Some parameterized Simpson’s type inequalities for differentiable convex functions involving generalized fractional integrals

In this paper, we establish some new inequalities of Simpson’s type for differentiable convex functions involving some parameters and generalized fractional integrals. The results given in this study are a generalization of results proved in (Du, Li and Yang in Appl. Math. Comput. 293:358–369, 2017).


Introduction
Simpson's rules are well-known ways for numerical integration and numerical estimation of definite integrals. This method is known as developed by Simpson (1710-1761). However, Kepler used the same approximation about 100 years ago, so that this method is also known as Kepler's rule. Simpson's rule includes the three-point Newton-Cotes quadrature rule, so estimation based on three-step quadratic kernel is sometimes called a Newtontype result.
There are a large number of estimations related to these quadrature rules in the literature; one of them is the following estimation known as Simpson's inequality.
Theorem 1 Suppose that F : [κ 1 , κ 2 ] → R is a four times continuously differentiable mapping on (κ 1 , κ 2 ), and let F (4) ∞ = sup κ∈(κ 1 ,κ 2 ) |F (4) (κ)| < ∞. Then we have the inequality 1 3 Integral-type inequalities have numerous applications in the study of qualitative theory of different classes of differential equations and partial differential equations; see, for instance, [3,4,8,11,14,15,[20][21][22] for more detail. In recent years, many authors have focused on Simpson's type inequalities for various classes of functions. Specifically, some mathematicians have worked on Simpson's and Newton's type results for convex mappings, because convexity theory is an effective and powerful method for solving a large number of problems that arise within different branches of pure and applied mathematics. For example, Dragomir et al. [9] presented new Simpson's type results and their applications to quadrature formulas in numerical integration. Moreover, some inequalities of Simpson's type for s-convex functions are deduced by Alomari et al. [2]. Afterwards, Sarikaya et al. [30] observed variants of Simpson's type inequalities based on convexity. In [25] and [26] the authors provided some Newton's type inequalities for harmonic convex and p-harmonic convex functions. Additionally, new Newton's type inequalities for functions whose local fractional derivatives are generalized convex are given by Iftikhar et al. [17]. For more recent developments, the reader can consult [1,5,12,27,31].

Generalized fractional integrals
In this section, we summarize the generalized fractional integrals defined by Sarikaya and Ertuğral [29].
Let a function ϕ : [0, ∞) → {0, ∞) satisfy the following condition: We define the following left-sided and right-sided generalized fractional integral operators, respectively, as follows: The most important feature of generalized fractional integrals is that they generalize some types of fractional integrals such as the Riemann-Liouville fractional integral, k-Riemann-Liouville fractional integral, Katugampola fractional integral, conformable fractional integral, Hadamard fractional integral, etc. These important particular cases of the integral operators (2.1) and (2.2) are mentioned below.

A lemma
In this section, we propose a parameterized identity involving the ordinary first derivative via generalized fractional integrals. Lemma 1 Let F : [κ 1 , κ 2 ] → R be a differentiable function on (κ 1 , κ 2 ). If F is continuous on [κ 1 , κ 2 ], then for λ, μ ≥ 0, we have the identity Proof Applying fundamental rules of integration, we have

Simpson's inequalities for generalized fractional integrals
In this section, we establish some new Simpson's type inequalities for differentiable convex functions via generalized fractional integrals.

Theorem 3
We assume that the conditions of Lemma 1 hold. If the mapping |F | is convex on [κ 1 , κ 2 ], then we have the following inequality for generalized fractional integrals: Proof By taking the modulus in Lemma 1 and using the properties of the modulus, we obtain that Since the mapping |F | is convex on [κ 1 , κ 2 ], we have which ends the proof.

Corollary 4 In Theorem
, then we obtain the following parameterized Simpson's type inequality for k-Riemann-Liouville fractional integrals: , and α k
Proof Reusing inequality (4.2), by the power mean inequality we have Using the convexity of |F | p 1 , we have which finishes the proof.

Corollary 6
If we assume that ϕ(τ ) = τ α k k k (α) in Theorem 4, then we have the following parameterized Simpson's type inequality for k-Riemann-Liouville fractional integrals: Proof Reusing inequality (4.2), by the well-known Hölder inequality we have Since |F | r 1 is convex, we have which completes the proof.

Particular cases
In this section, we give some particular cases of our main results.
Remark 5 From Lemma 1 we get the following identities.

Remark 8 From Theorem 3 we have the following new inequalities.
(1) For λ = 1 6 and μ = 5 6 , we have the following Simpson's type inequality for generalized fractional integrals: (2) For λ = μ = 1 2 , we have the following trapezoidal-type inequality for generalized fractional integrals: (3) For λ = 0 and μ = 1, we have the following midpoint-type inequality for generalized fractional integrals: Remark 9 From Corollary 3 we have the following inequalities.