Fractional Hermite–Hadamard–Fejer Inequalities for a Convex Function with Respect to an Increasing Function Involving a Positive Weighted Symmetric Function

: There have been many different deﬁnitions of fractional calculus presented in the literature, especially in recent years. These deﬁnitions can be classiﬁed into groups with similar properties. An important direction of research has involved proving inequalities for fractional integrals of particular types of functions, such as Hermite–Hadamard–Fejer (HHF) inequalities and related results. Here we consider some HHF fractional integral inequalities and related results for a class of fractional operators (namely, the weighted fractional operators), which apply to function of convex type with respect to an increasing function involving a positive weighted symmetric function. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities.


Introduction
First of all, we recall the basic notation in convex analysis. A set V ⊂ R is said to be convex if ε ϑ 1 + (1 − ε)ϑ 2 ∈ V for each ϑ 1 , ϑ 2 ∈ V and ε ∈ [0, 1]. Based on a convex set V, we say that a function : V → R is convex, if the inequality holds. We say that is concave if − is convex. Theory and application of convexity play an important role in the field of fractional integral inequalities due to the behavior of its properties and definition, especially in the past few years. There is a strong relationship between theories of convexity and symmetry. Whichever one we study, we can apply it to the other one; see, e.g., [1]. There are plenty of well-known integral inequalities that have been established for the convex functions (1) in the literature; for example, Ostrowski type integral inequalities [2], Simpson type integral inequalities [3], Hardy type integral inequalities [4], Olsen type integral inequalities [5], Gagliardo-Nirenberg type integral inequalities [6], Opial type type integral inequalities [7,8] and Rozanova type integral inequalities [9]. However, the most common integral inequalities are the Hermite-Hadamard type integral inequalities: the classical and fractional Hermite-Hadamard type integral inequalities [10,11] are, respectively, given by: and where : V → R is supposed to be a positive convex function, ∈ L 1 (ϑ 1 , ϑ 2 ) with ϑ 1 < ϑ 2 , and RL J ϑ 1 + and RL J ϑ 2 − stand for the left-sided and right-sided Riemann-Liouville fractional integrals of order > 0, respectively, and these are defined by [12,13]: The HH type inequality (2) has been applied to numerous types of convex functions, including s-geometrically convex functions [14], GA-convex functions [15], MT-convex function [16] and (α, m)-convex functions [17], and many other types can be found in [18]. Besides, the HH type inequality (3) has been applied to a huge number of convex functions, such as F-convex functions [19], λ ψ -convex functions [20], MT-convex functions [21] and (α, m)-convex functions [22], a new class of convex functions [23], and many other types can be found in the literature. Meanwhile, it has been applied to other models of fractional calculus, such as standard RL-fractional operators [24], conformable fractional operators [25,26], generalized fractional operators [27], ψ-RL-fractional operators [28,29], tempered fractional operators [30] and AB and Prabhakar fractional operators [31].

Definition 2.
Let (ϑ 1 , ϑ 2 ) ⊆ R and σ(x) be an increasing positive and monotonic function on the interval (ϑ 1 , ϑ 2 ] with a continuous derivative σ (x) on the interval (ϑ 1 , Then, the left-side and right-side of the weighted fractional integrals of a function with respect to another function σ(x) on [ϑ 1 , ϑ 2 ] are defined by [35]:

Remark 1. From the Definition 2, one can observe that
• If σ is specialized by σ(x) = x and w(x) = 1, then the weighted fractional integral operators (8) reduce to the classical Riemann-Liouville fractional integral operators (4).

•
If w(x) = 1, we get the fractional integral operators of a function with respect to another function σ(x), which is defined in [36,37] as follows: This study investigates several inequalities of HHF type via the weighted fractional operators (8) with positive weighted symmetric functions in the kernel.
The rest of the study is structured in the following way: In Section 2, we prove the necessary and auxiliary lemmas that are useful in the next section. Section 3 contains our main results which consists of proving several HHF fractional integral inequalities and some related results. In Section 4, we discuss our results and give the comparison between our results and the existing results, and we point out the future work. Section 5 is for the conclusions.

Auxiliary Results
Here, we shall prove analogues of the fractional HH inequalities (2) and (3) and HHF inequalities (6) and (7) for weighted fractional integrals with positive weighted symmetric function kernels. The main results here are Theorem 1 (a generalization of HH inequalities (2) and (3) and HHF inequality (6), and a reformulation of HHF inequality (7)) and Lemma 2 (a consequence of Theorem 1). First, we need the following fact.

Remark 2. Throughout this study w
Example 1. Consider the following integrable and positive weighted function One can easily show that Thus, w(x) = w(1 − x) and hence the given weighted function is symmetric on [0, 1] with respect to 1 2 .
By making use of (15) and (17) in (14), we get The first inequality of (12) is proved.
On the other hand, we will prove the second inequality of (12). By making use of the convexity of , we get We multiply both sides of (19) by ε −1 w(εϑ 1 + (1 − ε)ϑ 2 ) and integrate with respect to ε over [0, 1] to get Then, by using (10) and (17) in (20), we get This completes the proof of our theorem.

Remark 4.
From Remark 3, we can observe that the HH inequality (3) and the HHF inequality (6) are essentially particular cases of our HHF inequality (12). Additionally, the HHF inequality (21) can be seen as a reformulation of HHF inequality (12), even though it is about weighted fractional and RL-fractional integrals rather than RL-fractional integrals explicitly.

Remark 6.
From Remark 5 (i), we can observe that our result Lemma 2 is essentially a reformulation of the result of ( [34], lemma 2.4), even though it is about weighted fractional and RL-fractional integrals rather than RL-fractional integrals explicitly. Additionally, from Remark 5 (ii) and (iii), we can observe that the results of ( [11], lemma 2) and ( [38], lemma 2.1) are basically particular cases of our result Lemma 2.

Main Results
In view of Lemma 2, we can obtain the following HHF inequalities.