New inequalities for F -convex functions pertaining generalized fractional integrals

In this paper, the authors, utilizing F -convex functions which are defined by B. Samet, establish some new Hermite-Hadamard type inequalities via generalized fractional integrals. Some special cases of our main results recaptured the well-known earlier works.


Introduction
Let f : I ⊆ R → R be a convex function on the interval I of real numbers and a, b ∈ I with a < b. If f is a convex function then the following double inequality, which is well known in the literature as the Hermite-Hadamard inequality, holds [17]: Both inequalities in (1) hold in the reversed direction if f is concave. Over the last decade, this classical double inequality has been improved and generalized in a number of ways, see [5,7,8,13,18], [23]- [25] and the references therein. Also, many types of convexities have been defined, such as quasi-convex in [6], pseudo-convex in [14], strongly convex in [20], ε-convex in [11], s-convex in [10], h-convex in [28], etc. Recently, Samet in [21], has defined a new concept of convexity that depends on a certain function satisfying some axioms, that generalizes different types of convexity.
, be a given function. We say that f is a convex function with respect to some F ∈ F (or F -convex function), if and only if: Remark 1. 1) Let ε ≥ 0, and let f : [a, b] → R, (a, b) ∈ R 2 , a < b, be an ε-convex function, see [11], that is Define the functions F : R × R × R× [0, 1] → R by (2) F (e 1 , e 2 , e 3 , e 4 ) = e 1 − e 4 e 2 − (1 − e 4 )e 3 − ε it is clear that F ∈ F and that is f is an F -convex function. Particularly, taking ε = 0, we show that if f is a convex function then f is an F -convex function with respect to F defined above.
2) Let h : J → [0, +∞) be a given function which is not identical to 0, where J is an interval in R such that (0, 1) ⊆ J. Let f : [a, b] → [0, +∞), (a, b) ∈ R 2 , a < b, be an h-convex function, see [28], that is Define the functions F : R × R × R× [0, 1] → R by and T F,w : R × R × R → R by For L w = 0, it is clear that F ∈ F and that is, f is an F -convex function.
Samet in [21], established the following Hermite-Hadamard type inequalities using the new convexity concept: respectively. Here, Γ(α) is the Gamma function and Then k-fractional integrals of order α, k > 0 are defined by where Γ k (·) stands for the k-gamma function. For k = 1, the k-fractional integrals yield Riemann-Liouville integrals. For α = k = 1, the k-fractional integrals yield classical integrals. For more details, see [9,12,15,19].
It is remarkable that Sarikaya et al. in [26], first give the following interesting integral inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
, then the following inequalities for fractional integrals hold: with α > 0.
Now we summarize the generalized fractional integrals defined by Sarikaya and Ertuğral in [22].
The following left-sided and right-sided generalized fractional integral operators are defined respectively, as follows: The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integrals, conformable fractional integral, Hadamard fractional integrals, etc.
Sarikaya and Ertuğral in [22], establish the following Hermite-Hadamard inequality and lemmas for the generalized fractional integral operators: then the following inequalities for fractional integral operators hold: where the mapping Λ : [0, 1] → R is defined by Budak et al. prove the following Hermite Hadamard type inequalities for F -convex functions.

Theorem 5 ([4]
). Let I ⊆ R be an interval, f : I • ⊆ R → R be a mapping on I • , a, b ∈ I • , a < b. If f is F -convex on [a, b] for some F ∈ F, then we have New inequalities for F -convex functions and T F,w 1 Ψ (1) [ (1) . Motivated by the above literatures, the main objective of this article is to establish some new Hermite-Hadamard type inequalities via generalized fractional integrals utilizing F -convex functions. Some special cases of our main results recaptured the well-known earlier works. At the end, a briefly conclusion will be given as well.

Main results
In this section, we establish some inequalities of Hermite-Hadamard type including generalized fractional integrals via F -convex functions.
Theorem 6. Let I ⊆ R be an interval, f : I • ⊆ R → R be a mapping on I • , a, b ∈ I • , a < b and let F be linear with respect to the first three variables. If f is F -convex on [a, b] for some F ∈ F, then we have tΛ (1) and the function Λ : [0, 1] → R is defined by Kanıt. Since f is F -convex, we have we have for all t ∈ [0, 1] . Multiplying this inequality by tΛ (1) and using axiom (A3), we get for all t ∈ (0, 1). Integrating over (0, 1) with respect to the variable t and using axiom (A1), we obtain Using the facts that we obtain New inequalities for F -convex functions which gives (18). On the other hand, since f is F -convex, we have Using the linearity of F, we get for all t ∈ (0, 1). Integrating over (0, 1) and using axiom (A2), we have The proof of Theorem 6 is completed.
Remark 2. If we choose ϕ(t) = t in Theorem 6, then we have the following inequalities where w(t) = 1.
kΓ k (α) in Theorem 6, then we have the following inequalities for k-Riemann-Liouville fractional integrals

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New inequalities for F -convex functions Theorem 7. Let I ⊆ R be an interval, f : I • ⊆ R → R be a mapping on I • , a, b ∈ I • , a < b and let F be linear with respect to the first three variables. If f is F -convex on [a, b] for some F ∈ F, then we have we have tΛ (1) and using axiom (A3), we get for all t ∈ (0, 1). Integrating over (0, 1) with respect to the variable t and using axiom (A1), we obtain Using the facts that we obtain which gives (22). On the other hand, since f is F -convex, we have Using the linearity of F, we get Applying the axiom (A3) for w(t) = (1) , we obtain for all t ∈ (0, 1). Integrating over (0, 1) and using axiom (A2), we have The proof of Theorem 7 is completed.
Corollary 2. If we take ϕ(t) = t α k kΓ k (α) in Theorem 7, then we have the following inequalities for k-Riemann-Liouville fractional integrals:  Remark 6. One can obtain several results for convexity, ε-convexity, h-convexity, etc by special choice of the function F in Theorems 6 and 7.

Conclusion
In the development of this work, using the definition of F -convex functions some new Hermite-Hadamard type inequalities via generalized fractional integrals have been deduced. We also give several results capturing Riemann-Liouville fractional integrals and k-Riemann-Liouville fractional integrals as special cases. The authors hope that these results will serve as a motivation for future work in this fascinating area.