New Simpson type inequalities for twice differentiable functions via generalized fractional integrals

Abstract: Fractional versions of Simpson inequalities for differentiable convex functions are extensively researched. However, Simpson type inequalities for twice differentiable functions are also investigated slightly. Hence, we establish a new identity for twice differentiable functions. Furthermore, by utilizing generalized fractional integrals, we prove several Simpson type inequalities for functions whose second derivatives in absolute value are convex.


Introduction
It is well known that Simpson's inequality is used in several branches of mathematics in the literature. For four times continuously differentiable functions, the classical Simpson's inequality is expressed as follows: (4) (x) < ∞. Then, the following inequality holds: The convex theory is an available way to solve a large number of problems from various branches of mathematics. Hence, many authors have researched on the results of Simpson-type for convex functions. More precisely, some inequalities of Simpson's type for s-convex functions is proved by using differentiable functions [1]. In the paper [2], it is investigated the new variants of Simpson's type inequalities based on differentiable convex mapping. For more information about Simpson type inequalities for various convex classes, we refer the reader to Refs. [3][4][5][6][7] and the references therein.
The first and second results on fractional Simpson inequality for twice differentiable functions were established in [34] and [35], respectively. With the help of these articles, the aim of this paper is to extend the results of given in [33] for twice differentiable functions to generalized fractional integrals. The general structure of the paper consists of four chapters including an introduction. The remaining part of the paper proceeds as follows: In Section 2, after giving a general literature survey and definition of generalized fractional integral operators, we give an equality for twice differentiable functions involving generalized fractional integrals. In Section 3, for utilizing this equality, it is considered several Simpson type inequalities for mapping whose second derivatives are convex. In the last section, some conclusions and further directions of research are discussed.
The generalized fractional integrals were introduced by Sarikaya and Ertugral as follows: Let us note that a function ϕ : [0, ∞) → [0, ∞) satisfies the following condition: We consider the following left-sided and right-sided generalized fractional integral operators respectively.
The first result on fractional Simpson inequality for twice differentiable functions was proved by Budak et al. in [34] as follows: . Then, we have the following inequality Here, .
The other version of fractional Simpson inequality for twice differentiable functions was proved in [35] as follows: 2) If α > 1 2 , then there exist a real number c α such that 0 < c α < 1 and we obtain the following equality . Assume also that the mapping | | is convex on [a, b]. Then, we have the following inequality Here, Ω 1 (α) is defined by

Some equalities for twice differentiable functions
In this section, we give an identity on twice differentiable functions for using the main results.
. Then, the following equality Proof. By using integration by parts, we obtain With help of the Eq (2.2) and using the change of the variable x = 1+t 2 b + 1−t 2 a for t ∈ [0, 1] , it can be rewritten as follows Similarly, we get From Eqs (2.3) and (2.4), we have Multiplying the both sides of (2.5) by (b−a) 2 8Υ(1) , we obtain Eq (2.1). This ends the proof of Lemma 2.

New Simpson's type inequalities for twice differentiable functions
In this section, we establish several Simpson type inequalities for mapping whose second derivatives are convex.
Theorem 4. Let us consider that the assumptions of Lemma 2 are valid. Let us also consider that the mapping | | is convex on [a, b]. Then, we get the following inequality where Ψ ϕ 1 is defined by Proof. By taking modulus in Lemma 2, we have By using convexity of | |, we obtain This finishes the proof of Theorem 4. Corollary 1. If we assign ϕ (t) = 1 kΓ k (α) t α k in Theorem 4, then there exist a real number c k α so that 0 < c k α < 1 and the following inequality holds: Here, Θ 1 (α, k) is defined by

(3.3)
Theorem 5. Let us note that the assumptions of Lemma 2 hold. If the mapping | | q , q > 1 is convex on [a, b], then we have the following inequality Here, 1 p + 1 q = 1 and Ψ ϕ (p) is defined by Proof. By using the Hölder inequality in inequality (3.2), we obtain With the help of the convexity of | | q , we get This completes the proof of Theorem 5.
Theorem 6. Let us note that the assumptions of Lemma 2 hold. If the mapping | | q , q ≥ 1 is convex on [a, b], then we have the following inequality Here, Ψ ϕ 2 is defined by in Theorem 4 and Ψ ϕ 2 is defined by Proof. By applying power-mean inequality in (3.2), we obtain Then, we obtain the desired result of Theorem 6.

Conclusions
Fractional versions of Simpson inequalities for differentiable convex functions are investigated extensively. On the other hand, Simpson type inequalities for twice differentiable functions are also considered slightly. Hence, Simpson type inequality for twice differentiable functions by generalized fractional integrals are established in this paper. Furthermore, we prove that our results generalize the inequalities obtained by Sarikaya et al. [33] and Hezenci et al. [35]. In the future studies, authors can try to generalize our results by utilizing different kind of convex function classes or other type fractional integral operators.