A Family of Mappings Associated with Hadamard’s Inequality on a Hypercube

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


Introduction
For a convex function f defined on an open interval I of real numbers and a, b ∈ I with a < b, 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 H is convex, increasing on 0, 1 , and for all t ∈ 0, 1 , we have which is a refinement to the left side of 1.1 .

ISRN Mathematical Analysis
In 2 , Yang and Hong established the following mapping: then the mapping G is convex, increasing on 0, 1 , and for all t ∈ 0, 1 , 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 A 1 is convex, increasing on 0, 1 , and for all t ∈ 0, 1 we have Furthermore, there is a point ξ ∈ 0, 1 so that A 1 ξ 1/ b − a b a f x .Thus, 1.1 has been proved based on the properties of A 1 , 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 n-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 n-dimensional Euclidean space.The motivation for the present work is from many generalizations and refinements of the classical inequality 1.1 in 1-8 .

Properties of the Mapping
The main properties of the mapping A 1 defined by 1.6 are embodied in the following theorem.ii one has the bounds: iv there is a point ξ ∈ 0, 1 so that v one obtains, the classical inequality 1.1 and has a refinement of 1.1 : Proof.i Let t 1 , t 2 ∈ 0, 1 and α, β ≥ 0 with α β 1, then we have which proves the convexity of A 1 on 0, 1 .
ii By the convexity of f on the interval a, b , we have the bounds 2.1 hold.
iii Let 0 < t 1 < t 2 ≤ 1.By the convexity of the mapping A 1 , we have Since we have proved that A 1 t 1 ≥ A 1 0 for all t 1 ∈ 0, 1 in ii , the monotonicity of A 1 is established.iv We define the function φ on 0, 1 by According to the Rolle's mean-value theorem, there is a point ξ ∈ 0, 1 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 .

The Hadamard's Inequality on a Square
We consider a convex function with two variables defined on a square in two-dimensional Euclidean space.
and the mapping A 2 is defined by which is a parameter curvilinear integral with respect to arc length, then one has i the mapping A 2 is convex on 0, 1 , ii one has the bounds iii the mapping A 2 is increasing on 0, 1 , iv there is a point ξ ∈ 0, 1 so that f x, y dx dy, 3.4 v one has the Hadamard's type inequality 1.8 and a refinement

3.7
Similarly, we have

3.8
Thus, which proves the convexity of A 2 on 0, 1 .
ii By the convexity of f and 3.6 , we have the last inequality holds owing to 1.1 ; as the convexity of A 2 we have and note that

Remark
In recent years, several authors have discussed the problem of extending the classical inequality 1.1 to a convex function f on a general convex body Ω of n-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 c is the barycenter of Ω, and μ Ω denotes n-dimensional Lebesgue's measure of Ω.
It is a much more delicate problem to extend the right side of 1.1 on a general convex body of n-dimensional space 7 .On a closed hyperball of the n-dimensional space, de la Cal and Cárcamo 7 obtained the inequality where Vol K denotes the volumes of K, and Area K * denotes the areas of the hypersphere K * , for n 2, 3 which was early obtained by Dragomir 4, 5 based on Calculus.When the convex function f was supposed to be differentiable on a closed convex body Ω, the following inequality was proved in 9 based on Stokes' Formula where ∂Ω denotes the piecewisely smooth boundary of Ω, θ denotes the barycenter of ∂Ω, and π x denotes the exterior unit normal vector in the point x.
For a closed hyperball Ω, the inequality 4.2 is able to be deduced from 4.3 ; and for a hypercube Ω, the inequality 1.9 is also able to be deduced from 4.3 ; however, these are on the condition that f is differentiable.Now, let f be a convex function defined on an open region D ⊆ R n , the hypercube {x : x − c ∞ ≤ h/2} ⊆ D, the mapping A n is defined by which is a parameter surface integral with respect to area of n − 1 dimension, and let φ t x−c ∞ ≤th/2 f x dx − t n x−c ∞ ≤h/2 f x dx, 4.5 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 A 1 and subsequently established A 2 and proved 1.8 based on 1.1 .Thus, using the induction, we will be able to establish the mapping A n and prove Hadamard's inequality on a hypercube.

Theorem 2 . 1 .
If f is a convex function defined on an open interval I of real numbers, a, b ∈ I with a < b, and the mapping A 1 is defined by 1.6 , then, one has i the mapping A 1 is convex on 0, 1 , For a fixed t ∈ 0, 1 , let u z if |v| h/2, and v z if |u| h/2, we have , t 2 ∈ 0, 1 and α, β ≥ 0 with α β 1.By the convexity of f, we have