Some Simpson-like Inequalities Involving the ( s , m ) -Preinvexity

: In this article, closed Newton–Cotes-type symmetrical inequalities involving four-point functions whose second derivatives are ( s , m ) -preinvex in the second sense are established. Some applications to quadrature formulas as well as inequalities involving special means are provided.


Introduction
One of the important concepts of real analysis and mathematical programming is convexity.Due to its various applications in various fields such as applied mathematics, engineering sciences and other fields, this notion has been extended and generalized in several directions and in various ways.
The following inequalities are well known in the literature as Simpson's inequalities: C (4) (x) .
In [32], Chiheb et al. established some Simpson-type inequalities for functions whose second derivatives are prequasi-invex, the results of which are based on the following identity: Lemma 1 ([32]).We let C : [l, l + η(e, l)] → R be a function such that C is absolutely continuous and C is integrable on [l, l + η(e, l)], then the following equality holds: (e, l, C) = η 2 (e,l) where (e, l, C) = 1 6 C(l) + 2C 3l+η(e,l) 3 l+η(e,l) l C(u)du. ( In this paper, using the identity declared in [32], we establish some Simpson-type inequalities for twice differentiable (s, m)-preinvex functions.Some special cases are derived.Applications to quadrature formulas and inequalities involving means are provided.
where we used the facts that and The proof is finished.Theorem 2. Under the assumptions of Theorem 1, if |C | q where q > 1, is (s, m)-preinvex in the second sense for some fixed s, m ∈ (0, 1], with 1 p + 1 q = 1, we have where (l, e, C) is defined as in (1.2).
. Corollary 5.In Corollary 4, using the discrete power mean inequality, we obtain Theorem 4.Under the assumptions of Theorem 2, we have the following inequality: , where (l, e, C) is defined as in (1.2) and B and 2 F 1 are beta and hypergeometric functions, respectively.

Applications
We let Υ be the partition of points l = and R(C, Υ) is the error of approximation.