New Simpson Type Integral Inequalities for s-Convex Functions and Their Applications

Department of Mathematics, Faculty of Technical Science, University Ismail Qemali, Vlora 9401, Albania Department of Mathematics, College of Education, University of Sulaimani, Sulaimani 46001, Kurdistan Region, Iraq Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia Department of Medical Research, China Medical University, Taichung 40402, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung 41354, Taiwan Department of Mathematics, Huzhou University, Huzhou 313000, China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China

Nowadays, the study of convexity is considered as an original icon in the investigation of theoretical behavior of mathematical inequalities (e.g., [12][13][14][15]). Recently, several works on integral inequalities for convex functions were conducted. In particular, much attention has been given to the theoretical studies of inequalities on different types of convex functions such as s-geometrically convex functions [16], GA-convex functions [17], quasiconvex function [18], MT-convex function [19,20], (α, m)-convex functions [21], F-convex functions [22], λ ψ -convex functions [23], and a new class of convex functions [24], and many other types can be found in [25]. Now we recall some useful definitions of convex functions that are essential for our present investigation.
In view of the above indices and literatures, we will prove a new Simpson's integral identity. From this, then we will obtain several Simpson's integral inequalities and some special cases. e rest of this article is organized in the following way: Section 2 contains the main results of the article which consists of proving several Simpson's integral inequalities and some related results. In Section 3, we give some applications of the Bessel functions, special means, and error estimation about the Simpson quadrature formula to support the main results. Section 4 is for the conclusions.

Main Results
We need the following Lemma in order to establish our main results.
Also, let f ′ ∈ L 1 , and then for any positive integer number n, we have where Proof. From the definition of ϖ(]), we can deduce that 2 Mathematical Problems in Engineering By applying the integration by parts to the right side of (6), we have and similarly, for all k � 0, 1, 2, . . . , n − 1. en, substituting (7) and (8) in the equality (6), we get By summing over k from 0 to n − 1, we can deduce the desired identity (4). □ Remark 1. Taking n � 1 in Lemma 1, we have the following Simpson's rule (see [43]): Mathematical Problems in Engineering where ϖ(]) is defined by (5).
Also, let f ′ ∈ L 1 and |f ′ | be an s-convex function on [ω 1 , ω 2 ], and then for any positive integer number n, we have where , Proof. By using Lemma 1, s-convexity of |f ′ | and properties of modulus, we can deduce which completes the proof.
□ Corollary 1. Choosing s � 1 in eorem 2, we get the following inequality for the convex function:
Also, let f ′ ∈ L 1 and |f ′ | q , q ≥ 1 be s-convex function on [ω 1 , ω 2 ], and then for any positive integer number n, we have where A i (s) for all i � 1, 2, 3, 4 and e n,k are as defined in eorem 2.
Proof. By using Lemma 1, the well-known power mean inequality, s-convexity of |f ′ | q , and properties of modulus, we have which completes the proof.

Corollary 5. Choosing s � 1 in eorem 4, we get the following inequality for the convex function:
where A i for all i � 1, 2, 3, 4 are defined as in Corollary 1. Corollary 6. Choosing n � 1 in eorem 4, we get

Special Means.
We consider the following two special means for different positive real numbers ω 1 and ω 2 : (i) e arithmetic mean: (ii) e generalized logarithmic mean: Proposition 1. Let n ∈ N and 0 < ω 1 < ω 2 . en, we have for a fixed s ∈ (0, 1] where A i (s) for all i � 1, 2, 3, 4 are as defined in eorem 2.
and consider the quadrature formula where is the Simpson version and E(f, P) denotes the associated approximation error. In this section, we are going to derive some new estimates for the Simpson quadrature formula.
Proof. By the same method used for Proposition 4 and by using eorem 4, one can prove this proposition directly.

Conclusion
We have obtained a new Simpson identity at first. By using this, we have established some interesting Simpson's type integral inequalities for s-convex functions. In order to illustrate the efficiency of our main results, some applications to the Bessel functions, special means, and error estimation about Simpson quadrature formula are provided. To the best of our knowledge, these results are new in the literature. Since the class of convex (s-convex) functions have large applications in many mathematical areas, they can be applied to obtain several results in convex analysis, quantum mechanics, special functions, mathematical inequalities, and related optimization theory and may stimulate further research in different areas of pure and applied sciences.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.