Hadamard and Fejér–Hadamard Inequalities for Further Generalized Fractional Integrals Involving Mittag-Leffler Functions

Abstract

In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

1. Introduction

First we give definitions of fractional integral operators which are useful in establishing the results of this paper. In the following, we give fractional integral operators defined by Andrić et al. in [1] via an extended generalized Mittag-Leffler function in their kernels.

Definition 1 (see [1]). Let , , with , and . Let and . Then, the generalized fractional integral operators and are defined bywhereis the extended generalized Mittag-Leffler function and is the extension of beta function which is defined as follows:where are positive real numbers.
Recently, Farid defined elegantly a unified integral operator in [2] (see, also [3]) as follows.

Definition 2. Let , be the functions such that be positive and and be a differentiable and strictly increasing function. Also, let be an increasing function on and , , with , , and .
Then, for , the integral operators and are defined byThe following definition of generalized fractional integral operators containing extended Mittag-Leffler function in the kernel can be extracted from Definition 2. It is generalization of Definition 1 by a monotonically increasing function.

Definition 3. Let , be the functions such that be positive and and be a differentiable and strictly increasing function. Also, let , , with , and . Then, for , fractional integral operators are defined byThe following remark provides connection of Definition 3 with existing fractional integral operators.

Remark 1. (i)If we set and in equations (5) and (6), then these reduce to fractional integral operators defined by Salim and Faraj in [4].(ii)If we set and in equations (5) and (6), then these reduce to the fractional integral operators and containing generalized Mittag-Leffler function defined by Rahman et al. in [5].(iii)If we take and in equations (5) and (6), then these reduce to fractional integral operators containing extended generalized Mittag-Leffler function introduced by Srivastava and Tomovski in [6].(iv)If we set and in equations (5) and (6), then these reduce to fractional integral operators defined by Prabhaker in [7].(v)For and in equations (5) and (6), these reduce to renowned Riemann–Liouville fractional integral operators [8].The Riemann–Liouville fractional integrals for a function of order are defined byAfter introducing generalized fractional integral operators, now we define notions of functions for which generalized fractional integral operators are utilized to get main results of this paper.

Definition 4 (see [9]). A function is said to be convex ifholds for all and .

Definition 5 (see [10]). Let be an interval such that . Then, a function is said to be harmonically convex, ifholds for all and .

Definition 6 (see [11]). Let be a real interval and . Then, a function is said to be -convex, ifholds for and .
It is easy to see that for and , the -convexity reduces to convexity and harmonical convexity, respectively.

Definition 7 (see [11]). Let . Then, a function is said to be -symmetric with respect to ifholds, for .
Convex functions are equivalently studied by the Hadamard inequality.

Theorem 1. Let be a convex function such that . Then, the following inequality holds:

The Fejér–Hadamard inequality is a weighted version of the Hadamard inequality given by Fejér in [12].

Theorem 2. Let be a convex function and be non-negative, integrable, and symmetric about . Then, the following inequality holds:

In recent decades, the Hadamard and the Fejér–Hadamard fractional integral inequalities have been studied extensively for different kinds of convex functions (see [1, 3, 13–21]). In this paper, we find Hadamard and Fejér–Hadamard inequalities for a generalized fractional integral operator involving an extended generalized Mittag-Leffler function.

In the upcoming section, we give two versions of the Hadamard inequality as well as two versions of the Fejér–Hadamard inequality. Their special cases are also discussed along with noticing connections with published results.

2. Main Results

First we give the following version of the Hadamard inequality.

Theorem 3. Let , Range be the functions such that be positive and , and be a differentiable and strictly increasing function. If is -convex, , then the following inequalities for fractional integral operators (5) and (6) hold:(i)If , then where and for all .(ii)If , then where and for all .

Proof. (i) Since is -convex over , for all , we haveSetting and in above inequality, we haveMultiplying both sides of (17) by and then integrating over , we haveBy choosing and in (18), we havewhere .
This impliesTo prove the second inequality of (14), again from -convexity of over and for , we haveMultiplying both sides of (21) by and then integrating over , we haveSetting and in (22), we haveBy combining (20) and (23), we get (14).
(ii) Proof is similar to the proof of (i).

Remark 2. (i)By setting and , Theorem 9 of [11] is obtained.(ii)By setting , , and , Theorem 2.1 of [22] is obtained.(iii)By setting and , Theorem 2.1 of [23] is obtained.(iv)By setting , , and , Theorem 4 of [18] is obtained.(v)By setting , Theorem 2.1 of [24] is obtained.(vi)By setting and , Corollary 2.3 of [24] is obtained.

Corollary 1. In (15), if we take , , and , then we get the following Hadamard inequality for the RL fractional integrals:

Now we obtain Fejér–Hadamard type fractional integral inequalities for -convex function via generalized fractional integral operators; for this, first we prove the following lemma.

Lemma 1. Let , Range be the functions such that be positive and , , and be a differentiable and strictly increasing function. If is -convex, , and , then for generalized fractional integral operators (5) and (6), we have(i)If , then with , .(ii)If , then with , .

Proof. (i) By definition of generalized fractional integral operators (5) and (6), we haveSetting in the above equation and using , we haveThis impliesBy adding on both sides of (29), we haveFrom equations (29) and (30), the required result can be obtained.
(ii) Proof is on the same lines as the proof of (i).

Theorem 4. Let , Range , Range be the functions such that be positive and , , and be a differentiable and strictly increasing function where is a non-negative and integrable function. If is -convex, , and , then the following inequalities for generalized fractional integral operators (5) and (6) hold:(i)If , then where and for all .(ii)If , then where and for all .

Proof. (i) Multiplying both sides of (17) by and then integrating over , we haveBy choosing , that is, , in (33) and using , we haveThis impliesUsing Lemma 1 (i) in above inequality, we haveTo prove the second inequality of (31), multiplying both sides of (21) by and then integrating over , we haveSetting and using in (37) and after simplification, we haveUsing Lemma 1 (i), inequality (38) becomesBy combining (36) and (39), we get (31).
(ii) Proof is similar to the proof of (i) by using (ii) of Lemma 1.

Remark 3. (i)By setting and , Theorem 2.2 of [25] is obtained.(ii)By setting and , Theorem 9 of [11] is obtained.(iii)By setting , , , and , Theorem 2.1 of [22] is obtained.(iv)By setting , , and , Theorem 2.1 of [23] is obtained.(v)By setting , , , and , Theorem 4 of [18] is obtained.(vi)By setting , Theorem 2.5 of [24] is obtained.

Corollary 2. If we put , , and in Theorem 4 (ii), we get the following Fejér–Hadamard inequalities for harmonically convex function via generalized fractional integral operators:

Now we give another version of the Hadamard inequality.

Theorem 5. Let , Range be the functions such that be positive and , , and be a differentiable and strictly increasing function. If is -convex, , then for generalized fractional integral operators (5) and (6), we have(i)If , then where and for all .(ii)If , then where and for all .

Proof. (i) Setting and in (12), we haveMultiplying both sides of (43) by and then integrating over , we haveBy choosing and in (44) and by (5) and (6), we get first inequality of (41).
To prove the second inequality of (41), again from -convexity of over and for , we haveMultiplying both sides of (45) by and then integrating over , we haveSetting and in (46) and using (5) and (6), we get second inequality of (41).
(ii) Proof is similar to the proof of (i).

Remark 4. (i)By setting , , and , Theorem 2.3 of [22] is obtained.(ii)By setting and , Theorem 2.3 of [23] is obtained.(iii)By setting , Theorem 2.7 of [24] is obtained.Now we obtain another Fejér–Hadamard type fractional integral inequality for -convex function via generalized fractional integral operators (5) and (6).

Theorem 6. Let , Range , Range be the functions such that be positive and , , and be a differentiable and strictly increasing function where is a non-negative and integrable function. If is -convex, , and , then the following inequalities for generalized fractional integral operators (5) and (6) hold:(i)If , then where and for all .(ii)If , then where and for all .

Proof. (i)
Multiplying (43) by and then integrating over , we haveBy choosing , that is, , in (49) and using the condition , one can get first inequality of (47).
To prove the second inequality of (47), multiplying both sides of (45) by and then integrating over , we haveSetting in (38) and using the condition , one can get second inequality of (47).
(ii) Proof is similar to the proof of (i).

Remark 5. (i)By setting , , and , Theorem 2.6 of [22] is obtained.(ii)By setting and , Theorem 2.6 of [23] is obtained.(iii)By setting , Theorem 2.10 of [24] is obtained.

Corollary 3. When we set , , and in Theorem 6, then we get the following inequalities via RL fractional integrals.

3. Conclusion

We have established Hadamard and Fejér–Hadamard fractional integral inequalities for generalized fractional integrals of -convex functions. The results of this paper hold simultaneously for convex and harmonically convex functions for different fractional integral operators containing Mittag-Leffler functions.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.