1. Introduction

Let be a convex mapping defined on the interval of real numbers and with , then

(1)

This doubly inequality is known in the literature as Hermite-Hadamard integral inequality for convex mapping.We note that Hadamard’s inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen’s inequality. For several recent results concerning the inequality (1) we refer the interested reader to ^{[3, 5, 6, 8, 9, 11, 18, 21, 22]} and the references cited therein.

Definition 1.1 The function is said to be convex if the following inequality holds:

for all and . We say that f is concave if is convex.

In ^[18] Pearce and Pečarić established the following result connected with the right part of (1).

Theorem 1.2 Let be a differentiable mapping on , with , and let If the mapping convex on , then

(2)

The classical Hermite- Hadamard inequality provides estimates of the mean value of a continuous convex function .

We give some necessary definitions and mathematical preliminaries of fractional calculus theory which are used throughout this paper.

Definition 1.3 Let . The Riemann-Liouville integrals and of oder with are defined by

and

respectively, where is the Gamma function and

In the case of , the fractional integral reduces to the classical integral. Properties concerning this operator can be found (^{[7, 12, 17]}).

For some recent result connected with fractional integral see (^{[4, 19, 20, 22]}).

In ^[19] Sarıkaya et al. proved the following Hadamard type inequalities for fractional integrals as follows.

Theorem 1.4 Let be a positive function with and If is a convex function on, then the following inequalities for fractional integrals hold:

(3)

with

Using the following identity Sarıkaya et al. in ^[17] established the following result which hold for differentiable functions.

Lemma 1.5 Let be a differentiable mapping on with . If then the following equality for fractional integrals holds:

(4)

Theorem 1.6 Let be a differentiable mapping on with . If is a convex function on , then the following inequalities for fractional integrals holds:

(5)

In recent years several extentions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson in ^[8]. Weir and Mond ^[23] introduced the concept of preinvex functions and applied it to the establisment of the sufficient optimality conditions and duality in nonlinear programming. Pini ^[24] introduced the concept of prequasiinvex as a generalization of invex functions. Later, Mohan and Neogy ^[24] obtained some properties of generalized preinvex functions. Noor ^{[13, 14, 15, 16]} has established some Hermite-Hadamard type inequalities for preinvex and log-preinvex functions. In recent papers Barani et al. in ^[2] presented some estimates of the right hand side of a Hermite-Hadamard type inequality in which some preinvex functions are involved.

In this paper we generalized the results in ^[2] and ^[19] for preinvex functions via fractional integrals. Now we recall some notions in invexity analysis which will be used throught the paper (see ^{[1, 23]} and references therein)

Let and where is a nonempty set in , be continuous functions.

Definition 1.7 The set is said to be invex with respect to , if for every and

The invex set is also called a connected set. It is obvious that every convex set is invex with respect to , but there exist invex sets which are not convex ^[1].

Definition 1.8 The function on the invex set is said to be preinvex with respect to if

The function is said to be preconcave if and only if is preinvex.

Mohan and Neogy ^[24] introduced condition C defined as follows

Condition C: Let be an invex subset with respect to We say that the function satisfies the condition C if for any and any

(6)

Note that for every and every from condition C, we have

we will use the condition in our main results.

In ^[16], Noor proved the Hermite-Hadamard inequality for the preinvex functions as follows:

Theorem 1.9 Let be a preinvex function on the interval of real numbers (the interior of K) and with . Then the following inequality holds:

(7)

In ^[2] Barani, Gahazanfari, and Dragomir proved the following theorems:

Theorem 1.10 Let be an open invex subset with respect to Suppose that is a differentiable function. If is preinvex on then, for every with the following inequalities holds

(8)

Theorem 1.11 Let be an open invex subset with respect to Suppose that is a differentiable function. Assume that with If is preinvex on then, for every with the following inequalities holds

(9)

2. Main Results

Throughout this section, Let be an open invex subset. In this section, firstly we will establish Hermite- Hadamard’s inequalities for preinvex functions via fractional integrals. Secondly we will introduce some generalizations of the right side of a Hermite- Hadamard type inequalities for functions whose first derivatives absolute values are preinvex via fractional integrals.

Theorem 2.1 Let be an open invex subset with respect to and with If is a preinvex function, and satisfies condition C then, the following inequalities for fractional integrals holds:

(10)

with

Proof. Since and is an invex set with respect to , for every , we have By preinvexity of , we have for every with

i.e. with from equality (6) we get

(11)

Multiplying both sides (11) by , then integrating the resulting inequality with respect to over we obtain

i.e.

and the fist inequality is proved.

For the proof of the second inequality in (11) we first note that if is a preinvex function on and the mapping satisfies condition C then for every from inequality (6) it yields

(12)

and similarly

By adding these inequalities we have

(13)

Then multiplying both (13) by and integrating the resulting inequality with respect to over we obtain

i.e.

Using the mapping satisfies condition C the proof is completed.

Remark 2.2 a) If in Theorem 2.1, we let , then inequality (10) become inequality (3) of Theorem 1.4.

b) If in Theorem 2.1, we let , then inequality (10) become inequality (7) of Theorem 1.9.

Now we give the following lemma which is a generalization of Lemma 1.5 to invex setting.

Lemma 2.3 Let be an open invex subset with respect to and with If is a differentiable function such that then, the following equality holds:

(14)

Proof. It suffices to note that

(15)

integrating by parts

(16)

and similarly we get,

(17)

Using (16) and (17) in (15), it follows that

Thus, by multiplying both sides by , we have conclusion (14).

Remark 2.4 If in Lemma 2.3, we let , then equality (14) become inequality (4) of Lemma 1.5.

Theorem 2.5 Let be an open invex subset with respect to and with . Suppose that is a differentiable function such that . If is preinvex function on then the following inequality for fractional integrals with holds:

(18)

Proof. Using lemma 2.3 and the preinvexity of we get

which completes the proof.

Remark 2.6 a) If in Theorem 2.5, we let , then inequality (18) become inequality (5) of Theorem 1.6.

b) If in Theorem 2.5, we let , then inequality (18) become inequality (8) of Theorem 1.10.

c) In Theorem 2.5, assume that satisfies condition C and using inequality (12) for we get

Theorem 2.7 Let be an open invex subset with respect to and with such that . Suppose that is a differentiable function. If is preinvex function on for some fixed then the following inequality holds:

(19)

where and

Proof. From lemma2.3 and using Hölder inequality with properties of modulus, we have

We know that for and ,

therefore

Since is preinvex on we have inequality (19), which completes the proof.

Remark 2.8 a) If in Theorem 2.7, we let and then inequality (19) become inequality (9) of Theorem1.11.

b) In Theorem 2.7, assume that satisfies condition C and using inequality (12) we get

Theorem 2.9 Let be an open invex subset with respect to and with Suppose that is a differentiable function such that . If is preinvex function on for some fixed then the following inequality holds:

(20)

where and

Proof. From lemma2.3 and using Hölder inequality with properties of modulus, we have

On the other hand, we have

Since is preinvex function on , we obtain

and

from here we obtain inequality (20) which completes the proof.

Remark 2.10 a) If in Theorem2.9, we let and then inequality (20)become inequality (2) Theorem1.2.

b) In Theorem2.9, assume that satisfies condition C, using inequality (12) we get

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