Research Article | Open Access

# Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

**Academic Editor:**J. Zhang

#### Abstract

A generalized lemmas is proved and several new inequalities for differentiable co-ordinated convex and concave functions in two variables are obtained.

#### 1. Introduction

Let be a convex function and with ; we have the following double inequality: This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping.

Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1â€“4]).

A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].

Let us consider the bidimensional interval in with and ; a mapping is said to be convex on if the inequality holds for all , , and .

A function is said to be coordinated convex on if the partial mappings , and , are convex for all and .

A formal definition for coordinated convex functions may be stated as follows.

*Definition 1. *A function is said to be convex on coordinates on if the inequality
holds for all , , , , and .

Dragomir in [5] established the following Hadamard-type inequalities for coordinated convex functions in a rectangle from the plane .

Theorem 2. *Suppose that is convex on the coordinates on . Then one has the inequalities as follows:
*

Some new integral inequalities that are related to the Hermite-Hadamard type for coordinated convex functions are also established by many authors.

In ([6], 2008), Alomari and Darus defined coordinated -convex functions and proved some inequalities based on this definition. In ([7], 2009), analogous results for -convex functions on the coordinates were proved by Latif and Alomari. In ([8], 2009), Alomari and Darus established some Hadamard-type inequalities for coordinated log-convex functions.

In ([9], 2012), Latif and Dragomir obtained some new Hadamard type inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables based on the following lemma.

Lemma 3. *Let be a partial differentiable mapping on in with and . If , then the following equality holds:
**
where
*

Theorem 4 (see [9]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on , then the following equality holds:
**
where
*

Theorem 5 (see [9]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , , , then the following equality holds:
**
where is as given in Theorem 4.*

Theorem 6 (see [9]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , then the following equality holds:
**
where is as given in Theorem 4.*

In ([10], 2012), analogous results which are related to the right-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables were proved by SarÄ±kaya et al. based on the following lemma.

Lemma 7. *Let be a partial differentiable mapping on in with and . If , then the following equality holds:
*

Theorem 8 (see [10]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on , then the following equality holds:
**
where
*

Theorem 9 (see [10]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , , , then the following equality holds:
**
where is as given in Theorem 8.*

Theorem 10 (see [10]). *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , then the following equality holds:
**
where is as given in Theorem 8.*

In [11], Ozdemir et al. established some Simpsonâ€™s inequalities for coordinated convex functions based on the following lemma.

Lemma 11. *Let be a partial differentiable mapping on in with and . If , then the following equality holds:
**
where
*

Theorem 12 (see [11]). *Let be a partial differentiable mapping on . If is convex on the coordinates on , then the following equality holds:
**
where
*

For recent results and generalizations concerning Hermite-Hadamard type inequality for differentiable coordinated convex functions see ([12], 2012) and the references given therein.

In this paper, a generalized lemma is proved and several new inequalities for differentiable coordinated convex and concave functions in two variables are obtained.

#### 2. Lemmas

To establish our results, we need the following lemma.

Lemma 13. *Let be a partial differentiable mapping on in with and . If and , then the following equality holds:
**
where
*

*Proof. *Since
thus, by integration by parts, it follows that

Similarly, we can get

Now
Multiplying both sides by and using the change of the variable and , which completes the proof.

*Remark 14. *Applying Lemma 13 for , , , we get the results of Lemmas 3, 7, and 11, respectively.

#### 3. Main Results

Theorem 15. *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , then the following equality holds:
**
where
*

*Proof. *From Lemma 13, we obtain
Because is a convex function on the coordinates on , then one has
On the other hand, we have
which completes the proof.

*Remark 16. *Applying Theorem 15 for , 1, , we get the results of Theorems 4, 8, and 12, respectively.

Theorem 17. *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , one gets the following inequality:
**
where and is as given in Theorem 15 and .*

*Proof. *From Lemma 13, we obtain
By using the well-known HÃ¶lder inequality for double integrals, then one has
Because is a convex function on the coordinates on , by (4), then one has
We note that
Hence, it follows that

*Remark 18. *Applying Theorem 17 for , 1, we get the results of Theorems 5 and 9, respectively.

Theorem 19. *Let be a partial differentiable mapping on in with and . If is convex on the coordinates on and , then
*