# New Simpson Type Integral Inequalities fors-Convex Functions and Their Applications

Authors: Artion Kashuri; Pshtiwan Othman Mohammed; Thabet Abdeljawad; Faraidun Hamasalh; Yuming Chu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

First, we consider a new Simpson’s identity. This identity investigates our main results that consist of some integral inequalities of Simpson’s type for thes–convex functions. From our main results, we obtain some special cases which are discussed in detail. Finally, some applications on the Bessel functions, special means of distinct positive real numbers, and error estimation about Simpson quadrature formula are presented to support our theoretical results.


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## Body

## 1. Introduction and Preliminaries

Integral inequalities are generally applicable in many branches of mathematics such as mathematical analysis, fractional calculus, discrete fractional calculus, and abstract spaces; for an overview, the reader should see the literature on integral inequalities, e.g., [1–11] and the references therein.Nowadays, the study of convexity is considered as an original icon in the investigation of theoretical behavior of mathematical inequalities (e.g., [12–15]). 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], λψ-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.Definition 1.
(see [25]). A function f:T⊆ℝ⟶ℝ is said to be convex, if(1)fνω1+1−νω2≤νfω1+1−νfω2holds for all ω1,ω2∈T and ν∈0,1. Likewise, f is concave if −f is convex.Definition 2.
(see [26]). Let s∈0,1 be fixed. A function f:T⊆ℝ⟶ℝ is said to be s–convex (in the second sense), if(2)fνω1+1−νω2≤νsfω1+1−νsfω2holds for all ω1,ω2∈T and ν∈0,1.
Recently, many well-known inequalities haven been established for the convex function and its types such as Hermite–Hadamard type inequalities [27, 28], Hermite–Hadamard–Fejer type inequalities [29], Ostrowski type inequalities [30], Hardy type inequalities [31], Olsen type inequalities [32], Gagliardo–Nirenberg type inequalities [33], Opial type inequalities [34, 35], and Rozanova type inequalities [36]. However, our focus in this paper is on another type of inequality, namely, the Simpson’s inequality, which is obtained by Dragomir et al. [37], and their main result is as follows.Theorem 1.
(see [37]). Assume that f:ω1,ω2⟶ℝ is a four times continuous and differentiable function on ω1,ω2 such that f4∞≔supx∈ω1,ω2f4x<+∞ with ω1<ω2. Then, the following integral inequality holds:(3)16fω1+4fω1+ω22+fω2−1ω2−ω1∫ω1ω2fxdx≤12880ω2−ω14f4∞.
Just like the aforementioned inequalities, several integral inequalities related to the Simpson’s integral inequality (3) has been found for the convex functions (see, e.g., [38–45]).
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.The rest of this article is organized in the following way: Section2 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.


## 2. Main Results

We need the following Lemma in order to establish our main results.Lemma 1.
Letf:T⊆ℝ⟶ℝ be an ℒ1 function on T such that ω1,ω2∈T with ω1<ω2. Also, let f′∈ℒ1, and then for any positive integer number n, we have(4)Tnf;ω1,ω2≔∑k=0n−116nfn−kω1+kω2n+4f2n−k−1ω1+2k+1ω22n+fn−k−1ω1+k+1ω2n−1ω2−ω1∫ω1ω2fxdx=ω2−ω1n2∑k=0n−1∫01ϖνf′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν,where(5)ϖν=ν−16,ν∈0,12,ν−56,ν∈12,1.Proof.
From the definition ofϖν, we can deduce that(6)ω2−ω1n2∫01ϖνf′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν=ω2−ω1n2∫01/2ν−16f′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν+∫1/21ν−56f′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν.
By applying the integration by parts to the right side of (6), we have(7)∫01/2ν−16f′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν=nω2−ω1ν−16f1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2n|t=01/2−∫01/2f1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν=nω2−ω113f2n−k−1ω1+2k+1ω22n+16fn−kω1+kω2n−∫01/2f1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν,
and similarly,(8)∫1/21ν−56f′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν=nω2−ω113f2n−k−1ω1+2k+1ω22n+16fn−k−1ω1+k+1ω2n−∫1/21f1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν,
for allk=0,1,2,…,n−1. Then, substituting (7) and (8) in the equality (6), we get(9)ω2−ω1n2∫01ϖνf′1−νn−kω1+kω2n+νn−k−1ω1+k+1ω2ndν=16nfn−kω1+kω2n+4f2n−k−11+2k+1ω22n+fn−k−1ω1+k+1ω2n−1ω2−ω1∫n−kω1+kω2/nn−k−1ω1+k+1ω2/nfxdx.
By summing overk from 0 to n−1, we can deduce the desired identity (4).Remark 1.
Takingn=1 in Lemma 1, we have the following Simpson’s rule (see [43]):(10)16fω1+4fω1+ω22+fω2−1ω2−ω1∫ω1ω2fxdx=ω2−ω1∫01ϖνf′νω2+1−νω1dν,where ϖν is defined by (5).Theorem 2.
Letf:T⊆0,+∞⟶ℝ be an ℒ1 function on T such that ω1,ω2∈T with ω1<ω2 and s∈0,1 be fixed. Also, let f′∈ℒ1 and f′ be an s–convex function on ω1,ω2, and then for any positive integer number n, we have(11)Tnf;ω1,ω2≤ω2−ω1n2∑k=0n−1A1s+A3sf′en,k+A2s+A4sf′en,k+1,where(12)A1s≔∫01/2ν−161−νsdν=161−5/6s+1s+1−1−5/6s+1s+1+1−5/6s+2s+2+5/6s+1−1/2s+1s+1−5/6s+2−1/2s+2s+2−165/6s+1−1/2s+1s+1,A2s≔∫01/2ν−16νsdν=16s+12−3s+16s+1+1s+23s+2−26s+2,A3s≔∫1/21ν−561−νsdν=561/2s+1−1/6s+1s+1−1/2s+1−1/6s+1s+1+1/2s+2−1/6s+2s+2+16s+1s+1−16s+2s+2−56s+2s+1,A4s≔∫1/21ν−56νsdν=565/6s+1−1/2s+1s+1−5/6s+2−1/2s+2s+2+1−5/6s+2s+2−561−5/6s+1s+1,en,k≔n−kω1+kω2n.Proof.
By using Lemma1, s–convexity of f′ and properties of modulus, we can deduce(13)Tnf;ω1,ω2≤ω2−ω1n2∑k=0n−1∫01ϖνf′1−νen,k+νen,k+1dν≤ω2−ω1n2∑k=0n−1∫01ϖν1−νsf′en,k+νsf′en,k+1dν=ω2−ω1n2∑k=0n−1A1s+A3sf′en,k+A2s+A4sf′en,k+1,which completes the proof.Corollary 1.
Choosings=1 in Theorem 2, we get the following inequality for the convex function:(14)Tnf;ω1,ω2≤ω2−ω1n2∑k=0n−1A1+A3f′en,k+A2+A4f′en,k+1,where(15)A1≔∫01/2ν−161−νdν=161−5/622−1−5/622+1−5/633+5/62−1/222−5/63−1/233−165/62−1/222,A2≔∫01/2ν−16νdν=291296,A3≔∫1/21ν−561−νdν=561/22−1/622−1/22−1/622+1/23−1/633+11296,A4≔∫1/21ν−56νdν=565/62−1/222−5/63−1/233+1−5/633−561−5/622.Corollary 2.
Choosingn=1 in Theorem 2, we get(16)16fω1+4fω1+ω22+fω2−1ω2−ω1∫ω1ω2fxdx≤ω2−ω1×A1s+A3sf′ω1+A2s+A4sf′ω2.Theorem 3.
Letf:T⊆0,+∞⟶ℝ be an ℒ1 function on T such that ω1,ω2∈T with ω1<ω2 and s∈0,1 be fixed. Also, let f′∈ℒ1 and f′q,q>1 be an s–convex function on ω1,ω2, and then for any positive integer number n, we have(17)Tnf;ω1,ω2≤ω2−ω1n22p+12p+1+16p+11/p1s+11/q∑k=0n−1f′en,kq+f′en,k+1q1/q,where en,k is as defined in Theorem 2 and 1/p+1/q=1.Proof.
By making use of Lemma1, Hölder’s inequality, s–convexity of f′q, and properties of modulus, we have(18)Tnf;ω1,ω2≤ω2−ω1n2∑k=0n−1∫01ϖνf′1−νen,k+νen,k+1dν≤ω2−ω1n2∫01ϖνpdν1/p∑k=0n−1∫01f′1−νen,k+νen,k+1qdν1/q≤ω2−ω1n22p+12p+1+16p+11/p∑k=0n−1∫011−νsf′en,kq+νsf′en,k+1qdν1/q=ω2−ω1n22p+12p+1+16p+11/p1s+11/q∑k=0n−1f′en,kq+f′en,k+1q1/q.
This ends our proof.Corollary 3.
Choosings=1 in Theorem 3, we get the following inequality for the convex function:(19)Tnf;ω1,ω2≤ω2−ω1n2q22p+12p+1+16p+11/p∑k=0n−1f′en,kq+f′en,k+1q1/q.Corollary 4.
Choosingn=1 in Theorem 3, we get(20)16fω1+4fω1+ω22+fω2−1ω2−ω1∫ω1ω2fxdx≤ω2−ω1×2p+12p+1+16p+11/pf′ω1q+f′ω2qs+11/q.Theorem 4.
Letf:T⊆0,+∞⟶ℝ be an ℒ1 function on T such that ω1,ω2∈T with ω1<ω2 and s∈0,1 be fixed. Also, let f′∈ℒ1 and f′q,q≥1 be s–convex function on ω1,ω2, and then for any positive integer number n, we have(21)Tnf;ω1,ω2≤ω2−ω1n25361−1/q×∑k=0n−1A1sf′en,kq+A2sf′en,k+1q1/q+A3sf′en,kq+A4sf′en,k+1q1/q,where Ais for all i=1,2,3,4 and en,k are as defined in Theorem 2.Proof.
By using Lemma1, the well-known power mean inequality, s–convexity of f′q, and properties of modulus, we have(22)Tnf;ω1,ω2≤ω2−ω1n2∑k=0n−1∫01ϖνf′1−νen,k+νen,k+1dν≤ω2−ω1n2∫01ϖνdν1−1/q∑k=0n−1∫01ϖνf′1−νen,k+νen,k+1qdν1/q≤ω2−ω1n25361−1/q∑k=0n−1∫01ϖν1−νsf′en,kq+νsf′en,k+1qdν1/q=ω2−ω1n25361−1/q×∑k=0n−1A1sf′en,kq+A2sf′en,k+1q1/q+A3sf′en,kq+A4sf′en,k+1q1/q,
which completes the proof.Remark 2.
Choosingq=1 in Theorem 4, we get Theorem 2Corollary 5.
Choosings=1 in Theorem 4, we get the following inequality for the convex function:(23)Tnf;ω1,ω2≤ω2−ω1n25361−1/q×∑k=0n−1A1f′en,kq+A2f′en,k+1q1/q+A3f′en,kq+A4f′en,k+1q1/q,where Ai for all i=1,2,3,4 are defined as in Corollary 1.Corollary 6.
Choosingn=1 in Theorem 4, we get(24)16fω1+4fω1+ω22+fω2−1ω2−ω1∫ω1ω2fxdx≤ω2−ω15361−1/q×A1sf′ω1q+A2sf′ω2q1/q+A3sf′ω1q+A4sf′ω2q1/q.


## 3. Examples and Applications

### 3.1. Bessel Functions

Consider the functionDτ:0,∞⟶1,∞ given by(25)Dτx=2τΓτ+1x−τℳτx,where ℳτ is the modified Bessel function of the first kind defined by (see (2) in [46], p. 77):(26)ℳτx=∑n=0∞x/2τ+2nn!Γτ+1+n,x∈ℝ.The first order derivative formula ofDτx is given by [46]:(27)Dτ′x=x2τ+1Dτ+1x,and the second derivative can be easily calculated from (27) to be(28)Dτ″x=x2Dτ+2x4τ+1τ+2+Dτ+1x2τ+1.Example 1.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 2 (note that all assumptions are satisfied) and the identities (27) and (28), we have(29)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω12A1s+A3sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1+A2s+A4sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1,where Ais,i=1,2,3,4 are as in Theorem 2.Example 2.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 4 (note that all assumptions are satisfied) and the identities (27) and (28), we have(30)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω12s+11/q2p+12p+1+16p+11/pω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+ω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q.Example 3.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 6 (note that all assumptions are satisfied) and the identities (27) and (28), we have(31)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω125361−1/qA1sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+A2sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q+A3sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+A4sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q,where Ais,i=1,2,3,4 are as in Theorem 2.


### 3.2. Special Means

We consider the following two special means for different positive real numbersω1 and ω2:(i)
The arithmetic mean:(32)Aω1,ω2=ω1+ω22.(ii)
The generalized logarithmic mean:(33)ℒrω1,ω2=ω2r+1−ω1r+1r+1ω2−ω11/r,r∈ℝ∖−1,0.Proposition 1.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(34)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω1×∑k=0n−1A1s+A3sAs−1n−kω1,kω2+A2s+A4sAs−1n−k−1ω1,k+1ω2,where Ais for all i=1,2,3,4 are as defined in Theorem 2.Proof.
By applying Theorem2 with fx=xs,x∈ω1,ω2 and s∈0,1, then we can obtain the desired result (34).Proposition 2.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(35)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω12p+12p+1+16p+11/p1s+11/q×∑k=0n−1Aqs−1n−kω1,kω2+Aqs−1n−k−1ω1,k+1ω21/q,where 1/p+1/q=1 with q>1.Proof.
By applying Theorem3 with fx=xs,x∈ω1,ω2 and s∈0,1, we can obtain the desired result (37) directly.Proposition 3.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(36)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω15361−1/q×∑k=0n−1A1sAqs−1n−kω1,kω2+A2sAqs−1n−k−1ω1,k+1ω21/q+A3sAqs−1n−kω1,kω2+A4sAqs−1n−k−1ω1,k+1ω21/q,where Ais for all i=1,2,3,4 are defined as in Theorem 2 and q≥1.Proof.
By applying Theorem4 with fx=xs,x∈ω1,ω2 and s∈0,1, we can directly obtain the desired result (43).


### 3.3. Simpson Quadrature Formula

LetP be the partition of the points ω1=x0<x1<⋯<xl=ω2 of the interval ω1,ω2 and consider the quadrature formula(37)∫ω1ω2fxdx=Snif,P+ℰf,P,where(38)Snif,P≔∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xiis the Simpson version and ℰf,P denotes the associated approximation error. In this section, we are going to derive some new estimates for the Simpson quadrature formula.Proposition 4.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′ is an s–convex function on ω1,ω2, then we have(39)ℰf,P≤∑i=0l−1xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sfen,k+1i,where Ais for all i=1,2,3,4 are as defined in Theorem 2 and(40)en,ji≔n−jxi+jxi+1n,forj=k,k+1.Proof.
By applying Theorem2 on the subintervals xi,xi+1i=0,…,l−1 of the partition P, we can deduce(41)|∫xixi+1fxdx−∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xi|≤xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sf′en,k+1i.
Therefore, from (41), it follows that(42)ℰf,P=∫ω1ω2fxdx−Snif,P≤∑i=0l−1|∫xixi+1fxdx−∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xi|≤∑i=0l−1xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sf′en,k+1i.
This gives our Proposition4.Proposition 5.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′q is an s–convex function on ω1,ω2, then we have for q>1 and 1/p+1/q=1:(43)ℰf,P≤2p+12p+1+16p+11/p1s+11/q∑i=0l−1xi+1−xin2∑k=0n−1f′en,kiq+f′en,k+1iq1/q,where en,ki and en,k+1i are as defined in Proposition 4.Proof.
By the same method used for Proposition4 and by using Theorem 3, one can prove this proposition directly.Proposition 6.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′q is an s–convex function on ω1,ω2, then we have for q≥1:(44)ℰf,P≤5361−1/q∑i=0l−1xi+1−xin2∑k=0n−1A1sf′en,kiq+A2sf′en,k+1iq1/q+A3sf′en,kiq+A4sf′en,k+1iq1/q,where en,ki and en,k+1i are as defined in Proposition 4.Proof.
By the same method used for Proposition4 and by using Theorem 4, one can prove this proposition directly.


## 3.1. Bessel Functions

Consider the functionDτ:0,∞⟶1,∞ given by(25)Dτx=2τΓτ+1x−τℳτx,where ℳτ is the modified Bessel function of the first kind defined by (see (2) in [46], p. 77):(26)ℳτx=∑n=0∞x/2τ+2nn!Γτ+1+n,x∈ℝ.The first order derivative formula ofDτx is given by [46]:(27)Dτ′x=x2τ+1Dτ+1x,and the second derivative can be easily calculated from (27) to be(28)Dτ″x=x2Dτ+2x4τ+1τ+2+Dτ+1x2τ+1.Example 1.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 2 (note that all assumptions are satisfied) and the identities (27) and (28), we have(29)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω12A1s+A3sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1+A2s+A4sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1,where Ais,i=1,2,3,4 are as in Theorem 2.Example 2.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 4 (note that all assumptions are satisfied) and the identities (27) and (28), we have(30)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω12s+11/q2p+12p+1+16p+11/pω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+ω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q.Example 3.
Let0<ω1<ω2 and τ>−1. Then, by applying Corollary 6 (note that all assumptions are satisfied) and the identities (27) and (28), we have(31)112τ+1ω1Dτ+1ω1+ω1+ω22Dτ+1ω1+ω22+ω2Dτ+1ω2−Dτω2−Dτω1ω2−ω1≤ω2−ω125361−1/qA1sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+A2sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q+A3sω12Dτ+2ω12τ+1τ+2+ω1Dτ+1ω1τ+1q+A4sω22Dτ+2ω22τ+1τ+2+ω2Dτ+1ω2τ+1q1/q,where Ais,i=1,2,3,4 are as in Theorem 2.


## 3.2. Special Means

We consider the following two special means for different positive real numbersω1 and ω2:(i)
The arithmetic mean:(32)Aω1,ω2=ω1+ω22.(ii)
The generalized logarithmic mean:(33)ℒrω1,ω2=ω2r+1−ω1r+1r+1ω2−ω11/r,r∈ℝ∖−1,0.Proposition 1.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(34)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω1×∑k=0n−1A1s+A3sAs−1n−kω1,kω2+A2s+A4sAs−1n−k−1ω1,k+1ω2,where Ais for all i=1,2,3,4 are as defined in Theorem 2.Proof.
By applying Theorem2 with fx=xs,x∈ω1,ω2 and s∈0,1, then we can obtain the desired result (34).Proposition 2.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(35)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω12p+12p+1+16p+11/p1s+11/q×∑k=0n−1Aqs−1n−kω1,kω2+Aqs−1n−k−1ω1,k+1ω21/q,where 1/p+1/q=1 with q>1.Proof.
By applying Theorem3 with fx=xs,x∈ω1,ω2 and s∈0,1, we can obtain the desired result (37) directly.Proposition 3.
Letn∈ℕ and 0<ω1<ω2. Then, we have for a fixed s∈0,1(36)|∑k=0n−113nAn−kω1+kω2ns,n−k−1ω1+k+1ω2ns+2nsA2n−k−1ω1,2k+1ω2−ℒssω1,ω2|≤2s−1sns+1ω2−ω15361−1/q×∑k=0n−1A1sAqs−1n−kω1,kω2+A2sAqs−1n−k−1ω1,k+1ω21/q+A3sAqs−1n−kω1,kω2+A4sAqs−1n−k−1ω1,k+1ω21/q,where Ais for all i=1,2,3,4 are defined as in Theorem 2 and q≥1.Proof.
By applying Theorem4 with fx=xs,x∈ω1,ω2 and s∈0,1, we can directly obtain the desired result (43).


## 3.3. Simpson Quadrature Formula

LetP be the partition of the points ω1=x0<x1<⋯<xl=ω2 of the interval ω1,ω2 and consider the quadrature formula(37)∫ω1ω2fxdx=Snif,P+ℰf,P,where(38)Snif,P≔∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xiis the Simpson version and ℰf,P denotes the associated approximation error. In this section, we are going to derive some new estimates for the Simpson quadrature formula.Proposition 4.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′ is an s–convex function on ω1,ω2, then we have(39)ℰf,P≤∑i=0l−1xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sfen,k+1i,where Ais for all i=1,2,3,4 are as defined in Theorem 2 and(40)en,ji≔n−jxi+jxi+1n,forj=k,k+1.Proof.
By applying Theorem2 on the subintervals xi,xi+1i=0,…,l−1 of the partition P, we can deduce(41)|∫xixi+1fxdx−∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xi|≤xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sf′en,k+1i.
Therefore, from (41), it follows that(42)ℰf,P=∫ω1ω2fxdx−Snif,P≤∑i=0l−1|∫xixi+1fxdx−∑k=0n−116nfn−kxi+kxi+1n+4f2n−k−1xi+2k+1xi+12n+fn−k−1xi+k+1xi+1nxi+1−xi|≤∑i=0l−1xi+1−xin2∑k=0n−1A1s+A3sf′en,ki+A2s+A4sf′en,k+1i.
This gives our Proposition4.Proposition 5.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′q is an s–convex function on ω1,ω2, then we have for q>1 and 1/p+1/q=1:(43)ℰf,P≤2p+12p+1+16p+11/p1s+11/q∑i=0l−1xi+1−xin2∑k=0n−1f′en,kiq+f′en,k+1iq1/q,where en,ki and en,k+1i are as defined in Proposition 4.Proof.
By the same method used for Proposition4 and by using Theorem 3, one can prove this proposition directly.Proposition 6.
Let0<ω1<ω2, s∈0,1 be fixed and f:ω1,ω2⟶ℝ be a differentiable function on ω1,ω2. If f′∈ℒ1ω1,ω2 and f′q is an s–convex function on ω1,ω2, then we have for q≥1:(44)ℰf,P≤5361−1/q∑i=0l−1xi+1−xin2∑k=0n−1A1sf′en,kiq+A2sf′en,k+1iq1/q+A3sf′en,kiq+A4sf′en,k+1iq1/q,where en,ki and en,k+1i are as defined in Proposition 4.Proof.
By the same method used for Proposition4 and by using Theorem 4, one can prove this proposition directly.


## 4. Conclusion

We have obtained a new Simpson identity at first. By using this, we have established some interesting Simpson’s type integral inequalities fors–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.



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