An Inequality of Simpson's type Via Quasi-Convex Mappings with Applications

In this paper, an inequality of Simpson type for quasi-convex mappings are proved. The constant in the classical Simpson's inequality is improved. Furthermore, the obtained bounds can be (much) better than some recently obtained bounds. Application to Simpson's quadrature rule is also given.


Introduction
Suppose f : [a, b] → R, is fourth times continuously differentiable mapping on (a, b) and f (4) ∞ := sup holds, and it is well known in the literature as Simpson's inequality. It is well known that if the mapping f is neither four times differentiable nor is the fourth derivative f (4) bounded on (a, b), then we cannot apply the classical Simpson quadrature formula.
In [11], Pečarić and Varošanec, obtained some inequalities of Simpson's type for functions whose n-th derivative, n ∈ {0, 1, 2, 3} is of bounded variation, as follow: Theorem 1. Let n ∈ {0, 1, 2, 3}. Let f be a real function on [a, b] such that f (n) is function of bounded variation. Then where, Here to note that, the inequality (1.2) with n = 0, was proved in literature by Dragomir [6]. Also, Ghizzetti and Ossicini [10], proved that if f ′′′ is an absolutely continuous mapping with total variation b a (f ), then (1.2) with n = 3 holds.
In recent years many authors were established several generalizations of the Simpson's inequality for mappings of bounded variation and for Lipschitzian, monotonic, and absolutely continuous mappings via kernels, for refinements, counterparts, generalizations and several Simpson's type inequalities see [4]- [17].
The notion of quasi-convex functions generalizes the notion of convex functions. More precisely, a function f : for all x, y ∈ [a, b] and λ ∈ [0, 1]. Clearly, any convex function is a quasi-convex function. Furthermore, there exist quasi-convex functions which are not convex nor continuous.

Example 1.
The floor function f loor (x) = ⌊x⌋, is the largest integer not greater than x, is an example of a monotonic increasing function which is quasi-convex but it is neither convex nor continuous.
In the same time, one can note that the quasi-convex mappings may be not of bounded variation, i.e., there exist quasi-convex functions which are not of bounded variation. For example, consider the function f : therefore, f is quasi-convex but not of bounded variation on [0,2]. Therefore, we cannot apply the above inequalities. For recent inequalities concerning quasi-convex mappings see [1]- [5].
In this paper, we obtain an inequality of Simpson type via quasi-convex mapping. This approach allows us to investigate Simpson's quadrature rule that have restrictions on the behavior of the integrand and thus to deal with larger classes of functions. In general, we show that our result is better than the classical inequality (1.1).

Inequalities of Simpson's type for quasi-convex functions
In order to prove our main results, we start with the following lemma: , then the following equality holds: where, Proof. We note that Integrating by parts, we get which gives the desired representation (2.1).
Therefore, we may state our main result as follows: Theorem 2. Let f ′′′ : I ⊆ R → R be an absolutely continuous mapping on , then the following inequality holds:

Proof. From Lemma 1, and since f is quasi-convex, we have
which completes the proof.
Corollary 1. Let f as in Theorem 2.
(1) If f is decreasing, then we have (2) If f is increasing, then we have  (1.1) and (1.2) are improved.

2-
The corresponding version of the inequality (2.2) for powers may be done by applying the Hölder inequality and the power mean inequality.

Applications to Simpson's Formula
Let d be a division of the interval [a, b], i.e., d : a = x 0 < x 1 < ... < x n−1 < x n = b, h i = (x i+1 − x i )/2 and consider the Simpson's formula where the approximation error E S (f, d) of the integral I by the Simpson's formula S (f, d) satisfies In the following we give a new estimation for the remainder term E S (f, d).
Proof. Applying Theorem 2 on the subintervals [x i , x i+1 ], (i = 0, 1, ..., n − 1) of the division d, we get Summing over i from 0 to n − 1 and taking into account that f (4) is quasi-convex, which completes the proof.

conclusion
For fourth times continuously differentiable mapping f on (a, b) and f (4) ∞ := sup x∈(a,b) f (4) (x) < ∞, the classical Simpson's inequality holds. In this paper, we relax the conditions on Simpson's inequality; namely, the proposed inequality (2.2) holds if f ′′′ : I ⊆ R → R is an absolutely continuous mapping on I • such that f ′′′ ∈ L[a, b] and f (4) is quasi-convex on [a, b]. In general, the inequality (2.2) is better than the classical Simpson's inequality (1.1) and the result(s) in Theorem 1, where we give an example shows that there exist a quasi-convex mapping which is not of bounded variation.