Vol. 4, No. 5, 2016, pp 135-139. doi: 10.12691/tjant-4-5-3 | Research Article

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

Received August 06, 2016; Revised September 12, 2016; Accepted September 24, 2016

- P. O. Mohammed. Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions.
*Turkish Journal of Analysis and Number Theory*. Vol. 4, No. 5, 2016, pp 135-139. http://pubs.sciepub.com/tjant/4/5/3

- Mohammed, P. O.. "Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions."
*Turkish Journal of Analysis and Number Theory*4.5 (2016): 135-139.

- Mohammed, P. O. (2016). Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions.
*Turkish Journal of Analysis and Number Theory*,*4*(5), 135-139.

- Mohammed, P. O.. "Inequalities of Type Hermite-Hadamard for Fractional Integrals via Differentiable Convex Functions."
*Turkish Journal of Analysis and Number Theory*4, no. 5 (2016): 135-139.

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The following definition is well known in the literature.

**De****fi****nition 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

(1) |

that is known as Hermite-Hadamard inequality ^{[1]}. For more systematic information, please refer to the monographs [3-8]^{[3]} 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*:

(2) |

*where*

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

(3) |

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

(4) |

*where*

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

(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]}.

We start with the following lemma:

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

(6) |

*where*

*and *

*Proof*. Denoting

(7) |

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

(8) |

and similarly

(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 satis**fi**ed. If** ** **is convex on** ** then for each** ** **the following inequality holds*:

(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*:

(11) |

*where*

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

(12) |

Since is convex, then we have

(13) |

and

(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 satis**fi**ed. If** ** **is convex on** ** **for** ** **then the following inequality holds*:

(15) |

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

(16) |

Since is convex, then we have

(17) |

and

(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).

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