_{i}A_{i} )\), we get the last inequality
$$ \frac{1}{n+1}f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)+\frac {n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}) \leq\sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
Bringing together all of the above, we obtain the multiple inequality
$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq& \sum _{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum_{i=1}^{n+1} \beta_{i} \frac{\int_{\mathcal {A}_{i}}f(x) \,dx}{\operatorname{vol}(\mathcal{A}_{i})} \\ \leq& \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} \leq \sum_{i=1}^{n+1} \alpha_{i}f(A_{i}), \end{aligned}$$
(29)
of which the most important part is the double inequality in formula (28). □
The inequality in formula (29) is a generalization and refinement of the Hermite-Hadamard inequality. Taking the coefficients \(\alpha_{i}=1/(n+1)\), in which case \(\beta_{i}=1/(n+1)\), we realize the five terms inequality
$$\begin{aligned} f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr) \leq& \frac{1}{n+1}\sum _{i=1}^{n+1}f \biggl(\frac{A_{i}+(n+2)\sum_{i \neq j=1}^{n+1}A_{j}}{(n+1)(n+1)} \biggr) \leq \frac{\int_{\mathcal{A}}f(x) \,dx}{\operatorname{vol}(\mathcal{A})} \\ \leq& \frac{1}{n+1}f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr) +\frac{n}{n+1} \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1} \leq \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1}, \end{aligned}$$
(30)
where the second and fourth terms refine the Hermite-Hadamard inequality. The third term is generated from all of \(n+1\) simplices \(\mathcal {A}_{i}\). In the present case, these simplices have the same volume equal to \(\operatorname{vol}(\mathcal{A})/(n+1)\).

The inequality in formula (30) excepting the second term was obtained in [2], Theorem 2. Similar inequalities concerning the standard n-simplex were obtained in [5, 6] and [18]. Special refinements of the left and right-hand side of the Hermite-Hadamard inequality were recently obtained in [19] and [20].

4 Generalization to the function barycenter

If μ is a positive measure on \(\mathbb{R}^{n}\), if \(\mathcal {S}\subseteq\mathbb{R}^{n}\) is a measurable set, and if \(g:\mathcal{S}\to\mathbb{R}\) is a nonnegative integrable function such that \(\int_{\mathcal{S}}g(x) \,d\mu(x)>0\), then the integral mean point
$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1}g(x) \,d\mu(x)}{\int_{\mathcal {S}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal{S}}x_{n}g(x) \,d\mu (x)}{\int_{\mathcal{S}}g(x) \,d\mu(x)} \biggr) $$
(31)
can be called the μ-barycenter of the function g. It is about the following measure. Introducing the measure ν as
$$ \nu(S)= \int_{\mathcal{S}}g(x) \,d\mu(x), $$
(32)
we get
$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\nu(x)}{\nu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\nu(x)}{\nu(\mathcal {S})} \biggr). $$
(33)
Thus the μ-barycenter of the function g coincides with the ν-barycenter of its domain \(\mathcal{S}\). So, the barycenter S belongs to the convex hull of the set \(\mathcal{S}\). By using the unit function \(g(x)=1\) in formula (31), it is reduced to formula (7).

Utilizing the function barycenter instead of the set barycenter, we have the following reformulation of Lemma 2.1.

Lemma 4.1

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\), and let \(g:\mathcal{A}\to\mathbb{R}\) be a nonnegative integrable function such that \(\int_{\mathcal{A}}g(x) \,d\mu(x)>0\). Let A be the μ-barycenter of g, and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be its unique convex combination by means of
$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1}g(x) \,d\mu(x)}{\int_{\mathcal {A}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal{A}}x_{n}g(x) \,d\mu (x)}{\int_{\mathcal{A}}g(x) \,d\mu(x)} \biggr) = \sum_{i=1}^{n+1}\alpha_{i}A_{i}. $$
(34)
Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \frac{\int_{\mathcal{A}}f(x)g(x) \,d\mu(x)}{ \int_{\mathcal{A}}g(x) \,d\mu(x)} \leq \sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(35)
The proof of Lemma 2.1 can be employed as the proof of Lemma 4.1 by using the measure ν in formula (32) or by utilizing the affinity of the hyperplanes \(h_{1}\) and \(h_{2}\) in the form of the equalities
$$ h_{1,2} \biggl(\frac{\int_{\mathcal{A}}x_{1}g(x) \,d\mu(x)}{\int _{\mathcal{A}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal {A}}x_{n}g(x) \,d\mu(x)}{\int_{\mathcal{A}}g(x) \,d\mu(x)} \biggr) =\frac{\int_{\mathcal{A}}h_{1,2}(x)g(x) \,d\mu(x)}{ \int_{\mathcal{A}}g(x) \,d\mu(x)}. $$
(36)

Lemma 4.1 is an extension of the Fejér inequality (see [21]) to multivariable convex functions. As regards univariable convex functions, using the Lebesgue measure on \(\mathbb{R}\) and a closed interval as 1-simplex in Lemma 4.1, we get the following generalization of the Fejér inequality.

Corollary 4.2

Let \([a,b]\) be a closed interval in \(\mathbb{R}\), and let \(g:[a,b]\to \mathbb{R}\) be a nonnegative integrable function such that \(\int _{a}^{b}g(x) \,dx>0\). Let c be the barycenter of g, and let \(\alpha a+\beta b\) be its unique convex combination by means of
$$ c=\frac{\int_{a}^{b}xg(x) \,dx}{\int_{a}^{b}g(x) \,dx}=\alpha a+\beta b. $$
(37)
Then each convex function \(f:[a,b]\to\mathbb{R}\) satisfies the double inequality
$$ f(\alpha a+\beta b) \leq\frac{\int_{a}^{b}f(x)g(x) \,dx}{\int _{a}^{b}g(x) \,dx} \leq\alpha f(a)+\beta f(b). $$
(38)
Fejér used a nonnegative integrable function g that is symmetric with respect to the midpoint \(c=(a+b)/2\). Such a function satisfies \(g(x)=g(2c-x)\), and therefore
$$ \int_{a}^{b}(x-c)g(x) \,dx=0. $$
As a consequence it follows that
$$ \frac{\int_{a}^{b}xg(x) \,dx}{\int_{a}^{b}g(x) \,dx} =\frac{\int_{a}^{b}(x-c)g(x) \,dx}{\int_{a}^{b}g(x) \,dx} +\frac{\int_{a}^{b}cg(x) \,dx}{\int_{a}^{b}g(x) \,dx}= \frac{a+b}{2}, $$
and formula (38) with \(\alpha=\beta=1/2\) turns into the Fejér inequality
$$ f \biggl(\frac{a+b}{2} \biggr) \leq\frac{\int_{a}^{b}f(x)g(x) \,dx}{\int_{a}^{b}g(x) \,dx} \leq \frac{f(a)+f(b)}{2}. $$
(39)

Using the barycenters of the restrictions of g onto simplices \(\mathcal{A}_{i}\) in formula (4), we have the following generalization of Theorem 3.4.

Theorem 4.3

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha_{i}\) satisfying \(0<\alpha _{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\). Let \(\mathcal{A}_{i}\) be the simplices defined by formula (4), and let \(g_{i}:\mathcal{A}_{i}\to\mathbb{R}\) be nonnegative integrable functions such that \(\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu (x)>0\) and
$$ C_{i}= \biggl(\frac{\int_{\mathcal{A}_{i}}x_{1}g_{i}(x) \,d\mu(x)}{\int _{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal {A}_{i}}x_{n}g_{i}(x) \,d\mu(x)}{\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu (x)} \biggr) =\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1}. $$
(40)
Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality
$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{ \int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)} \leq \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(41)

Proof

The first step of the proof is to apply Lemma 4.1 to the functions f and \(g_{i}\) on the simplex \(\mathcal{A}_{i}\) in the way of
$$ f \biggl(\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{\int_{\mathcal {A}_{i}}g_{i}(x) \,d\mu(x)} \leq \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1}. $$
Summing the products of the above inequalities with the coefficients \(\beta_{i}\), we obtain the double inequality that may be combined with formula (29), and so we obtain the multiple inequality
$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq& \sum _{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum_{i=1}^{n+1} \beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)} \\ \leq& \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} \leq \sum_{i=1}^{n+1} \alpha_{i}f(A_{i}) \end{aligned}$$
(42)
containing the double inequality in formula (41). □

The conditions in formula (40) require that the μ-barycenter of the function \(g_{i}\) coincides with the barycenter \(C_{i}= (A+\sum_{i \neq j=1}^{n+1}A_{j} )/(n+1)\) of the simplex \(\mathcal{A}_{i}\).

Using the Lebesgue measure and functions \(g_{i}(x)=1\), the inequality in formula (42) reduces to the inequality in formula (29).

Declarations

Acknowledgements

This work has been fully supported by Mechanical Engineering Faculty in Slavonski Brod, and the Croatian Science Foundation under the project HRZZ-5435. The author wishes to thank Velimir Pavić who graphically prepared Figure 1.

Open Access 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.

Authors’ Affiliations

(1)
Department of Mathematics, Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Slavonski Brod, 35000, Croatia

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