Journal of Applied Mathematics

Volume 2014 (2014), Article ID 248710, 8 pages

http://dx.doi.org/10.1155/2014/248710

Research Article

On Hermite-Hadamard Type Inequalities for -Convex Functions on the Coordinates via Riemann-Liouville Fractional Integrals

Feixiang Chen

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

Received 29 December 2013; Accepted 14 April 2014; Published 30 April 2014

Academic Editor: Fernando Simões

Copyright © 2014 Feixiang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We obtain some Hermite-Hadamard type inequalities for -convex functions on the coordinates via Riemann-Liouville integrals. Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also established.

1. Introduction

If is a convex function on the interval , then, for any with , we have the following double inequality:

This remarkable result is well known in the literature as the Hermite-Hadamard inequality.

Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–7]).

Definition 1 (see [8]). Let be a fixed real number. A function is said to be -convex in the second sense, or that belongs to the class , if the inequality,

holds for all and .

It can be easily seen that, for , -convexity reduces to ordinary convexity of functions defined on .

In [4], Dragomir defined convex functions on the coordinates as follows.

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 convex functions on the coordinates may be stated as follows.

Definition 2. A function is said to be convex on coordinates on if the inequality,

holds for all , , , , and .

In [4], Dragomir established the following Hadamard-type inequalities for convex functions on the coordinates in a rectangle from the plane .

Theorem 3 (see [4]). Suppose that is convex on the coordinates on . Then one has the inequalities:

The concept of -convex functions on the coordinates in the second sense was introduced by Alomari and Darus in [9].

Let us consider the bidimensional interval in with and ; a mapping is -convex on if the inequality,

holds for all , with and for some fixed .

A function is said to be -convex on the coordinates on in the second sense if the partial mappings , , and , are -convex in the second sense for all and with some fixed .

A formal definition for convex functions on the coordinates in the second sense may be stated as follows.

Definition 4. A function is said to be -convex on coordinates in the second sense on if the inequality,

holds for all , , , with and some fixed .

In [10], Alomari and Darus proved the following inequalities based on the above definition.

Theorem 5 (see [10]). Suppose that is s-convex on the coordinates in the second sense on . Then one has the inequalities:

It is remarkable that Sarikaya et al. [11] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 6 (see [11]). Let be a positive function with and . If is a convex function on , then the following inequalities for fractional integrals hold

with .

We remark that the symbol and denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order with which are defined by

respectively. Here, is the Gamma function defined by .

Definition 7 (see [12]). Let . The Riemann-Liouville fractional integrals , , and of order with are defined by

In [12], Sarıkaya proposed the following Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals by using convex functions of two variables on the coordinates.

Theorem 8 (see [12]). Let be convex functions on the coordinates on in with , , and . Then one has the inequalities:

In [12], Sarıkaya established some Hermite-Hadamard inequalities for convex functions on the coordinates in the second sense via fractional integrals based on the following lemma.

Lemma 9 (see [12]). Let be a partial differentiable mapping on in with , . If , then the following equality holds:

where

In this paper, we establish some Hermite-Hadamard type inequalities for -convex functions on the coordinates functions via Riemann-Liouville integrals. Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also given.

2. Fractional Inequalities for -Convex Functions on the Coordinates

Theorem 10. Suppose that is -convex function on the coordinates in the second sense on . Then one has the inequalities:

Proof. From (7), with , , , , and , we get

Multiplying both sides of above inequality by then integrating the resulting inequality with respect to over , we obtain

Using the change of the variable, we get

by which the first inequality is proved. For the proof of the second inequality, we note that is -convex on coordinates, then

Multiplying both sides of above inequalities by , then integrating the resulting inequality with respect to over , we obtain

Here, using the change of the variable, we have

The proof is completed.

Remark 11. Applying Theorem 10 for , we get Theorem 8.

We note that the Beta functions is defined by

3. Inequalities for Differentiable Functions

Theorem 12. Let be a partial differentiable mapping with , . If is -convex on the coordinates in the second sense on for some fixed . Then one has the inequalities:

where is as given in Lemma 9.

Proof. From Lemma 9, we obtain

Because is -convex function on the coordinates on , we obtain

which completes the proof.

Theorem 13. Let be a partial differentiable mapping with , . If is -convex on the coordinates in the second sense on for some fixed and . Then, one has the inequalities:

where and are as given in Lemma 9.

Proof. From Lemma 9, we obtain

By using the well-known Hölder's inequality for double integrals, then one has

Because is -convex function on the coordinates on , by (8), we have

So

which completes the proof.

Theorem 14. Let be a partial differentiable mapping with , . If is -concave on the coordinates in the second sense on for some fixed and , then one has the inequalities:

where and is as given in Lemma 9.

Proof. Similarly as in Theorem 13, we obtain

Because is -concave function on the coordinates on , by the reversed direction of (8), then one has

Then

We get the desired results.

4. Conclusion

In this paper, we obtain some Hermite-Hadamard type inequalities for coordinated -convex functions via Riemann-Liouville integrals. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for other kinds of convex functions on the coordinates via Riemann-Liouville integrals.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by Youth Project of Chongqing Three Gorges University of China (no. 13QN11).

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