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In this paper, we shall establish an inequality for differentiable co-ordinated convex functions on a rectangle from the plane. It is connected with the left side and right side of extended Hermite-Hadamard inequality in two variables. In addition, six other inequalities are derived from it for some refinements. Finally, this paper shows some examples that these inequalities are able to be applied to some special means.

Throughout this paper, let

derivative of second order

The inequality

which holds for all convex functions

For some results which generalize, improve, and extend the Inequality (1), please refer to [

Based on the convex functions on

Definition 1. A function

are convex.

Definition 2. A function

holds for all

Clearly, we can observe that every convex function

The following extended Hadamard’s inequality for co-ordinated convex functions on

Theorem 1. Suppose that

The above inequalities are sharp.

In [

Theorem 2. Let

(1) If

(2) If

(3) If

where

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

In [

Theorem 3. Let

(1) If

(2) If

(3) If

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

(II) For

(III) For

(IV) For

In this section, let the mapping

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

Lemma 1. Let

where

Proof. It suffices to note that

Integration by parts, we have

and

By summing the above four identities

Then, multiply both sides by

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

Theorem 1. Let

where

and

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

If

We denote

and

respectively, and then

By using the integration techniques, we have

and similary we get,

and

By summing the above four identities

On the other hand, by using the identity mappings

and

we have

By the convexit3 of

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

Corollary 1. Under the assumptions of Theorem 1 with

where

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

where

and

Remark 3. By using the convexity of

and then

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

Remark 4. Under the assumptions of Theorem 1 with

Corollary 3. Under the assumptions of Theorem 1 with

where

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

where

and

Remark 5. By using the convexity of

and then

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

Remark 6. Under the assumptions of Theorem 1 with

Example 1. Let the function

As in [

where

Proposition 1. Let

Proof. The proof is immediate from Corollary 4 with

Proposition 2. Suppose

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.