The Hermite–Hadamard–Jensen–Mercer Type Inequalities for Riemann–Liouville Fractional Integral

School of Mathematics and Statistics Changsha University of Science and Technology, Changsha 410114, China Government College of Management Sciences, Higher Education Department KPK, Hangu, Pakistan Department of Mathematics, University of Peshawar, Peshawar, Pakistan Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan Department of Mathematics, Abbotabad University of Science and Technology, Abbotabad, Pakistan


Introduction
e 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 ψ: for all x, y ∈ [m, M] and s ∈ [0, 1]. If the inequality in (1) is strict for x ≠ y, then ψ is said to be a strictly convex function, and if − ψ is convex, then ψ 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. e classical Jensen's inequality is among the most prominent inequalities stated as follows [4,5].
In [6], Mercer presented a type of Jensen's inequality called Jensen-Mercer inequality.
for each x i ∈ [m, M] and w i ∈ [0, 1](i � 1, 2, . . . , n) with n i�1 w i � 1. For a convex function, there exist at least one line lies on or below the graph of the function.
for all x 0 ∈ I and for each u ∈ [m, M] ⊂ I. Inequality (4) is said to be the support line inequality.
e following theorem connects the support line inequality with convex functions.
Theorem 2 (see [7]). e function ψ: [ e 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.
e 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][20][21][22].
en, the integrals J α m + ψ(x) and are called the left and right Riemann-Liouville fractional integrals of order α > 0 respectively. Here, Γ represents gamma function defined by Γ(α) � ∞ 0 e − s s α− 1 ds. In [25,26], authors used the following lemmas to obtain trapezoidal and midpoint type inequalities.
Lemma 1 (see [25]). Let ψ: Lemma 2 (see [26]). Let all the assumptions of Lemma 1 hold. en, 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. roughout this article, α represents a positive real number.
Now, we prove the other two inequalities of (11). As ψ is a convex function, we have Journal of Mathematics Multiplying with αs α− 1 and integrating, we obtain By changing of variable, (24) becomes and on combining (22) and (25), we obtain (11).

Bounds for the Difference of Hermite-Hadamard-Jensen-Mercer Type Inequalities
roughout this section, we consider ψ: [m, M] ⟶ R is a differentiable function. To give the bounds for the difference of Hermite-Hadamard-Jensen-Mercer type inequalities, first, we present the following lemmas.

Journal of Mathematics
Proof. Using techniques of integration, we have □ Remark 4. If we put α � 1, x � m, and y � M in (31), we obtain (9). We use Lemmas 3 and 4 and obtain bounds for the difference of the inequalities in (11). where Since |ψ ′ | is convex, using Mercer's inequality, we obtain where Journal of Mathematics Substituting these values in (37), we get (33).
Proof. From Lemma 3, we have (35). Applying power mean inequality, we obtain Since |ψ ′ | q is convex, using Mercer's inequality, we have is implies that Substituting the values of L 1 , L 2 , L 3 , N 1 , N 2 , and N 3 as given in the proof of eorem 4 in (42), we get (39). □ Remark 6. If we put α � 1, x � m, and y � M in (39), we obtain the inequality proved in eorem 1 of [28].
Proof. Using Lemma 3, we have (35). Applying Hölder's inequality, we obtain Journal of Mathematics 11 where Proof. Using Lemma 4, we have Since |ψ ′ | is convex, using Mercer's inequality, we obtain which implies that where Substituting these values in (49), we get (45). In next theorem, we use power mean inequality and derive midpoint type inequality. □ Theorem 8. Let |ψ ′ | q be a convex function for q ≥ 1 and let x, y ∈ (m, M) such that x < y. en, where and L 5 (α), L 6 (α), N 5 (α), and N 6 (α) are given in the proof of eorem 7.
Proof. Using power mean inequality in (47), we obtain

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
e authors declare that there are no conflicts of interest regarding the publication of this paper.