A Generalized Hermite-Hadamard Inequality for Coordinated Convex Function and Some Associated Mappings

Abstract

We have discussed the generalization of Hermite-Hadamard inequality introduced by Lupaş for convex functions on coordinates defined in a rectangle from the plane. Also we define that mappings are related to it and their properties are discussed.

1. Introduction

Convex functions have great importance in many areas of Mathematics. A real valued function , where is interval in , is said to be convex if for with In particular, if we have thatOne of the most important inequalities that has attracted many mathematician in this field in the last few decades is the famous Hermite-Hadamard inequality, which establishes a refinement of (2) and it involves the notions of convexity.

Let be a convex mapping defined on the interval of real numbers and with . The double inequalityis known as the Hermite-Hadamard inequality. This inequality was published by Hermite in 1883 and was independently proved by Hadamard in 1893 (see Mitrinović and Lacković [1] for the whole history). It gives us an estimation of the mean value of a convex function and it is important to note that (3) provides a refinement to the Jensen inequality. Fink [2] has worked on a best possible Hermite-Hadamard inequality and has deduced this inequality with least restrictions. Dragomir in [3, 4] worked on Hermite-Hadamard inequality and a mapping in connection to it. Dragomir and Milośević [5] gave some refinements of Hadamard's inequalities and its applications (see also [6–8] for more results). Dragomir and Pearce [9] wrote a monograph on selected topics on Hermite-Hadamard inequalities.

The aim of this paper is to discuss an analogue of the generalization of Hermite-Hadamard inequality introduced by Lupaş for convex functions defined in a rectangle from the plane. Also some related mappings are defined and their properties are discussed.

In [10] Dragomir has given the concept of convex functions on the coordinates in a rectangle from the plane and established the Hermite-Hadamard inequality for it.

Definition 1. Let with and . A function is called convex on coordinates if the partial mappings , and , are convex, defined for all and .

Note that every convex mapping is convex on the coordinates but the converse is not generally true.

Theorem 2. Suppose that is convex on the coordinates on . Then we have

In [10], Dragomir for , defined a mapping as and proved the properties of this mapping in following theorem.

Theorem 3. Suppose that is convex on the coordinates on .(i)The mapping is convex on the coordinates on .(ii)We have the bounds (iii)The mapping is monotonic nondecreasing on the coordinates.

Also, in [10] Dragomir gave the following mapping, which is closely connected with Hadamard's inequality, defined as and proved the following properties of this mapping.

Theorem 4. Suppose that is convex on the coordinates on . (i)We have the equalities (ii) is convex on the coordinates.(iii)We have the bounds (iv)The mapping is monotonic nondecreasing on and nondecreasing on for all . A similar property has the mapping for all .(v)We have the inequality A generalized form of Hermite-Hadamard inequality is given by Lupaş in [11] (see also [12, page 143]).

Theorem 5. Let p, q be given positive numbers and . Then the inequality,holds for , and all continuous convex functions iff .

2. Main Results

The following results comprise generalization of Hermite-Hadamard inequality introduced by Lupaş on coordinates in a rectangle from the plane.

Theorem 6. Let and , , , and be positive real numbers and Also let be convex on the coordinates on .

Proof. Since is convex on coordinates, it follows that the mapping , , is convex on for all , so by inequality (11) one has that is Observe that for , by considering two cases and we can easily verify that , so that is defined on . Integrating the above inequality on , we haveBy a similar argument applied for the mapping , , we getSumming the inequalities (16) and (17), we get the second and third inequality in (13). Since is convex on coordinates, using convexity of on first coordinate and inequality (11), we haveand, using convexity of on second coordinate, we haveAdding the inequalities (18) and (19) we get the first inequality in (13). Finally, by the same Hermite-Hadamard inequality introduced by Lupaş we can also stateMultiply (20) and (21) by and multiply (22) and (23) by then on addition we get the last inequality in (13).

3. Some Associated Mappings

In this section we will discuss some mappings associated with generalized Hermite-Hadamard inequality introduced by Lupaş for convex mappings on coordinates.

Let , , , and be positive real numbers and and where for . For mapping , we can define a mapping , where and , so that is defined on and . The properties of this mapping are embodied in the following theorem.

Theorem 7. Suppose that , and are positive real numbers and Also let be convex on the coordinates on . (i)The mapping is convex on the coordinates on .(ii)We have the bounds (iii)The mapping is monotonic nondecreasing on the coordinates.

Proof. (i) Fix , for all and with ; we have If is fixed then, for all , and with , we also have and the statement is proved.
(ii) Since is convex on coordinates, we have by Jensen's inequality for integrals that By the convexity of on the coordinates, we have By Hadamard’s inequality introduced by Lupas (11), we also have Thus by integration, we get that Also using the result we deduce the inequality for all and the second bound in (ii) is proved.
(iii) Firstly, we show that for all . By Jensen’s inequality for integrals, we have for all .
Now let . By the convexity of mapping for all , we have

The following theorem also holds.

Theorem 8. Suppose that , and are positive real numbers and Also let be convex on the coordinates on . (i)The mapping is convex on .(ii)Define the mapping , . Then is convex, monotonic nondecreasing on and one has the bounds

Proof. (i) Let , and with . Since is convex on , we have which shows is convex on .
(ii) Let and with . which shows the convexity of on .
We have, by the above theorem, that which proves the required bounds.
Now let . By the convexity of mapping , we have that and the theorem is proved.

Next, for positive real numbers and and where for , then for mapping we shall consider the mapping , which is given as where and , so that is defined on and .

The next theorem contains the main properties of this mapping.

Theorem 9. Suppose that , and are positive real numbers and Also let be convex on the coordinates on .(i)We have the equalities (ii) is convex on the coordinates.(iii)We have the bounds (iv)The mapping is monotonic nonincreasing on and nondecreasing on for all . A similar property has the mapping for all .(v)We have the inequality

Proof. (i) and (ii) are obvious.
(iii) By the convexity of in the first variable, we get that for all , , and . Integrating on , we get Similarly,Now integrating this inequality on and taking into account the above inequality, we deduce for . The first bound in (iii) is proved and the second bound goes on likewise.
(iv) The monotonicity of follows by a similar argument from (iii).
(v) By Jensen’s integral inequality, we have successively for all that Similarly, we can easily show , , and .
In addition, as for all , so we deduce .
The theorem is thus proved.

Theorem 10. Suppose that , , , and are positive real numbers and Also let be convex on the coordinates on . (i)The mapping is convex on .(ii)Define the mapping , . Then is convex, monotonic nonincreasing on and nondecreasing on and one has the bounds(iii)One has the inequality

Proof. (i) Let , and with . Since is convex on , we have which shows is convex on .
(ii) Let , and with . which shows the convexity of on .
As which proves the required bounds.
It is obvious that is monotonic nonincreasing on and nondecreasing on .
(iii) As we know and the theorem is proved.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.