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., [14]).

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 Sarıkaya 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 Simpson’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 Hölder 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 where and is as given in Theorem 15.

Proof. From Lemma 13, we obtain By using the well-known power mean inequality for double integrals, then one has Because is a convex function on the coordinates on , then one has Thus, it follows that