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Research Article | Open Access

Volume 2014 |Article ID 741291 | https://doi.org/10.1155/2014/741291

Feixiang Chen, "Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions", Chinese Journal of Mathematics, vol. 2014, Article ID 741291, 12 pages, 2014. https://doi.org/10.1155/2014/741291

# Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

Accepted16 Jan 2014
Published19 Mar 2014

#### 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., ).

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

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

In (, 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 ). 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 ). 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 ). 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 (, 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 ). 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 ). 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 ). 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 , 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 ). 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 (, 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