Midpoint Inequalities in Fractional Calculus Deﬁned Using Positive Weighted Symmetry Function Kernels

: The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results conﬁrm the generality of our attempt.

In addition, in 2013, the HH integral inequality (2) was generalized and reformulated by Sarikaya et al. [25] in terms of RL fractional integrals. Their result is given by: where u : J → R is assumed to be a positive convex function, continuous on the closed interval [c 1 , c 2 ], and for Lebesgue, almost all x ∈ [c 1 , c 2 ] when u(x) ∈ L 1 [c 1 , c 2 ] with c 1 < c 2 , where RL I ν c 1 + and RL I ν c 2 − are the left-and right-sided RL fractional integrals of order ν > 0, defined by [12]: respectively.
The inequality (3) is also known as the endpoint HH inequality due to using the ends c 1 , c 2 of the interval.
On the other hand, the endpoint HH inequality (3) has been applied for various classes of convexity such as λ ψ -convexity [26], F-convexity [27], (α, m)-convexity [28], MT-convexity [29]. The reader can find other types of convexity in the literature, which in particular, is true for [30]. In the mean time, applying the end-point HH inequality to other models of fractional calculus has received a huge amount of attention. For example, this is true for RL fractional models [31], conformable fractional models [32,33], generalized fractional models [34], ψ RL fractional models [35,36], tempered fractional models [37], and ABand Prabhakar fractional models [38].
After extending the important field of the integral inequalities in (2) and (3), a new version of the endpoint HH inequality (3) was found by Sarikaya and Yildirim [39], namely the midpoint HH inequality due to using the midpoint c 1 +c 2 2 of the interval, which is given by where the function u : [c 1 , c 2 ] → R is convex and continuous.
Based on above definition, in [41], Fejér found a new extension of the HH type inequality (2), namely the HHF type inequality, and the result is as follows: where g is the integrable function, and Işcan [42] found the endpoint version of (7) in the sense of RL fractional integrals, which is also the extension of (3). The result is as follows: where u is convex and continuous and the function g belongs to L 1 [c 1 , c 2 ] and is symmetric (see Definition 1). It is worth mentioning that the midpoint version of (8) has not been found yet, even though many related inequalities of midpoint type were obtained in [43].
Recently, Mohammed et al. [44] found a new endpoint HHF-inequality in terms of weighted fractional integrals with positive weighted symmetric function in a kernel, and their result is as follows: Here, u is a convex and continuous function, (x) a monotone increasing function from the interval (c 1 , c 2 ] onto itself with a continuous derivative (x) on the open interval (c 1 , c 2 ), and w : [c 1 , c 2 ] → (0, ∞) is an integrable function, which is symmetric with respect to (c 1 + c 2 )/2, where c 1 < c 2 .

Remark 1.
From Definition 2, we can obtain the following special cases.
• If w(x) = 1, we obtain the fractional integrals of the function u with respect to the function (x), which is defined by [13,14]: In this article, we will investigate the midpoint version of (9) and some related HHF inequalities by using the weighted fractional integrals (10) with positive weighted symmetric functions in the kernel.
The rest of our article is structured in the following way: In Section 2, we will prove the necessary and auxiliary lemmas, including the midpoint version of (9). In Section 3, we will prove our main results, including new midpoint fractional HHF integral inequalities with some related results. We will present some concluding remarks in Section 4.

Auxiliary Results
In this section, we prove analogues of the fractional HH inequalities (2)-(3) and HHF inequalities (7)-(8) for weighted fractional integral operators with positive weighted symmetric function kernels. Here, the main results are as follows: Theorem 1 (it is a generalisation of HH inequalities (2)-(3) and HHF inequality (7), and a reformulation of HHF inequality (8)) and Lemma 2 (it is a consequence of Theorem 1).
At first, we need the following lemma.
Then, by making use of the assumptions and Definition 1, we can obtain (12). (ii) The symmetry property of w leads to

From above and setting (x)
which completes the desired equality (13).
→ R be an L 1 convex function and w : [c 1 , c 2 ] → R be an integrable, positive and weighted symmetric function with respect to c 1 +c 2 2 . If, in addition, is an increasing and positive function from [c 1 , c 2 ) onto itself such that its derivative (x) is continuous on (c 1 , c 2 ), then for ν > 0, the following inequalities are valid: Multiplying both sides of (15) by κ ν−1 w κ 2 c 1 + 2−κ 2 c 2 and integrating the resulting inequality with respect to κ over [0, 1],, we obtain 2u From the left-hand side of the inequality in (16), we use (13) to obtain It follows that 2u By evaluating the weighted fractional operators, we see that where we used Setting by using (12) dκ .
It follows that By making use of (17) and (19) in (16), we get Thus, the proof of the first inequality of (14) is completed.
On the other hand, we can prove the second inequality of (14) by making use of the convexity of u to get Multiplying both sides of (21) by κ ν−1 w κ 2 c 1 + 2−κ 2 c 2 and integrating with respect to κ over [0, 1] to get Then, by using (12) and (19) in (22), we get This ends our proof.

Remark 3.
From Theorem 1, we can obtain some special cases as follows: where RL c 1 + I ν w and RL w I ν c 2 − are the left-and right-weighted RL fractional integrals, respectively, given by RL  (c 1 , c 2 ), then for ν > 0, the following equality is valid: Proof. Let us set By integrating by parts, using Lemma 1, and (10) and (11), we obtain Analogously, we get Thus, we deduce: which completes the proof of Lemma 2.

Remark 4.
From Lemma 2, we can obtain some special cases as follows: where RL c 1 +c 2 2 + I ν w and RL w I ν c 1 +c 2 2 − are as defined in Remark 3.

Main Results
By the help of Lemma 2, we can deduce the following HHF inequalities.
u (κ)dκ, and let w : [c 1 , c 2 ] → R be an integrable, positive and weighted symmetric function with respect to c 1 +c 2 2 . If, in addition, |u | is convex on [c 1 , c 2 ], and is an increasing and positive function from [c 1 , c 2 ) onto itself such that its derivative (x) is continuous on (c 1 , c 2 ), then for ν > 0 the following inequalities are valid: Proof. By making use of Lemma 2 and properties of the modulus, we obtain Since |u | is convex on [c 1 , c 2 ], we get for κ ∈ −1 (c 1 ), −1 (c 2 ) : Hence, we obtain This completes our proof.

(iii)
If (x) = x, w(x) = 1 and ν = 1, then inequality (27) becomes  u (κ)dκ, and let w : [c 1 , c 2 ] → R be an integrable, positive and weighted symmetric function with respect to c 1 +c 2 2 . If, in addition, |u | q is convex on [c 1 , c 2 ] with q ≥ 1, and is an increasing and positive function from [c 1 , c 2 ) onto itself such that its derivative (x) is continuous on (c 1 , c 2 ), then for ν > 0, the following inequalities are valid: By making use of Lemma 2, power mean inequality and convexity of |u | q , we get where it is easily seen that Hence, the proof is completed.

Concluding Remarks
In the present article, we have investigated a midpoint fractional HHF integral inequality by using the weighted fractional integrals with positive weighted symmetric function kernels, which is also the midpoint version of (9). Moreover, we have investigated some related results.
The existing versions of HHF integral inequalities (7) and (8) have been successfully applied to other classes of convex functions, see [46][47][48]. Therefore, our present results can be applied to those classes of convex functions as well.
Furthermore, one can observe that our results in this article are very generic and can be extended to give further potentially useful and interesting HHF integral inequalities of end-midpoint version, like the following one

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in our manuscript: HH Hermite-Hadamard HHF Hermite-Hadamard-Fejér RL Riemann-Liouville