Abstract

In this paper, certain Hermite–Hadamard–Mercer-type inequalities are proved via Riemann–-Liouville fractional integral operators. We established several new variants of Hermite–Hadamard’s inequalities for Riemann–Liouville fractional integral operators by utilizing Jensen–Mercer inequality for differentiable mapping whose derivatives in the absolute values are convex. Moreover, we construct new lemmas for differentiable functions , , and and formulate related inequalities for these differentiable functions using variants of Hölder’s inequality.

1. Introduction

The study of convex functions started over the period 1905 to 1906 by thought provoking ideas and fascinating work of Jensen. It is used as a major tool to solve optimization problems in analysis. However, inequalities involving convex functions are quite stimulating in the expansion of numerous sections of mathematics, for example, mathematical finance, economics, management sciences, and optimization theory. That is why the study of such inequalities have been given great importance in the literature [1–4].

Let and let be nonnegative weights such that . The famous Jensen inequality (see [5]) in the literature states that if is a convex function on the interval , thenfor all and all .

Mercer gave a variant of Jensen’s inequality [6] as follows.

Theorem 1. If is a convex function on , then and .

Matkovic and Pecaric worked on Jensen’s inequality of Mercer’s type for operators with applications [7]. Later, Mercer’s result was generalized to higher dimensions by Niezgoda [8]. In recent years, notable contributions have been made on Jensen–Mercer’s type inequality. Kian gave concept of Jensen inequality for superquadratic functions [9]. Furthermore, Anjidani and Changalvaiy worked on reverse Jensen–Mercer-type operator inequalities and Jensen–Mercer operator inequalities for superquadratic functions (see [10, 11]). Ali and Khan generalized integral Mercer’s inequality and integral means in [12].

Another important inequality that characterize convex function is Hermite–Hadamard inequality, that is, if a mapping is a convex function on and , , then

Fractional calculus is the study of fractional order derivatives and integrals. Fractional calculus has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics (see [13–15]). Now, we give necessary definition of fractional calculus theory, which is used throughout this paper.

Definition 1. Let . The Riemann–Liouville integrals and of order with are defined byRespectively, where is Euler Gamma function and .
In the case of , the fractional integral reduces to classical integral. Several researchers have focused on new integral inequalities involving Riemann–Liouville fractional integrals in recent years (see [16–18]) and references therein.
In this article, by using the Jensen–Mercer inequality, we proved Hermite–Hadamard’s inequalities for fractional integrals and established some new Riemann–Liouville fractional integrals connected with the left and right sides of Hermite–Hadamard-type inequalities for differentiable mappings whose derivatives in the absolute value are convex. Moreover, there will be further equalities for , , and and related inequalities for these differentiable functions using Hölder’s inequality.

2. Variant of Hermite–Hadamard–Mercer-Type Inequalities

By using Jensen–Mercer inequalities, Hermite–Hadamard-type inequalities can be expressed in Riemann–Liouville fractional integral form as follows:

Theorem 2. Let be a convex function. Then, the following fractional integral inequalities hold:where and is the Gamma function.

Proof. Using convexity of , we have.
By change of variables and we obtainMultiply both sides by and then integrate the resulting inequality with respect to over . Let and , and we haveso the first inequality of (5) is proved.
Now, for the proof of second inequality of (5), we first note that if is convex function, then for it givesBy adding the inequalities of (11) and (12), we haveMultiplying both sides by and then integrating the resulting inequality with respect to over we haveSimplifying the integrals givesCombining (10) and (15), we will get (5). Now, in order to prove (6), we employ Jensen–Mercer inequality aswhere .
Now, by change of variables and , and , in (16), we haveMultiplying both sides by and then integrating the resulting inequality with respect to over we haveso the first inequality of (6) is proved.
Now, for the proof of second inequality of (6), we first note that if is convex function, then for ,Multiplying both sides by and then integrating the resulting inequality with respect to over we haveMultiplying by ,Adding both sides in (21), we get the second inequality of (6).

Remark 1. Under the assumption of Theorem 2 for inequality (5) with , one hasInequality (22) is proved by Kian and Moslehian in [19].

In this section, we prove new lemmas for , , and , which plays a key part in proving our inequalities. Throughout the rest of the paper, we need the following assumption.

() Let , , , and is the Gamma function.

Lemma 1. Let be a differentiable mapping on with along with assumption . If , then the following equality holds:

Proof. It suffices to note thatwhereCombining (25) and (26) with (24), we get (22).

Corollary 1. For in Lemma 1, we will obtain

Remark 2. Taking , , and in Lemma 1, we get Lemma 2.1 in [20] and the following inequality holds:

Theorem 3. Suppose that is a differentiable mapping on with and along with assumption . If is differentiable function on then the following inequality for Riemann–Liouville fractional integral holds:

Proof. By using Lemma 1 and applying mean value theorem for the function , we havewhere . This leads us to

Theorem 4. Suppose that is a differentiable mapping on with and along with assumption . If is convex function on then the following inequality for Riemann–Liouville fractional integral holds:

Proof. By using Lemma 1 and Jensen–Mercer inequality, we haveSolving the integrals gives the required result.

Corollary 2. For in Theorem 4, we obtain

Remark 3. Taking , , and in Theorem 4, we get (Theorem 2.2, [21]), and following inequality holds:Now we state our results for twice differentiable functions :

Lemma 2. Let be a differentiable mapping on with along with assumption . If , then the following equation holds:

Proof. It suffices to note thatwhereandCombining (39) and (40) with (38), we get (33).

Remark 4. Taking and in Lemma 2 gives Lemma 1 of [22].

Remark 5. For , , and in Lemma 2, it reduces to Lemma 2 proved in [22].

Theorem 5. Suppose that is a differentiable mapping on with and along with assumption . If is convex function on then the following inequality for Riemann–Liouville fractional integral holds:

Proof. By using Lemma 2 and Jensen–Mercer inequality, we haveBy simplifying the integrals, we get the required result.

Remark 6. For and in Theorem 5, we will get Theorem 5 proved in [22] given as follows:

Theorem 6. Suppose that is a three times differentiable function on along with assumption . If and is convex function, where , , then

Proof. By using Lemma 2 and well-known Hölder’s inequality with Jensen–Mercer inequality on the fact that is convex function, we haveFinally, we state our results for three times differentiable functions :

Lemma 3. Let be a differentiable mapping on with along with assumption . If , then the following equation holds:

Proof. Proof. It suffices to note thatwhereReplacing the values of the integrals and in (47), we get (46).

Remark 7. For and in Lemma 3, it reduces to Lemma 1 proved in [23].

Remark 8. For , , and in Lemma 3, it reduces to Lemma 2.1 proved in [24].

Theorem 7. Suppose that is a differentiable mapping on with and along with assumption . If is convex function on then the following inequality for Riemann–Liouville fractional integral holds:

Proof. By using Lemma 3 and Jensen–Mercer inequality, we have

Remark 9. For and in Theorem 7, we will get Theorem 4 proved in [23].

Theorem 8. Suppose that is a three times differentiable function on along with assumption . If and is a convex function, where , , then

Proof. By using Lemma 3 with Jensen–Mercer inequality and well-known Hölder’s inequality on the fact that is a convex function, we haveAfter some further simplification of integrals, we get the required results.

4. Conclusion

In this paper, we have obtained some new Hermite–Jensen–Mercer-type integral inequalities for Riemann–Liouville Fractional integral operators. The results presented here would provide elegant connections and generalizations of well-known results established recently. These results can be applied in convex analysis, optimization, and different areas of pure and applied sciences. The authors hope that these results will serve as a motivation for future work in this fascinating area. This research is open for further work by investigating such results for other classes of convex functions. However, it is not easy to give Jensen–Mercer inequality for other classes of convex functions.

Data Availability

The data used to support the findings of the study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This paper is supported by HEC Pakistan.