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

(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 satisfied. If is convex on , then for each the following inequality holds:

(3)

Theorem 1.2. The assupmtions of Lemma 1.1 are satisfied. 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].

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:

(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 satisfied. 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 satisfied. 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).

References

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