Abstract

This paper provides a method to generalize and refine the Hadamard's inequality on a hypercube in -dimensional Euclidean space by establishing a family of mappings.

1. Introduction

For a convex function defined on an open interval of real numbers and with , then the following inequality: is known as the classical Hadamard’s inequality for the convex function.

In [1], Dragomir established the following mapping: then the mapping is convex, increasing on , and for all , we have which is a refinement to the left side of (1.1).

In [2], Yang and Hong established the following mapping: then the mapping is convex, increasing on , and for all , we have which is a refinement to the right side of (1.1).

In this paper, we first establish a mapping in Section 2, defined by then the mapping is convex, increasing on , and for all we have Furthermore, there is a point so that . Thus, (1.1) has been proved based on the properties of , and the inequality (1.7) is a refinement to the both sides of (1.1).

In Section 3, we also use an analogue of (1.6) to obtain the following Hadamard’s type inequality on a square: which is a generalization of (1.1) to two-dimensional Euclidean space.

In Section 4, by establishing another analogue of (1.6) on a hypercube, the following Hadamard’s inequality is also pointed out: which is a generalization of (1.1) to -dimensional Euclidean space.

All the above Hadamard’s inequalities (1.1), (1.8), and (1.9) have a uniform format

Thus, by establishing a family of the mappings, we have provided a method to get Hadamard’s inequality of the convex function on a hypercube in -dimensional Euclidean space. The motivation for the present work is from many generalizations and refinements of the classical inequality (1.1) in [1–8].

2. Properties of the Mapping

The main properties of the mapping defined by (1.6) are embodied in the following theorem.

Theorem 2.1. If is a convex function defined on an open interval of real numbers, with , and the mapping is defined by (1.6), then, one has (i)the mapping is convex on ,(ii)one has the bounds: (iii)the mapping is increasing on ,(iv)there is a point so that (v)one obtains, the classical inequality (1.1) and has a refinement of (1.1):

Proof. (i) Let and with , then we have which proves the convexity of on .
(ii) By the convexity of on the interval , we have the bounds (2.1) hold.
(iii) Let . By the convexity of the mapping , we have Since we have proved that for all in (ii), the monotonicity of is established.
(iv) We define the function on by then According to the Rolle’s mean-value theorem, there is a point so that Thus, (2.2) holds.
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s inequality (1.1) and the inequality (2.3).

3. The Hadamard’s Inequality on a Square

We consider a convex function with two variables defined on a square in two-dimensional Euclidean space.

Theorem 3.1. If is a convex function defined on an open region , the square and the mapping A₂ is defined by which is a parameter curvilinear integral with respect to arc length, then one has (i)the mapping is convex on ,(ii)one has the bounds (iii)the mapping is increasing on ,(iv)there is a point so that (v)one has the Hadamard’s type inequality (1.8) and a refinement

Proof. For a fixed , let if , and if , we have which indicates the mapping is continuous on .
(i) Let and with . By the convexity of , we have Similarly, we have Thus, which proves the convexity of on .
(ii) By the convexity of and (3.6), we have the last inequality holds owing to (1.1); as the convexity of we have and note that the bounds (3.3) hold.
(iii) The monotonicity of follows as in the proof of Theorem 2.1, and we omit the detail.
(iv) We define the function on by then Let and , we have According to the Rolle’s mean-value theorem, there is a point so that thus, (3.4) holds.
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s type inequality (1.8) and the inequality (3.5).

4. Remark

In recent years, several authors have discussed the problem of extending the classical inequality (1.1) to a convex function on a general convex body of -dimensional space [6–9]. For the left side of (1.1) (i.e., Jensen’s inequality), the following inequality was showed in [6, 9] (also see [7]): where is the barycenter of , and denotes -dimensional Lebesgue’s measure of .

It is a much more delicate problem to extend the right side of (1.1) on a general convex body of -dimensional space [7]. On a closed hyperball of the -dimensional space, de la Cal and Cárcamo [7] obtained the inequality where denotes the volumes of , and denotes the areas of the hypersphere , for which was early obtained by Dragomir [4, 5] based on Calculus.

When the convex function was supposed to be differentiable on a closed convex body , the following inequality was proved in [9] based on Stokes’ Formula where denotes the piecewisely smooth boundary of , denotes the barycenter of , and denotes the exterior unit normal vector in the point .

For a closed hyperball , the inequality (4.2) is able to be deduced from (4.3); and for a hypercube , the inequality (1.9) is also able to be deduced from (4.3); however, these are on the condition that is differentiable.

Now, let be a convex function defined on an open region , the hypercube , the mapping is defined by which is a parameter surface integral with respect to area of dimension, and let in the same way as the proof of Theorem 3.1, by the induction we will be able to obtain the inequality (1.9).

Some mappings associated to Hadamard’s inequality have been established in [1–5], which refine the classical Hadamard’s inequality. These mappings are based on Hadamard’s inequality. However, our method in this paper to establish mappings is different from theirs: we first proved (1.1) by the properties of and subsequently established and proved (1.8) based on (1.1). Thus, using the induction, we will be able to establish the mapping and prove Hadamard’s inequality on a hypercube.