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American Journal of Public Health Research
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The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
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Anne Lanham, Lillian Mwanri

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Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions

P. O. Mohammed

Department of Mathematics, College of Education, Sulaimani University, Sulaimani, Kurdistan Region, Iraq

Abstract: In this paper, we established some new Hermite-Hadamard-type inequalities for differentiable convex functions via Reimann-Liouville fractional integrals. Moreover, our results improve and extend the corresponding ones in the literature.

Keywords: integral inequalities, Riemann-Liouville fractional integral, Hermite-Hadamard inequality, convex function, Hölder inequality

Citation: P. O. Mohammed, Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions. Turkish Journal of Analysis and Number Theory, 4 (5). 135-139. January 2016. doi:10.12691/tjant-4-5-3

1. Introduction

The following definition is well known in the literature.

Definition 1.1. [7,8] A function: is said to beconvex on the interval if for all and satisfies the following inequality:

Many significant inequalities have been studied for the class of convex functions, but the most important is the following inequality

[picture] | (1)

that is known as Hermite-Hadamard inequality [1]. For more systematic information, please refer to the monographs [3-8] and closely related references therein.

In what follows we recall the following definition [9].

Definition 1.2. Let The left Riemann-Liouville fractional integrals and of order are defined by

and

respectively. Here, is the gamma function and

Tomar et. al. [2] established the following Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals:

Lemma 1.1. Let be twice differentiable mapping on the interior of an interval such that where with a < b. Then for each and the following equality holds:

[picture] | (2)

where

Theorem 1.1. The assupmtions of Lemma 1.1 are satisfied. If is convex on , then for each the following inequality holds:

[picture] | (3)

Theorem 1.2. The assupmtions of Lemma 1.1 are satisfied. If is convex on for then the following inequality holds:

[picture] | (4)

where

Theorem 1.3. Let the assupmtions of Lemma 1.1 be satisfied. If is convex on for then the following inequality holds:

[picture] | (5)

The main objective of this article is to establish some Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. Our results improve and generalize some types of Hermite-Hadamard inequalities in [2].

2. Main Results

We start with the following lemma:

Lemma 2.1. Let be twice differentiable mapping on the interior of an interval such that where with Then for each and the following equality holds:

[picture] | (6)

where

and

Proof. Denoting

[picture] | (7)

Integrating by parts twice and changing variable of definite integral, we have

[picture] | (8)

and similarly

[picture] | (9)

Using equations (8)-(9) in (7) and by the simple calculations, we obtain the desired result.

Remark 2.1. If we take in Lemma 2.1, then the identity (6) reduces to the identity (2) which was proved by Tomar et. al. [2].

Theorem 2.1. The assupmtions of Lemma 1.1 are satisfied. If is convex on then for each the following inequality holds:

[picture] | (10)

Proof. From Lemma 2.1, we have

Now, since is convex, we have

and this ends the proof.

Remark 2.2. If we take in Theorem 2.1, then inequality (10) reduces to (3).

Theorem 2.2. The assupmtions of Lemma 1.1 are satisfied. If is convex on for then the following inequality holds:

[picture] | (11)

where

Proof. From Lemma 2.1 and using the well known Hölder integral inequality, we have

[picture] | (12)

Since is convex, then we have

[picture] | (13)

and

[picture] | (14)

Thus, if we use (13) and (14) in (12), we obtain

this completes the proof of first part of the Theorem.

Let and for Using the fact that for and we find

which completes the proof of last part of the Theorem. This completes the proof.

Remark 2.3. If we choose in Theorem 2.2, then inequality (11) reduces to (4).

Theorem 2.3. Let the assumptions of Lemma 1.1 be satisfied. If is convex on for then the following inequality holds:

[picture] | (15)

Proof. From Lemma 2.1 and using the well known power mean inequality, we have

[picture] | (16)

Since is convex, then we have

[picture] | (17)

and

[picture] | (18)

Combining equations (16), (17) and (18), we obtain inequality (15) as required.

Remark 2.4. If we choose in Theorem 2.3, then inequality (15) reduces to (5).

References

[1]   | J. Hadamard, Étude sur les propriétés des fonctions entières en particulier d'une function considérée par Riemann, J Math Pures Appl, 58 (1893), 171-215.
[2]   | M. Tomar, E. Set and M. Z. Sarıkaya, Hermite-Hadamard type Riemann-Liouville fractional integral inequalities for convex functions, AIP Conference Proceedings 1726, 020035 (2016).
[3]   | Z. Dahmani, On Minkowski and Hermite-Hadamad integral inequalities via fractional integration, Annals of Functional Analysis, 1(1) (2010), 51-58.
[4]   | M. Tunç, Y. Subas and I. Karabayir, On some Hadamard type inequalities for MTconvex functions, Int. J. Open Probl. Comput. Sci. Math., 6 (2013), 102-113.
[5]   | S. Qaisar and S. Hussain, Generalization of integral inequalities of the type of Hermite-Hadamard through invexity, Malaya J. Mat., 3(1) (2015), 99-109.
[6]   | J. Park, Fractional Hermite-Hadamard-like Type Inequalities for Convex Functions, International Journal of Mathematical Analysis, 9(29) (2015), 1415-1429.
[7]   | Z. Lin, J. Wang and W. Wei, Fractional Hermite-Hadamard inequalities through r-convex functions via power means, Ser. Math. Inform., 30(2) (2015), 129-145.
[8]   | M. Z. Sarikaya, H. Yaldiz and H. Budak, Some integral inequalities for convex stochastic processes, Acta Math. Univ. Comenianae, LXXXV(1) (2016), 155-164.
[9]   | I. Podlubni, Fractional Differential Equations, Academic Press, San Diego, 1999.