Throughout this paper, let be double intervals with, in, and a partial

derivative of second order is denoted by for brevity.

The inequality

which holds for all convex functions is known as Hermite-Hadamard’s inequality [1] or simply Hadamard’s inequality.

For some results which generalize, improve, and extend the Inequality (1), please refer to [2] -[17] .

Based on the convex functions on, Dragomir proposed the concept of co-ordinated convex functions in [3] , defined as follows:

Definition 1. A function is said to be convex on the co-ordinates on if the partial mappings

are convex.

Definition 2. A function is said to be convex on the co-ordinates on if the inequality

holds for all and, , ,.

Clearly, we can observe that every convex function is convex on the co-ordinates, but in some special cases, some co-ordinated convex functions are not convex (please refer to [3] ). For more relevant co- ordinated convex functions, please refer to [5] [6] [8] -[10] [12] .

The following extended Hadamard’s inequality for co-ordinated convex functions on in two variables was proved in [3] :

Theorem 1. Suppose that is co-ordinated convex on. Then the following inequalities hold:

The above inequalities are sharp.

In [10] , Latif and Dragomir established the following Hadamard-type inequalities that gave an estimate of the difference in the left side of the Inequalities (3) for differentiable co-ordinated convex functions on.

Theorem 2. Let be a partial differentiable mapping on.

(1) If is convex on the co-ordinates on, then the following inequality holds:

(2) If is convex on the co-ordinates on and, , then the following inequality holds:

(3) If is convex on the co-ordinates on and, then the following inequality holds:

where

Remark 1. The Inequality (6) shows the result of giving the Inequality (5) an improved and simplified constant.

In [12] , Sarikaya et al. established the following results that gave an estimate of the difference in the right side of the Inequalities (3) for differentiable co-ordinated convex functions on.

Theorem 3. Let be a partial differentiable mapping on.

(1) If is convex on the co-ordinates on, then the following inequality holds:

(2) If is convex on the co-ordinates on and, , then the following inequality holds:

(3) If is convex on the co-ordinates on and, then the following inequality holds:

where

Remark 2. The Inequality (9) shows the result of giving the Inequality (8) an improved and simplified constant.

The goal of this paper is to establish an inequality which could be connected with the left side and right side of the extended Hadamard’s Inequality (3) and improve and generalize the Theorem 2 and Theorem 3. Also, the paper aims to note some consequent applications to special means.

In order to show our main results, we need the following identities (I)-(VI):

(I) For, , the following four identities hold:

(II) For, , the following four identities hold:

(III) For, , the following four identities hold:

(IV) For, , the following four identities hold:

In this section, let the mapping for all be defined as follows:

In order to prove our main results, we need the following lemma:

Lemma 1. Let be a partial differentiable mapping on. Then the following inequality holds:

where

Proof. It suffices to note that

Integration by parts, we have

and

By summing the above four identities, , and and simplifying the result, it follows that

Then, multiply both sides by in (12). From (12) and (13), we get the equations and. This proof of the identity 11 is complete.

Now, we are ready to state and prove the main results.

Theorem 1. Let be defined as Lemma 1. If and the mapping is convex on the co-ordinates on then

where

and

Proof. By using the identity (11), we have

If, , it follows from the power mean inequality that

We denote and by

and

respectively, and then

By using the integration techniques, we have

and similary we get,

and

By summing the above four identities, , and and simplifying the result. Then according to (16), we get the estimated bound.

On the other hand, by using the identity mappings

and

we have

By the convexit3 of on the co-ordinates on and the Inequality (2) in, , and, then we have

and

By applying the identities (I), (II), (III) and (IV) to the above four inequalities and then simplifying the results, we get the estimated bound and the Inequality (14) for. If, then the Inequality (14) follows from (15) and (17). The proof of the Inequality (14) is complete.

Corollary 1. Under the assumptions of Theorem 1 with, , , , and, we have

where is as given in Theorem 2,

and

The Corollary 1 shows that we get the new estimated bound of the Inequality (6).

Corollary 2. Under the assumptions of Corollary 1 with, we have

where is as given in Theorem 2,

and

Remark 3. By using the convexity of on the co-ordinates on, we have the inequality

and then

Hence the Inequality (19) improves the Inequality (6).

Remark 4. Under the assumptions of Theorem 1 with, , , , , and, we get the new estimated bound and it could improve the Inequality (4).

Corollary 3. Under the assumptions of Theorem 1 with, , we have

where is as given in Theorem 3,

and

The Corollary 3 shows that we get the new estimated bound of the Inequality (9).

Corollary 4. Under the assumptions of Corollary 3 with, we have

where is as given in Theorem 3,

and

Remark 5. By using the convexity of on the co-ordinates on, we have the inequality

and then

Hence the Inequality (21) improves the Inequality (9).

Remark 6. Under the assumptions of Theorem 1 with, and, we get the new estimated bound and it could improve the Inequality (7).

Example 1. Let the function be,. Then the result of the right-hand side of (6) or (9) is, whereas the right-hand side of (19) and (21) are and, respectively.

As in [11] we shall consider extensions of arithmetic, logarithmic and generalized logarithmic means from posi- tive real numbers. We take

where is the set of integers.

Proposition 1. Let, , , , , , , , and, , and. Then, for, we have

Proof. The proof is immediate from Corollary 4 with, , , ,.

Proposition 2. Suppose, , , , , , ,. Then, for, we have

Proof. The result follows from Corollary 4 with

Remark 7. The Corollary 2 could also be applied to some special means.

The author is very grateful to the reviewers for carefully reading this paper and giving constructive comments.