Abstract
In this paper, we give Hermite–Hadamard type inequalities of the Jensen–Mercer type for Riemann–Liouville fractional integrals. We prove integral identities, and with the help of these identities and some other eminent inequalities, such as Jensen, Hölder, and power mean inequalities, we obtain bounds for the difference of the newly obtained inequalities.
1. Introduction
The concept of convex functions plays a vital role in both pure and applied mathematics. Convex functions also have many applications in other branches of science such as finance, economics, and engineering.
Definition 1. (see [1]). A function is convex iffor all and .
If the inequality in (1) is strict for , then is said to be a strictly convex function, and if is convex, then is said to be a concave function [2, 3].
Many important inequalities such as Jensen, Jensen–Mercer, Hermite–Hadamard, and support line inequalities hold for convex functions. The classical Jensen’s inequality is among the most prominent inequalities stated as follows [4, 5].
If is convex, thenfor all and with .
In [6], Mercer presented a type of Jensen’s inequality called Jensen–Mercer inequality.
Theorem 1. If is convex, thenfor each and with .
For a convex function, there exist at least one line lies on or below the graph of the function.
Definition 2. (see [7]). A function has a support at iffor all and for each . Inequality (4) is said to be the support line inequality.
The following theorem connects the support line inequality with convex functions.
Theorem 2. (see [7]). The function is convex if and only if has at least one line of support at each .
The Hermite–Hadamard inequality is one of the most investigated inequality in the theory of convex functions due to its geometrical significance and applications. Because of the importance of Hermite–Hadamard inequality, there is an ample amount of research work dedicated to the extensions, generalizations, refinements, and applications of the Hermite–Hadamard inequality. The Hermite–Hadamard inequality is given below [8].
Let be a convex function, where is an interval and such that . Then,
If is concave, then (5) holds in the reversed direction. For more results associated with Hermite–Hadamard inequality, see [9–18].
The Hermite–Hadamard inequality has been extended by means of fractional integral operators. Most popular of them is the Riemann–Liouville fractional operator given in the following definition [19–22].
Definition 3. (see [23, 24]). Let be an integrable function defined on . Then, the integrals and defined byare called the left and right Riemann–Liouville fractional integrals of order respectively. Here, represents gamma function defined by .
In [25, 26], authors used the following lemmas to obtain trapezoidal and midpoint type inequalities.
Lemma 1. (see [25]). Let (where is the interior of ) be a differentiable function and such that . If , then
Lemma 2. (see [26]). Let all the assumptions of Lemma 1 hold. Then,In this article, we establish fractional Hermite–Hadamard–Jensen–Mercer type inequalities. We give identities involving fractional integrals, and from these identities, we derive trapezoidal and midpoint type inequalities.
Throughout this article, represents a positive real number.
2. Main Results
We begin this section with our first main result.
Theorem 3. Suppose is a convex function and such that . Then,
Proof. Since is convex, it has support line at each point , that is,for each . Substituting and , where , in inequality (12), we obtainMultiplying (13) with and integrate with respect to , we obtainUsing Mercer’s inequality, we obtainSince is convex, we have and (15) becomesSubstituting in (16), we obtainNow, we prove the second inequality of (10). Put and in (12), we obtainMultiplying the above inequality with and integrating and using and , we obtainPut , and we obtainAdding on both sides of (20), we obtainand on combining (17) and (21), we obtain (10).
Now, we prove the inequalities in (11). Let and (14) becomeNow, we prove the other two inequalities of (11). As is a convex function, we haveMultiplying with and integrating, we obtainBy changing of variable, (24) becomesand on combining (22) and (25), we obtain (11).
Remark 1. If we put in Theorem 3 and in the obtained expressions substitute = and , respectively, we obtainrespectively. Inequalities (26) and (27) have been proved in [27].
Remark 2. Substituting , , and in (11), one can obtain Hermite–Hadamard inequality.
3. Bounds for the Difference of Hermite–Hadamard–Jensen–Mercer Type Inequalities
Throughout this section, we consider is a differentiable function. To give the bounds for the difference of Hermite–Hadamard–Jensen–Mercer type inequalities, first, we present the following lemmas.
Lemma 3. Let such that and let . Then,
Proof. Using the techniques of integration, we have
Remark 3. Substituting , , and in (28), we obtain (8).
Lemma 4. Let all the assumptions of Lemma 3 hold. Then,
Proof. Using techniques of integration, we have
Remark 4. If we put , , and in (31), we obtain (9).
We use Lemmas 3 and 4 and obtain bounds for the difference of the inequalities in (11).
Theorem 4. Let be a convex function defined on and let such that . Then,where
Proof. From Lemma 3, we haveSince is convex, using Mercer’s inequality, we obtainequivalent towhereSubstituting these values in (37), we get (33).
Remark 5. If we put , , and in (33), we get the inequality given in Theorem 2.2 of [25].
Theorem 5. Let be a convex function for and let such that . Then,where , , and are the same as defined in Theorem 4.
Proof. From Lemma 3, we have (35). Applying power mean inequality, we obtainSince is convex, using Mercer’s inequality, we haveThis implies thatSubstituting the values of , , , , , and as given in the proof of Theorem 4 in (42), we get (39).
Remark 6. If we put , , and in (39), we obtain the inequality proved in Theorem 1 of [28].
In the following theorem, we derive trapezoidal type inequality.
Theorem 6. Let , and be a convex function, and let such that . Then,where such that .
Proof. Using Lemma 3, we have (35). Applying Hölder’s inequality, we obtain
Remark 7. Substituting , , and in Theorem 6, we obtain Theorem 2.3 of [25].
Theorem 7. Let be a convex function and let such that . Then,where
Proof. Using Lemma 4, we haveSince is convex, using Mercer’s inequality, we obtainwhich implies thatwhereSubstituting these values in (49), we get (45).
In next theorem, we use power mean inequality and derive midpoint type inequality.
Theorem 8. Let be a convex function for and let such that . Then,whereand , , , and are given in the proof of Theorem 7.
Proof. Using power mean inequality in (47), we obtainSince is convex, using Mercer’s inequality, we obtain
Remark 8. Substituting , , and in Theorem 8, we obtain the following midpoint type inequality:Another midpoint type inequality is presented in the following theorem.
Theorem 9. Let , and be convex, and let such that . Then,where and such that .
Proof. Applying Hölder’s inequality in (47), we haveAs is convex, applying Mercer’s inequality, we obtain
Remark 9. Substituting , , and in Theorem 9, we obtain
Theorem 10. Let such that . If is concave on , thenwhere and are given in the proof of Theorem 4 and
Proof. Lemma 3 implies thatSince is concave, using Jensen’s inequality, we obtain
Theorem 11. Let be a concave function, and let such that . Then,where
Proof. From Lemma 4, we have (47). As is concave, using Jensen’s inequality, we obtain
4. Conclusion
In this paper, we establish the fractional Hermite–Hadamard type inequalities of Mercer type by using support line inequality. We expect that this work will lead to the new fractional integral studies for Hermite–Hadamard inequality. It is an open problem to prove inequalities (10) and (11) by any other method.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to this paper. All authors read and approved the final manuscript.