Hermite–Hadamard type inequalities for F-convex function involving fractional integrals

In this study, the family F and F-convex function are given with its properties. In view of this, we establish some new inequalities of Hermite–Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F-convex. We also show that through the notion of F-convex we can find some new Hermite–Hadamard type and trapezoid type inequalities for the Riemann–Liouville fractional integrals and classical integrals.


Introduction
A function f : I ⊆ R → R is said to be convex on the interval I, if for all x, y ∈ I and t ∈ (0, 1) it satisfies the following inequality: Convex functions play an important role in the field of integral inequalities. For convex functions, many equalities and inequalities have been established, but one of the most important ones is the Hermite-Hadamard' integral inequality, which is defined as follows [1]: Let f : I ⊆ R → R be a convex function with a < b and a, b ∈ I. Then the Hermite-Hadamard inequality is given by In recent years, a number of mathematicians have devoted their efforts to generalizing, refining, counterparting, and extending the Hermite-Hadamard inequality (2) for different classes of convex functions and mappings. The Hermite-Hadamard inequality (2) is established for the classical integral, fractional integrals, conformable fractional integrals and most recently for generalized fractional integrals; see for details and applications [2][3][4][5][6][7][8] and the references therein.
(i) Let f : [a, b] → R be an ε-convex function, that is [18], Define the functions F : and For it will be seen 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. (ii) Let f : [a, b] → R be λ ϕ -preinvex function according to ϕ and bifunction η, 0 ≤ ϕ ≤ π 2 , λ ∈ (0, 1 2 ], that is [16], and For L w = 0, it will be seen that F ∈ F and Define the functions F : and For L w = (1w)ε, it will be seen that F ∈ F and Recently Samet [17] established some integral inequalities of Hermite-Hadamard type via F-convex functions.
As consequences of the above theorems, the author obtained some integral inequalities for ε-convexity, α-convexity, and h-convexity.
In the sequel, we recall the concepts of the left-sided and right-sided Riemann-Liouville fractional integrals of the order α > 0.
, for some F ∈ F and the function t ∈ (0, 1) → L w(t) belongs to L 1 (a, b), where w(t) = |(1t) αt α |. Then we have the inequality The following definitions will be useful for this study [20].

Definition 2.3
The Euler beta function is defined as follows: The incomplete beta function is defined by Note that, for x = 1, the incomplete beta function reduces to the Euler beta function. Also, the following three lemmas are important to obtain our main results.
, then the following equality for the fractional integral holds: , then the following equality for the fractional integral holds: In this study, using the λ ϕ -preinvexity of the function, we establish new inequalities of Hermite-Hadamard type for differentiable function and some trapezoid type inequalities for function whose second derivatives absolutely values are F-convex.

Hermite-Hadamard type inequalities for differentiable functions
In this section, we establish some inequalities of Hermite-Hadamard type for F-convex functions in fractional integral forms.

Theorem 9 Let I ⊆ R be an open invex set with respect to bifunction
Multiplying this inequality by Integrating over [0, 1] and using axiom (A2), we get But from Lemma 1 we have Because T F,w is nondecreasing with respect to the first variable so that This proves (10).
Remark 2 If we choose η(b, a) = ba and ϕ = 0 in Theorem 9, we get

Corollary 1 Under the assumptions of Theorem
for u 1 , u 2 , u 3 ∈ R. Hence, by Theorem 9, we have This completes the proof.
which is given by [18].

Corollary 2 Under the assumptions of Theorem
Proof Using (7) with Hence, by Theorem 9, we have η(b,a)) -f (a) This leads to Thus, the proof is done.

Corollary 3 Under the assumptions of Theorem
for u 1 , u 2 , u 3 ∈ R. So, by Theorem 9, we have iϕ η(b, a)) 2 η(b, a) η(b, a) This leads to Thus, the proof is done.  1 (a, b). Then

Theorem 10 Let I ⊆ R be an open invex set with respect to bifunction
where With w(t) = 1 in (A2), we have Using Lemma 1 and the Hölder inequality, we get η(b, a) + e iϕ η(b, a)) 2 Because T F,1 is nondecreasing with respect to the first variable, we get Thus, the proof is completed. η(b, a) = ba and ϕ = 0 in Theorem 10, we get

Corollary 4 Under the assumptions of Theorem
From (4) with w(t) = 1, we have for u 1 , u 2 , u 3 ∈ R. Hence, by Theorem 10, we have This leads to This completes the proof.   η(b, a) Proof Using (7) with w(t) = 1, we have for u 1 , u 2 , u 3 ∈ R. So, by Theorem 10, we have This leads to Thus, the proof is done.  η(b, a)) η(b, a)

Remark 7 In Corollary 5, if we choose
Proof From (9) with w(t) = 1, we have for u 1 , u 2 , u 3 ∈ R. So, by Theorem 9, we have This completes the proof.  1 (a, b). Then

Theorem 11 Let I ⊆ R be an open invex set with respect to bifunction
where iϕ η(b, a)) 2 η(b, a) Integrating over [0, 1] and using axiom (A2), we obtain Using Lemma 1 and the power mean inequality, we get iϕ η(b, a)) 2 η(b, a)) α J α a + f a + e iϕ η(b, a) + J α (a+e iϕ η(b,a)) -f (a) Because T F,w is nondecreasing with respect to the first variable, we find This completes the proof. η(b, a) = ba and ϕ = 0 in Theorem 11, we get

Corollary 7 Under the assumptions of Theorem
.
This implies that This completes the proof.

Corollary 8 Under the assumptions of Theorem
Proof Using (7) with w(t) = |(1t) αt α |, we have for u 1 , u 2 , u 3 ∈ R. Now, by Theorem 11, we have This leads to Thus, the proof is done.

Corollary 9
Under the assumptions of Theorem 11. If |f | p p-1 is h-convex, we have This completes the proof.