Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

: Fractional integral inequality plays a signiﬁcant role in pure and applied mathematics ﬁelds. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the ﬁeld of fractional integral inequalities due to the behavior of its deﬁnition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, speciﬁcally in the last few decades. First, we recall the deﬁnition of ϕ -Riemann–Liouville fractional integral operators and the recently deﬁned class of convex functions, namely the ˘ σ -convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a ˘ σ -convex function with respect to an increasing function involving the ϕ -Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.


Introduction
First of all, we recall the basic notations in convex analysis. A set χ ⊂ R is said to be convex if for each u, v ∈ χ and η ∈ [0, 1]. Based on a convex set χ, we say that a function Υ : χ → R is convex, if the inequality holds for all u, v ∈ χ. If −Υ is convex, then we say that Υ is concave.
Recently, in Reference [30], Wu et al. introduced a new class of convex sets and convex functions namedσ-convex sets andσ-convex functions which are explained in the following definitions: Definition 1. Letσ ⊆ R be a strictly monotone and continuous function, and denote for each u, v ∈ Q.
holds for each u, v ∈ Q. Remark 1. The function Υ is called: is true as a strict inequality for all η ∈ (0, 1) and u, v ∈ Q with u = v.
Furthermore, in Reference [30], many inequalities of Hermite-Hadamard's type have been established using the notion ofσ-convexity.
be an increasing and positive function on (u, v] and ϕ (x) be continuous on (u, v). Then, the left-sided and right-sided ϕ−RL fractional integrals of a function Υ with respect to the function ϕ(x) on [u, v] are respectively defined by [10,[31][32][33]: One can observe that if ϕ is specialized by ϕ(x) = x, then ϕ-Riemann-Liouville fractional integral operators (7) reduce to the classical Riemann-Liouville fractional integral operators (4).

Hermite-Hadamard's Type Inequalities forσ-Convex Functions
Our main results depend on the following lemmas: Multiplying both sides of (10) by η ℘−1 , then integrating the resulting inequality with respect to η over [0, 1], we get By changing the variables z =σ −1 (ησ(u) This completes the proof of our first inequality in (8). In order to prove the second inequality in (8), we use theσ-convexity of Υ; that is By adding these two inequalities we get Multiplying both sides of (11) by η ℘−1 and then integrating with respect to η over [0, 1], we can obtain This completes the proof of our Lemma 1.
If the functionσ is increasing and positive on (u, v] andσ (x) is continuous on (u, v). Then, we have for ℘ > 0: Proof. Again, since Υ is aσ-convex function, we can substituting x =σ −1 η 2σ (u) + 2−η 2σ (v) and Multiplying both sides of (13) by η ℘−1 , then integrating the resulting inequality with respect to η over [0, 1], we get By changing the variables z =σ −1 η This completes the proof of our first inequality in (12). In order to obtain the second inequality in (12), we use theσ-convexity of Υ as: By adding these two inequalities we get Multiplying both sides of (14) by η ℘−1 and then integrating with respect to η over [0, 1], we get This completes the proof of our Lemma 2.

Remark 3.
Especially, in Lemma 2, if we take (i)σ(x) = x, then inequality (12) reduces to the following inequality: which was already established in Reference [34].

Further Consequences
As consequences for the Lemmas 1 and 2, we can obtain the following theorems.
If the functionσ is increasing and positive on (u, v] andσ (x) is continuous on (u, v). Then, we have for ℘ > 0: (15) Proof. By Definition 3 and integrating by parts one can find Analogously, we get From the identities (16) and (17), it yields This completes the proof of Theorem 3.
As a particular case of Theorem 3, ifσ is specialized byσ(z) = z, then we have the following corollary, which has been studied by Sarikaya et al. in Reference [9]. Corollary 1. Under the same assumptions of Theorem 3, ifσ(z) = z, then we have Proof. By puttingσ(z) = z into Theorem 3, we directly obtain the desired equality (18). To prove the inequality (19), we change the variable z = (1 − η)u + ηv in (18), we have Then, we obtain (19) as in the proof of [9, Theorem 3].

Proof. By Definition 3 and integrating by parts one can find
Similarly, we have From the identities (21) and (22), one has This ends the proof of our Theorem 4.
Additionally, whenσ(z) = z, then our result Theorem 4 becomes to the following corollary, which has been already explored by Sarikaya et al. in [34].

Special Means
We consider the special means of positive numbers u = v:

Conclusions
In the study, we have considered a new class of convex functions and the definition of ϕ−RL fractional integral operators. In our present investigation, we have established new fractional Hermite-Hadamard's integral inequalities associated to increasing functions. The results obtained here are very useful in obtaining other type of inequalities. Also, these results are very generic and can be specified to give further potentially useful and interesting integral inequalities involving other type of fractional integral operators.