Some parameterized Simpson-, midpoint- and trapezoid-type inequalities for generalized fractional integrals

In this paper, we first obtain an identity for differentiable mappings. Then, we establish some new generalized inequalities for differentiable convex functions involving some parameters and generalized fractional integrals. We show that these results reduce to several new Simpson-, midpoint- and trapezoid-type inequalities. The results given in this study are the generalizations of results proved in several earlier papers.

In recent years, many writers have focused on Simpson-type inequalities in various categories of work. Specifically, some mathematicians have worked on the results of the Simpson-and Newton-type inequalities by using convex mappings, because convexity theory is an effective and powerful way to solve a large number of problems from different branches of pure and applied mathematics. For example, Dragomir et al. [11] presented new Simpson-type results and their applications to quadrature formulas in numerical integration. Also, new Newton-type inequalities for functions whose local fractional derivatives are generalized convex are given by Iftikhar et al. in [20]. For more recent developments, one can consult [1-5, 9, 12-15, 19, 28-30, 35].
The aim of this paper is to obtain several generalized inequalities for differentiable mappings by utilizing generalized fractional integrals and some nonnegative parameters. By special choice of parameters, the obtained results reduce some well-known Simpson-, midpoint-and trapezoid-type inequalities obtained by several authors in [10,16,17,23,33,34].

Generalized fractional integral operators
In this section, we mention the generalized fractional integrals defined by Sarikaya and Ertuğral in [32].

An identity for generalized fractional integrals
In this section, we offer a parameterized identity involving an ordinary first derivative via generalized fractional integrals.
If is continuous and integrable on [κ 1 , κ 2 ], then for ρ, σ ≥ 0, one has the identity where the mapping Proof Applying the fundamental rules of integration, we have By adding (3.2) and (3.3), we obtain the required equality (3.1).

Corollary 1
If we assume ϕ(t) = t in Lemma 1, then we obtain the following equality:

Corollary 2
In Lemma 1, if we set ϕ(t) = t α (α) , then we obtain the following new identity for the Riemann-Liouville fractional integral: , then we obtain the following new identity for the k-Riemann-Liouville fractional integral:

Some parameterized inequalities for generalized fractional integral operators
In this section, we establish some new generalized inequalities for differentiable convex functions via generalized fractional integrals.

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

Corollary 4
Under the assumption of Theorem 3 with ϕ(t) = t, we obtain the following inequality: Corollary 5 Under the assumption of Theorem 3 with ϕ(t) = t α (κ 1 ) , we obtain the following inequality for Riemann-Liouville fractional integrals: , , then we obtain the following inequality for k-Riemann-Liouville fractional integrals: , Theorem 4 We assume that the conditions of Lemma 1 hold. If the mapping | | p 1 , p 1 > 1, is convex on [κ 1 , κ 2 ], then we have the following inequality for generalized fractional integrals: Proof Reutilizing inequality (4.2) and from the power mean inequality, we have Using the convexity of | | p 1 , we have which finishes the proof.

Corollary 7
If we assume that ϕ(t) = t in Theorem 4, then we obtain the following inequality: where 1 (τ ) and 2 (τ ) are defined as in Corollary 4.

Corollary 10
In Theorem 5, if we set ϕ(t) = t, then we obtain the following inequality:

Corollary 12
In Theorem 5, if we set ϕ(t) = t α k k k (α) , then we obtain the following inequality for k-Riemann-Liouville fractional integrals:

Special cases
In this section, we give some special cases of our main results.
Remark 1 From Lemma 1, we give the following identities: 1. For ρ = σ = 2 3 , we have the following identity: which is given by Ertuğral and Sarikaya in [16].
Remark 5 From Theorem 3, we have the following new inequalities: 1. For ρ = σ = 2 3 , we have the following inequality: which is given by Ertuğral and Sarikaya in [16]. 2. For ρ = σ = 0, we have the following inequality: which is given by Ertuğral et al. in [17].