# Simpson’s Integral Inequalities for Twice Differentiable Convex Functions

Authors: Miguel Vivas-Cortez; Thabet Abdeljawad; Pshtiwan Othman Mohammed; Yenny Rangel-Oliveros
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Integral inequality is an interesting mathematical model due to its wide and significant applications in mathematical analysis and fractional calculus. In the present research article, we obtain new inequalities of Simpson’s integral type based on theφ-convex and φ-quasiconvex functions in the second derivative sense. In the last sections, some applications on special functions are provided and shown via two figures to demonstrate the explanation of the readers.


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## Body

## 1. Introduction

Integral inequality is a modern model of approximation theory that describes the growth rate of competing mathematical analysis. This model is also used in various fields such as ordinary differential equations [1–5] and fractional calculus [6–17].Among the several known inequalities, the most simple is Simpson’s type, which has been successfully applied in several models of ordinary differential equations [18–29] and fractional differential equations [30–32]. Simpson’s integral inequality is as follows: for any four times continuously differentiable function F¯:ξ1,ξ2⟶R on ξ1,ξ2, Simpson’s integral inequality is defined as follows:(1)13F¯ξ1+F¯ξ12+2F¯ξ1+ξ22−1ξ2−ξ1∫ξ1ξ2F¯xdx≤12880F¯4∞ξ2−ξ14,where F¯4∞=supx∈ξ1,ξ2F¯4x<∞.If the functionF¯ is neither four times differentiable nor is the fourth derivative F¯4 bounded on ξ1,ξ2, then we cannot apply the classical Simpson quadrature formula.The following literature results obtained by Alomari et al. [18] and Sarikaya et al. [23] become a special case in our findings in Sections 2 and 3.Lemma 1.
(see [18]). Let F¯:J⟶R be twice differentiable function on J with F¯″∈L1ξ1,ξ2, then we have(2)F¯ξ1+F¯ξ22−1ξ2−ξ1∫ξ1ξ2F¯xdx=ξ2−ξ122∫01t1−tF¯″tξ1+1−tξ2dt.Lemma 2.
(see [23]). Let F¯:J⟶R be twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2, then we have(3)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx=ξ2−ξ12∫01ℏtF¯″tξ2+1−tξ1dt,where(4)ℏt=t213−t;ift∈0,12,1−tt2−13;ift∈12,1.
Through this paper,R represents the set of real numbers and J be an interval in R and φ:R×R⟶R be a bifunction apart from some special cases.
This paper deals with the notations ofφ-convex and φ-quasiconvex functions which were introduced by Gordji et al. [33] as follows.Definition 1.
A functionF¯:J⟶R is called convex with respect to φ (or briefly φ-convex), if(5)F¯tξ1+1−tξ2≤F¯ξ2+tφF¯ξ1,F¯ξ2,for all ξ1,ξ2∈J and t∈0,1. Furthermore, F¯ is called φ-quasiconvex, if(6)F¯tξ1+1−tξ2≤maxF¯ξ2,F¯ξ2+φF¯ξ1,F¯ξ2,for all ξ1,ξ2∈J and t∈0,1.Remark 1.
(i)
It is easy to see the definition that everyφ-convex function is φ-quasiconvex(ii)
If we takeφξ1,ξ2=ξ1−ξ2 in Definition 1, then the definitions of φ-convex and φ-quasiconvex are reduced to the definition of convex function and quasiconvex function, respectively
Next, we will give examples for the above definitions.Example 1.
LetF¯x=x2, then F¯ is convex and φ-convex with φγ1,γ2=2γ1+γ2; indeed,(7)F¯tξ1+1−tξ2=tξ1+1−tξ22≤ξ22+tξ12+2t1−tξ1ξ2≤ξ22+tξ12+t1−tξ12+ξ22≤ξ22+tξ12+ξ12+ξ22=F¯ξ2+tφF¯ξ1,F¯ξ2.Example 2.
LetF¯x=x3, then F¯ is not convex but is φ-convex with φγ1,γ2=3γ22γ1−γ2+3γ2γ1−γ22+γ1−γ23; indeed,(8)F¯tξ1+1−tξ2=tξ1+1−tξ23=ξ2+tξ1−ξ23=ξ23+3ξ22tξ1−ξ2+3ξ2t2ξ1−ξ22+t3ξ1−ξ23=F¯ξ2+t3ξ22ξ1−ξ2+3ξ2tξ1−ξ22+t2ξ1−ξ23≤F¯ξ2+t3ξ22ξ1−ξ2+3ξ2ξ1−ξ22+ξ1−ξ23=F¯ξ2+tφF¯ξ1,F¯ξ2.Example 3.
LetF¯:ξ1,ξ2⟶R,0<ξ1<ξ2, with F¯x=1/x2. We observe that F¯ is convex on ξ1,ξ2 and therefore φ-quasiconvex with φγ1,γ2=γ1−γ2.Example 4.
LetF¯:ξ1,ξ2⟶R,0<ξ1<ξ2, with F¯x=2/x3. We observe that F¯ is convex on ξ1,ξ2 and therefore φ-quasiconvex with φγ1,γ2=γ1−γ2.Example 5.
LetF¯:ξ1,ξ2⟶R,0<ξ1<ξ2, with F¯x=2. We obviously see that F¯ is φ-quasiconvex with φγ1,γ2=γ1−γ2.
The essential object of this study is to establish new Simpson’s integral inequalities for theφ-convex and φ-quasiconvex functions in the second derivative sense at certain powers.


## 2. Simpson’s Inequality forφ-Convex

In this section, we give a new refinement of Simpson integral inequality for twice differentiable functions.Theorem 1.
LetF¯:J⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-convex on ξ1,ξ2, then we have(9)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ1281F¯″ξ1+12φF¯″ξ1,F¯″ξ2.Proof.
By making the use of Lemma2 and the φ-convexity of F¯″, we find that(10)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01ktF¯″tξ2+1−tξ1dt≤ξ2−ξ12∫01/2t213−tF¯″ξ1+tφF¯″ξ1,F¯″ξ2dt+ξ2−ξ12∫1/211−tt2−13F¯″ξ1+tφF¯″ξ1,F¯″ξ2dt≔qξ2−ξ12⊤1+⊤2,where(11)⊤1≔∫01/2t213−tF¯″ξ1+tφF¯″ξ1,F¯″ξ2dt=∫01/2t213−tF¯″ξ1dt+∫01/2t213−ttφF¯″ξ1,F¯″ξ2dt=F¯″ξ1∫012t213−tdt+φF¯″ξ1,F¯″ξ2∫01/2t213−ttdt=F¯″ξ1∫01/3t213−tdt−F¯″ξ1∫131/2t213−tdt+φF¯″ξ1,F¯″ξ2∫01/3t213−ttdt−φF¯″ξ1,F¯″ξ2∫1/31/2t213−ttdt=1162F¯″ξ1+5931104φF¯″ξ1,F¯″ξ2,⊤2≔∫1/211−tt2−13F¯″ξ1+tφF¯″ξ1,F¯″ξ2dt=∫1/211−tt2−13F¯″ξ1dt+∫1/211−tt2−13tφF¯″ξ1,F¯″ξ2dt=F¯″ξ1∫1/211−tt2−13dt+φF¯″ξ1,F¯″ξ2∫1/211−tt2−13tdt=1162F¯″ξ1+13331104φF¯″ξ1,F¯″ξ2.
A simple rearrangement gives us the proof.Corollary 1.
Theorem1 with F¯ξ1=F¯ξ1+ξ2/2=F¯ξ2 gives the following new inequality:(12)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1281F¯″ξ1+12φF¯″ξ1|,|F¯″ξ2.Remark 2.
Inequality (9) with φF¯″ξ1,F¯″ξ2=F¯″ξ2−F¯″ξ1 becomes(13)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12162F¯″ξ1+F¯″ξ2.
Moreover, inequality (12) with φF¯″ξ1,F¯″ξ2=F¯″ξ2−F¯″ξ1 becomes(14)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ12162F¯″ξ1+F¯″ξ2.
These are both obtained by Sarikaya et al. [23] in Theorem 2.2 and Corollary 2.3, respectively.Theorem 2.
LetF¯:J⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-convex on ξ1,ξ2 and q≥1, then we have(15)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ1211621−1/q1162F¯″ξ1q+5931104φF¯″ξ1q,F¯″ξ2q1/q+1162F¯″ξ1q+13331104φF¯″ξ1q,F¯″ξ2q1q,where 1/p+1/q=1.Proof.
Letq≥1, then by using Lemma 2, we have(16)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01ktF¯″tξ2+1−tξ1dt=ξ2−ξ12∫01/2t213−tF¯″tξ2+1−tξ1dt+ξ2−ξ12∫1/211−tt2−13F¯″tξ2+1−tξ1dt.
By making the use of the Hölder’s inequality for the above integrals, we have(17)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01/2t213−tdt1−1/q∫01/2t213−tF¯″tξ2+1−tξ1qdt1/q+ξ2−ξ12∫1/211−tt2−13dt1−1/q∫1/211−tt2−13F¯″tξ2+1−tξ1qdt1/q.
Byφ-convexity of F¯″q for the last two integrals, we have(18)∫01/2t213−tF¯″tξ2+1−tξ1qdt≤∫01/2t213−tF¯″ξ1q+tφF¯″ξ1q,F¯″ξ2qdt=F¯″ξ1q∫01/2t213−tdt+φF¯″ξ1q,F¯″ξ2q∫01/2t213−ttdt=1162F¯″ξ1q+5931104φF¯″ξ1q,F¯″ξ2q,(19)∫1/211−tt2−13F¯″tξ2+1−tξ1qdt≤∫1/211−tt2−13F¯″ξ1q+tφF¯″ξ1q,F¯″ξ2qdt=F¯″ξ1q∫1211−tt2−13dt+φF¯″ξ1q,F¯″ξ2q∫1/211−tt2−13tdt=1162F¯″ξ1q+13331104φF¯″ξ1q,F¯″ξ2q.
By substituting (18) and (19) into (17), we have(20)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01/2t213−tdt1−1/q1162F¯″ξ1q+5931104φF¯″ξ1q,F¯″ξ2q1/q+ξ2−ξ12∫1/211−tt2−13dt1−1/q1162F¯″ξ1q+13331104φF¯″ξ1q,F¯″ξ2q1/q=ξ2−ξ1211621−1/q1162F¯″ξ1q+5931104φF¯″ξ1q,F¯″ξ2q1/q+1162F¯″ξ1q+13331104φF¯″ξ1q,F¯″ξ2q1/q,where we used the identity(21)∫01/2t213−tdt=∫1/211−tt2−13dt=1162.
Thus, we are done.Corollary 2.
Theorem2 with F¯ξ1=F¯ξ1+ξ2/2=F¯ξ2 gives the following new inequality:(22)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1211621−1/q1162F¯″ξ1q+5931104φF¯″ξ1q,F¯″ξ2q1/q+1162F¯″ξ1q+13331104φF¯″ξ1q,F¯″ξ2q1/q.Remark 3.
Inequality (15) with φF¯″ξ1q,F¯″ξ2q=F¯″ξ2q−F¯″ξ1q becomes(23)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ1211621−1/q13331104F¯″ξ1q+5931104F¯″ξ2q1/q+5931104F¯″ξ1q+13331104F¯″ξ2q1/q.
Moreover, inequality (22) with φF¯″ξ1q,F¯″ξ2q=F¯″ξ2q−F¯″ξ1q becomes(24)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1211621−1/q13331104F¯″ξ1q+5931104F¯″ξ2q1/q+5931104F¯″ξ1q+13331104F¯″ξ2q1/q.
These are both obtained by Sarikaya et al. [23] in Theorem 2.5 and Corollary 2.6, respectively.Remark 4.
Theorem2 and Corollary 2 with q=1 become Theorem 1 and Corollary 1, respectively.


## 3. Simpson’s Inequality forφ-Quasiconvex

Theorem 3.
LetF¯:J⟶R be a twice differentiable function on J provided F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-quasiconvex on ξ1,ξ2, then we have(25)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ1281maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1.Proof.
By making use ofφ-quasiconvexity of F¯″ and Lemma 2, we get(26)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01ktF¯″tξ2+1−tξ1dt≤ξ2−ξ12∫01/2t213−tmaxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1dt+ξ2−ξ12∫1/211−tt2−13maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1dt=ξ2−ξ12⊤1¯+⊤2¯,where(27)⊤1¯=∫01/2t213−tmaxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1dt=maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1∫01/2t213−tdt=maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1∫01/3t213−tdt−∫1/31/2t213−tdt=1162maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1,⊤2¯=∫1/211−tt2−13maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1dt=maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1∫1/211−tt2−13dt=1162maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1.
A simple rearrangement completes the proof.Corollary 3.
Theorem3 with F¯ξ1=F¯ξ1+ξ2/2=F¯ξ2 becomes(28)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1281maxF¯″ξ1,F¯″ξ1+φF¯″ξ2,F¯″ξ1.Theorem 4.
LetF¯:J⟶R be a twice differentiable function on J provided F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-quasiconvex on ξ1,ξ2 and q≥1, then we have(29)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ1281maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q,where 1/p+1/q=1.Proof.
Letq≥1, then by using Lemma 2, we have(30)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01ktF¯″tξ2+1−tξ1dt=ξ2−ξ12∫01/2t213−tF¯″tξ2+1−tξ1dt+ξ2−ξ12∫1/211−tt2−13F¯″tξ2+1−tξ1dt.
By making the use of the Hölder’s inequality for the above integrals, we have(31)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01/2t213−tdt1−1/q∫01/2t213−tF¯″tξ2+1−tξ1qdt1/q+ξ2−ξ12∫1211−tt2−13dt1−1/q∫1/211−tt2−13F¯″tξ2+1−tξ1qdt1/q.
Byφ-quasiconvexity of F¯″q for the last two integrals, we have(32)∫01/2t213−tF¯″tξ2+1−tξ1qdt≤∫01/2t213−tmaxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1qdt=maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q∫01/2t213−tdt=1162maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q(33)∫1/211−tt2−13F¯″tξ2+1−tξ1qdt≤∫1/211−tt2−13maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1qdt=maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q∫1/211−tt2−13dt=1162maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q.
By substituting (32) and (33) into (31), we have(34)16F¯ξ1+4F¯ξ1+ξ22+F¯ξ2−1ξ2−ξ1∫ξ1ξ2F¯xdx≤ξ2−ξ12∫01/2t213−tdt1−1/q1162maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q+ξ2−ξ12∫1/211−tt2−13dt1−1/q1162maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q=2ξ2−ξ1211621−1/q11621/qmaxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q=ξ2−ξ1281maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q.where we used the following identity(35)∫01/2t213−tdt=∫1/211−tt2−13dt=1162.
Thus we are done.Corollary 4.
Theorem4 with F¯ξ1=F¯ξ1+ξ2/2=F¯ξ2 becomes(36)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1281maxF¯″ξ1q,F¯″ξ1q+φF¯″ξ2q,F¯″ξ1q1/q.Remark 5.
Theorem4 and Corollary 4 with q=1 become Theorem 3 and Corollary 3, respectively.Corollary 5.
Theorem4 with F¯ξ1=F¯ξ1+ξ2/2=F¯ξ2 becomes(37)1ξ2−ξ1∫ξ1ξ2F¯xdx−F¯ξ1+ξ22≤ξ2−ξ1281maxF¯″ξ12,F¯″ξ12+φF¯″ξ22,F¯″ξ121/2.


## 4. Applications

Some applications for our findings are presented.


### 4.1. Applications to Special Means

The special means are itemized as follows:(i)
The arithmetic mean:(38)A=Aξ1,ξ2=ξ1+ξ22,ξ1,ξ2≥0.(ii)
The harmonic mean:(39)ℋ=ℋξ1,ξ2=2ξ1ξ2ξ1+ξ2,ξ1,ξ2>0.(iii)
The logarithmic mean:(40)ℒ=ℒξ1,ξ2=ξ2−ξ1lnξ2−lnξ1;ifξ1≠ξ2,ξ1;ifξ1=ξ2,forξ1,ξ2>0.(iv)
Thep-logarithmic mean:(41)ℒp=ℒpξ1,ξ2=ξ2p+1−ξ1p+1p+1ξ2−ξ11/p;ifξ1≠ξ2,ξ1;ifξ1=ξ2,forp∈R∖−1,0;ξ1,ξ2>0.We know thatℒp is a monotonic nondecreasing function over p∈R with ℒ−1=ℒ. In particular, we can say that ℋ≤ℒ≤A.Now, using our findings in Section2, we conclude the following new inequalities.Proposition 1.
Letξ1,ξ2∈R with 0<ξ1<ξ2. Then, we have(42)13Aξ14,ξ24+23A4ξ1,ξ2−ℒ55ξ1,ξ2≤ξ2−ξ12278ξ12+ξ22.Proof.
The assertion follows from Theorem1 with F¯x=x4/12,x∈ξ1,ξ2 and a simple computation, where F¯″ is φ-convex function with φx,y=2x+y (see Example 1).Proposition 2.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, we have(43)13Aξ15,ξ25+23A5ξ1,ξ2−ℒ66ξ1,ξ2≤10ξ2−ξ12812ξ13+ξ19−ξ29.Proof.
The assertion follows from Theorem1 and a simple computation applied to F¯x=x5/20,x∈ξ1,ξ2, where F¯″ is φ-convex function with φx,y=3y2x−y+3yx−y2+x−y3 (see Example 2).
The following proposition is a particular case of Corollary 11 in [34] when λ=1/3 (see Remark 12 in [34]).Proposition 3.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, we have(44)13Aξ12,ξ22+23A2ξ1,ξ2−ℒ33ξ1,ξ2≤2ξ2−ξ1281.Proof.
The assertion follows from Theorem3 and a simple computation applied to F¯x=x2,x∈ξ1,ξ2, where F¯″x=2 is φ-quasiconvex function with φx,y=x−y (see Example 5).Proposition 4.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, for all q>1, we have(45)13ℋ−1ξ1,ξ2+23A−1ξ1,ξ2−ℒ−1ξ1,ξ2≤ξ2−ξ1281max2qξ13q,2qξ23q.Proof.
The assertion follows from Theorem4 and a simple computation applied to F¯x=1/x,x∈ξ1,ξ2, where F¯″x=2/x3 is φ-quasiconvex function with φx,y=x−y (see Example 4).


### 4.2. Applications to Simpson’s Formula

LetP be a partition of the interval ξ1,ξ2; that is P:ξ1=s0<s1<⋯<sn−1<sn=ξ2; hi=si+1−si/2 and consider Simpson’s formula:(46)SF¯,P=∑i=1n−1F¯si+4F¯si+hi+F¯si+16si+1−si.We know that ifF¯:ξ1,ξ2⟶R is differentiable such that F¯qx exists on ξ1,ξ2 and K=maxx∈ξ1,ξ2F¯qx<∞. Then, we have(47)I=∫ξ1ξ2F¯sds=SF¯,P+EsF¯,P,where the approximation error EsF¯,L satisfies(48)EsF¯,L≤K90∑i=0n−1si+1−si5.It is clear that if the functionF¯ is not four times differentiable or F¯4 is not bounded on ξ1,ξ2, then (47) cannot be applied.Theorem 5.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-convex on ξ1,ξ2, then for every division P of ξ1,ξ2 we have(49)EsF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1.Proof.
By applying Theorem1 on the subintervals si,si+1,i=0,1,2,…,n−1 of the division P to get(50)si+1−si6F¯si+4F¯si+1−si2+F¯si+1−∫sisi+1F¯sds≤si+1−si36F¯″si+12φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 and taking into account that F¯″ is φ-convex to get(51)SF¯,P−∫ξ1ξ2F¯sds≤∑i=0n−1si+1−si381F¯″si+12φF¯″si,F¯″si+1,which completes our proof.Corollary 6.
Theorem5 with φx,y=y−x becomes(52)EsF¯,P≤1162∑i=0n−1si+1−si3F¯″si+F¯″si+1.Theorem 6.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-quasiconvex on ξ1,ξ2, then for every division P of ξ1,ξ2 we have(53)SF¯,P−∫ξ1ξ2F¯sds≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1.Proof.
By applying Theorem3 and by the same method used for proof of the previous theorem, we can produce the desired result.Proposition 5.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″qq≥1 is φ-convex on ξ1,ξ2, then we have(54)EsF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,where(55)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.Proof.
The proof follows from Theorem2 directly.Proposition 6.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″qq≥1 is φ-quasiconvex on ξ1,ξ2, then we have(56)EsF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.Proof.
The proof follows from Theorem4 directly.


### 4.3. Applications to the Midpoint Formula

LetP be a partition as before. Here we consider the midpoint formula:(57)MF¯,P=∑i=0n−1si+1−siF¯si+si+12.Suppose that the functionF¯:ξ1,ξ2⟶R is differentiable with F¯″x existing on ξ1,ξ2 and K=supx∈ξ1,ξ2F¯″x<∞, and then, we have(58)I=∫ξ1ξ2F¯sds=MF¯,P+EMF¯,P,where the approximation error EMF¯,P satisfies(59)EMF¯,P≤K24∑i=0n−1si+1−si3.Proposition 7.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-convex on ξ1,ξ2, then for any division P of ξ1,ξ2, we have(60)EMF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1.Proof.
By applying Corollary1 on the subintervals si,si+1,i=0,1,…,n−1 of the division P, to get(61)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381F¯″si+12φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 to get(62)EMF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1,which completes our proof.Proposition 8.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-convex on ξ1,ξ2 and q≥1, then for any division P of ξ1,ξ2, we have(63)EMF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,where(64)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.Proof.
By applying Corollary2 on the subintervals si,si+1,i=0,1,…,n−1 of the division P to get(65)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤11621−1/qsi+1−si3KφqF¯″si,F¯″si+1,where(66)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.
By summing overi from 0 to n−1 to get(67)EMF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,which completes our proof.Proposition 9.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-quasiconvex on ξ1,ξ2, then for any division P of ξ1,ξ2, we have(68)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1.Proof.
By applying Corollary3 on the subintervals si,si+1,i=0,1,…,n−1 of the division P to get(69)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381maxF¯″si,F¯″si+φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 to get(70)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1,which completes our proof.Proposition 10.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-quasiconvex on ξ1,ξ2 and q≥1, then in (30), for every division P of ξ1,ξ2, we have(71)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.Proof.
By applying Corollary4 on the subintervals si,si+1,i=0,1,…,n−1 of the division P, we get(72)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.
By summing overi from 0 to n−1 to get(73)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq,which rearranges to the proof.


## 4.1. Applications to Special Means

The special means are itemized as follows:(i)
The arithmetic mean:(38)A=Aξ1,ξ2=ξ1+ξ22,ξ1,ξ2≥0.(ii)
The harmonic mean:(39)ℋ=ℋξ1,ξ2=2ξ1ξ2ξ1+ξ2,ξ1,ξ2>0.(iii)
The logarithmic mean:(40)ℒ=ℒξ1,ξ2=ξ2−ξ1lnξ2−lnξ1;ifξ1≠ξ2,ξ1;ifξ1=ξ2,forξ1,ξ2>0.(iv)
Thep-logarithmic mean:(41)ℒp=ℒpξ1,ξ2=ξ2p+1−ξ1p+1p+1ξ2−ξ11/p;ifξ1≠ξ2,ξ1;ifξ1=ξ2,forp∈R∖−1,0;ξ1,ξ2>0.We know thatℒp is a monotonic nondecreasing function over p∈R with ℒ−1=ℒ. In particular, we can say that ℋ≤ℒ≤A.Now, using our findings in Section2, we conclude the following new inequalities.Proposition 1.
Letξ1,ξ2∈R with 0<ξ1<ξ2. Then, we have(42)13Aξ14,ξ24+23A4ξ1,ξ2−ℒ55ξ1,ξ2≤ξ2−ξ12278ξ12+ξ22.Proof.
The assertion follows from Theorem1 with F¯x=x4/12,x∈ξ1,ξ2 and a simple computation, where F¯″ is φ-convex function with φx,y=2x+y (see Example 1).Proposition 2.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, we have(43)13Aξ15,ξ25+23A5ξ1,ξ2−ℒ66ξ1,ξ2≤10ξ2−ξ12812ξ13+ξ19−ξ29.Proof.
The assertion follows from Theorem1 and a simple computation applied to F¯x=x5/20,x∈ξ1,ξ2, where F¯″ is φ-convex function with φx,y=3y2x−y+3yx−y2+x−y3 (see Example 2).
The following proposition is a particular case of Corollary 11 in [34] when λ=1/3 (see Remark 12 in [34]).Proposition 3.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, we have(44)13Aξ12,ξ22+23A2ξ1,ξ2−ℒ33ξ1,ξ2≤2ξ2−ξ1281.Proof.
The assertion follows from Theorem3 and a simple computation applied to F¯x=x2,x∈ξ1,ξ2, where F¯″x=2 is φ-quasiconvex function with φx,y=x−y (see Example 5).Proposition 4.
Letξ1,ξ2∈R, 0<ξ1<ξ2. Then, for all q>1, we have(45)13ℋ−1ξ1,ξ2+23A−1ξ1,ξ2−ℒ−1ξ1,ξ2≤ξ2−ξ1281max2qξ13q,2qξ23q.Proof.
The assertion follows from Theorem4 and a simple computation applied to F¯x=1/x,x∈ξ1,ξ2, where F¯″x=2/x3 is φ-quasiconvex function with φx,y=x−y (see Example 4).


## 4.2. Applications to Simpson’s Formula

LetP be a partition of the interval ξ1,ξ2; that is P:ξ1=s0<s1<⋯<sn−1<sn=ξ2; hi=si+1−si/2 and consider Simpson’s formula:(46)SF¯,P=∑i=1n−1F¯si+4F¯si+hi+F¯si+16si+1−si.We know that ifF¯:ξ1,ξ2⟶R is differentiable such that F¯qx exists on ξ1,ξ2 and K=maxx∈ξ1,ξ2F¯qx<∞. Then, we have(47)I=∫ξ1ξ2F¯sds=SF¯,P+EsF¯,P,where the approximation error EsF¯,L satisfies(48)EsF¯,L≤K90∑i=0n−1si+1−si5.It is clear that if the functionF¯ is not four times differentiable or F¯4 is not bounded on ξ1,ξ2, then (47) cannot be applied.Theorem 5.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-convex on ξ1,ξ2, then for every division P of ξ1,ξ2 we have(49)EsF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1.Proof.
By applying Theorem1 on the subintervals si,si+1,i=0,1,2,…,n−1 of the division P to get(50)si+1−si6F¯si+4F¯si+1−si2+F¯si+1−∫sisi+1F¯sds≤si+1−si36F¯″si+12φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 and taking into account that F¯″ is φ-convex to get(51)SF¯,P−∫ξ1ξ2F¯sds≤∑i=0n−1si+1−si381F¯″si+12φF¯″si,F¯″si+1,which completes our proof.Corollary 6.
Theorem5 with φx,y=y−x becomes(52)EsF¯,P≤1162∑i=0n−1si+1−si3F¯″si+F¯″si+1.Theorem 6.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-quasiconvex on ξ1,ξ2, then for every division P of ξ1,ξ2 we have(53)SF¯,P−∫ξ1ξ2F¯sds≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1.Proof.
By applying Theorem3 and by the same method used for proof of the previous theorem, we can produce the desired result.Proposition 5.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″qq≥1 is φ-convex on ξ1,ξ2, then we have(54)EsF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,where(55)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.Proof.
The proof follows from Theorem2 directly.Proposition 6.
LetF¯:J⊆0,∞⟶R be a twice differentiable function on J such that F¯″∈L1ξ1,ξ2, where ξ1,ξ2∈J with ξ1<ξ2. If F¯″qq≥1 is φ-quasiconvex on ξ1,ξ2, then we have(56)EsF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.Proof.
The proof follows from Theorem4 directly.


## 4.3. Applications to the Midpoint Formula

LetP be a partition as before. Here we consider the midpoint formula:(57)MF¯,P=∑i=0n−1si+1−siF¯si+si+12.Suppose that the functionF¯:ξ1,ξ2⟶R is differentiable with F¯″x existing on ξ1,ξ2 and K=supx∈ξ1,ξ2F¯″x<∞, and then, we have(58)I=∫ξ1ξ2F¯sds=MF¯,P+EMF¯,P,where the approximation error EMF¯,P satisfies(59)EMF¯,P≤K24∑i=0n−1si+1−si3.Proposition 7.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-convex on ξ1,ξ2, then for any division P of ξ1,ξ2, we have(60)EMF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1.Proof.
By applying Corollary1 on the subintervals si,si+1,i=0,1,…,n−1 of the division P, to get(61)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381F¯″si+12φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 to get(62)EMF¯,P≤181∑i=0n−1si+1−si3F¯″si+12φF¯″si,F¯″si+1,which completes our proof.Proposition 8.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-convex on ξ1,ξ2 and q≥1, then for any division P of ξ1,ξ2, we have(63)EMF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,where(64)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.Proof.
By applying Corollary2 on the subintervals si,si+1,i=0,1,…,n−1 of the division P to get(65)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤11621−1/qsi+1−si3KφqF¯″si,F¯″si+1,where(66)KφqF¯″si,F¯″si+1=1162F¯″siq+5931104φF¯″siq,F¯″si+1q1/q+1162F¯″siq+13331104φF¯″siq,F¯″si+1q1/q.
By summing overi from 0 to n−1 to get(67)EMF¯,P≤11621−1/q∑i=0n−1si+1−si3KφqF¯″si,F¯″si+1,which completes our proof.Proposition 9.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″ is φ-quasiconvex on ξ1,ξ2, then for any division P of ξ1,ξ2, we have(68)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1.Proof.
By applying Corollary3 on the subintervals si,si+1,i=0,1,…,n−1 of the division P to get(69)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381maxF¯″si,F¯″si+φF¯″si,F¯″si+1.
By summing overi from 0 to n−1 to get(70)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″si,F¯″si+φF¯″si,F¯″si+1,which completes our proof.Proposition 10.
LetF¯:J⟶R be a twice differentiable function on J, ξ1,ξ2∈J with ξ1<ξ2. If F¯″q is φ-quasiconvex on ξ1,ξ2 and q≥1, then in (30), for every division P of ξ1,ξ2, we have(71)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.Proof.
By applying Corollary4 on the subintervals si,si+1,i=0,1,…,n−1 of the division P, we get(72)si+1−siF¯si+1+si2−∫sisi+1F¯sds≤si+1−si381maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq.
By summing overi from 0 to n−1 to get(73)EMF¯,P≤181∑i=0n−1si+1−si3maxF¯″siq,F¯″siq+φF¯″si+1q,F¯″siq,which rearranges to the proof.


## 5. Illustrative Plots

Finally, we present two three-dimensional plots to demonstrate the validity of the inequalities (42) and (44) in the case of φ-convex and φ-quasiconvex functions, respectively.From inequality (42), we can define(74)vx,y≔q13Ax4,y4+23A4x,y−ℒ55x,yVx,y≔qy−x2278x2+y2.Thus, Figure1 represents the plot of inequality (42) and Vx,y−vx,y.Figure 1
Plot illustrations for inequality (42). (a) For v (x; y) and V (x; y). (b) For V (x; y) − v (x; y).
(a)(b)From inequality (44), we can define(75)wx,y=13Ax2,y2+23A2x,y−ℒ33x,yWx,y=2y−x281.Thus, Figure2 represents the plot of inequality (44) and Wx,y−wx,y.Figure 2
Plot illustrations for inequality (44). (a) For w (x; y) and W (x; y). (b) For Wx,y−wx,y.
(a)(b)


## 6. Conclusion

In this study, we have considered Simpson’s type integral inequalities for theφ-convex and φ-quasiconvex functions in the second derivative sense. Some special cases of our findings are investigated to show the powerfulness of our results. Also, the proposed inequalities can be applied to other mathematical and statistical models, as we have shown in Section 4.



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