# Bounds for the Remainder in Simpson’s Inequality vian-Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals

Authors: Yu-Ming Chu; Muhammad Uzair Awan; Muhammad Zakria Javad; Awais Gul Khan
Journal: Journal of Mathematics
 (2020)

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## Abstract

The goal of this paper is to derive some new variants of Simpson’s inequality using the class ofn-polynomial convex functions of higher order. To obtain the main results of the paper, we first derive a new generalized fractional integral identity utilizing the concepts of Katugampola fractional integrals. This new fractional integral identity will serve as an auxiliary result in the development of the main results of this paper.


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## Body

## 1. Introduction and Preliminaries

The following inequality known in the literature as Simpson’s inequality [1].(1)∫abΛtdt−b−a6Λa+4Λa+b2+Λb≤b−a42880Λ4∞,where Λ:a,b⟶ℝ is a four times continuously differentiable function on a,b and(2)Λ4∞=supt∈a,bΛ4t<∞.Simpson’s inequality plays a significant role in analysis [2–4]. Over the years, it has been extended and generalized in different directions using novel and innovative approaches. The survey by Dragomir et al. [5] is very informative regarding the developments of Simpson’s inequality and its applications.In recent years, the fractional calculus [6–10] is often known as noninteger calculus which has become a powerful tool in mathematics because it provides a good tool to describe physical memory. Fractional calculus has wide applications in real life through its help in solving different physical problems [11–20]. The classic definition of Riemann–Liouville fractional integrals is one of the most basic concepts in fractional calculus which is defined as:Definition 1.
LetΛ∈La,b. Then Riemann–Liouville integrals Ja+αΛ and Jb−αΛ of order α>0 with a≥0 are defined by(3)Ja+αΛx=1Γα∫axx−tα−1Λtdt,x>a,Jb−αΛx=1Γα∫xbt−xα−1Λtdt,x<b,where(4)Γα=∫0∞e−xxα−1dx,is a well known gamma function.
In past few decades, several successful attempts have been made in generalizing the classical concepts of fractional calculus. Erdelyi–Kober operator is a significant generalization of fractional integrals introduced and was studied by Arthur Erdelyi and Hermann Kober. But there is a drawback that one cannot get the Hadamard version of the derivatives and integrals from Erdelyi–Kober operators. Katugampola [21] gave a well-defined concept of fractional integrals as:Definition 2.
Leta,b⊂ℝ be a finite interval. Then the left and right sides of Katugampola fractional integrals of order α>0 of Λ∈Xcϱa,b are defined by (5)Ia+αϱΛx=ϱ1−αΓα∫axtϱ−1xϱ−tϱ1−αΛtdt,Ia+αϱΛx=ϱ1−αΓα∫xbtϱ−1tϱ−xϱ1−αΛtdt,if the integral exists.
If we takeρ=1, then we can recapture Riemann–Liouville fractional integrals from the Katugampola fractional integrals. It worth to mention here that Erdelyi–Kober operators and Katugampola fractional integrals are not equivalent to each other.
Sarikaya et al. [22] are the first authors to utilize the concepts of Riemann–Liouville fractional integrals in obtaining the fractional analogues of Hermite–Hadamard’s inequality. This idea inspired several inequalities expert, and resultantly huge number of articles have been written on the fractional analogues of classical inequalities. For example, Hu et al. [23] obtained some new fractional analogues of integral inequalities using Katugampola fractional integrals. Nie et al. [24] obtained k-fractional analogues of Simpson’s inequality. Peng et al. [25] also obtained some new fractional analogues of Simpson’s inequality. Set [26] obtained fractional analogues of Ostrowski’s inequality. Wu et al. [27] obtained fractional analogues of inequalities using k-th order differentiable functions. Kermausuor [28] obtained new Simpson type inequalities involving Katugampola fractional integrals essentially using the class of Breckner type s-convex function.
Recently, Toplu et al. [29] introduced the notion of n-polynomial convex functions as follows.Definition 3.
Letn∈ℕ. Then a nonnegative function Λ:I⊂ℝ⟶ℝ is said to be a n-polynomial convex function if the inequality(6)Λtx+1−ty≤1n∑s=1n1−1−tsΛx+1n∑s=1n1−tsΛy,holds for every x,y∈I and t∈0,1.
Many researchers have also derived several new Hermite–Hadamard’s like inequalities [30–36] using the concept of n-polynomial convex functions. We would like to point out here that the class of n-polynomial convex functions generalize the class of convex functions if we take n=1, then we have the class of 1-polynomial convex functions which is just the classical convex functions. Also we can get other type of convexities: for example, for n=2, we have 2-polynomial convexity. Another point of pondering here is that every n-polynomial convex function is an h-convex function with the function ht=1/n∑s=1n1−1−ts. So more generally, every nonnegative convex function is also an n-polynomial convex function.
The idea behind the study of this article is to extend the notion ofn-polynomial convex functions with the introduction of higher order n-polynomial convex functions. We derive a new fractional integral identity using the concepts of Katugampola fractional integrals. This new identity will serve as an auxiliary result in the development of some new fractional analogues of Simpson’s inequalities using the concept of n-polynomial convex functions of higher order.
Before we move to our main results, we would like to introduce the notion ofn-polynomial convex functions of higher order.Definition 4.
Letn∈ℕ. Then a nonnegative function Λ:I⊂ℝ⟶ℝ is said to be a higher order n-polynomial convex function if the inequality(7)Λtx+1−ty≤1n∑s=1n1−1−tsΛx+1n∑s=1n1−tsΛy−μtσ1−t+t1−tσy−xσ,holds for every x,y∈I, σ>0, and t∈0,1.Remark 1.
Note that ifσ=0 in (7), then the class of n-polynomial convex functions of higher order reduces to the class of n-polynomial convex functions. If σ=2, then we have a new class of strongly n-polynomial convex functions. If n=1, then we have the class of higher-order convex functions [37]. And along with n=1, if we have σ=2, then the class of n-polynomial convex functions of higher order reduces to the class of strongly convex functions [38]. From this, it is evident that the class of n-polynomial convex functions of higher order is quite unifying one as it relates several other unrelated classes of convexity [39–45].


## 2. Main Results

In this section, we discuss our main results.


### 2.1. A Key Lemma

We now derive the main auxiliary result of the paper.Lemma 1.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ, then(8)1i+1i+2Λaϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1+Λbϱ−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1=ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt−∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt.Proof.
Consider(9)I=ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt−∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdtI=ϱbϱ−aϱi+1I1−I2.Integrating by parts I1, we have(10)I1=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=−i+1ϱbϱ−aϱ1i+2−tαϱi+1Λi+tϱi+1aϱ+1−tϱi+1bϱ10−αbϱ−aϱ∫01tαϱ−1Λi+tϱi+1aϱ+1−tϱi+1bϱdt,=Λaϱϱbϱ−aϱi+2+i+1ϱi+2bϱ−aϱΛiaϱ+bϱi+1−i+1αϱα−1Γα+1bϱ−aϱα+1ϱIa+αΛiaϱ+bϱi+1.
Similarly,(11)I2=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=−Λbϱϱbϱ−aϱi+2−i+1ϱi+2bϱ−aϱΛaϱ+ibϱi+1+i+1αϱα−1Γα+1bϱ−aϱα+1ϱIb−αΛaϱ+ibϱi+1.
By substituting the values ofI1 and I2 in I, we get the required result.
This completes the proof.


### 2.2. Results and Discussions

We now derive the main results using Lemma1.Theorem 1.
Letα,ϱ>0 and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′ is higher-order n-polynomial convex function of order σ>0, then(12)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21n∑s=1ni+1s2s−i+1+is+1−1i+1ss+1Λ′aϱ+Λ′bϱ−4μσ+1σ+2bϱ−aϱσ.Proof.
By using Lemma1 and because Λ′ is higher-order n-polynomial convex function of order σ>0, we have(13)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I1+I2.
Note that(14)I1=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫01Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫011n∑s=1n1−1−i+xi+1sΛ′aϱ+1n∑s=1n1−i+xi+1sΛ′bϱ−μxσ1−x+x1−xσbϱ−aϱσdx=1ϱi+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱ+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱ−2μσ+1σ+2bϱ−aϱσ1/q.
Similarly,(15)I2=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫01Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫011n∑s=1n1−1−1−xi+1sΛ′aϱ+1n∑s=1n1−1−xi+1sΛ′bϱ−μxσ1−x+x1−xσbϱ−aϱσdx=1ϱi+21n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱ+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱ−2μσ+1σ+2bϱ−aϱσ,where we have used the fact that 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I1 and I2 in (13), we get the required result.Theorem 2.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′q is higher-order n-polynomial convex function of order σ>0, where r−1+q−1=1 and q>1, then(16)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q.Proof.
Using Lemma1, Holder’s inequality and because Λ′q is a higher-order n-polynomial convex function of order σ>0, we have(17)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I3+I4.
Note that(18)I3=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1rdx1/r∫01Λ′i+xi+1aϱ+1−xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1rdx1/r∫011n∑s=1n1−1−i+xi+1sΛ′aϱq+1n∑s=1n1−i+xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q.
Similarly,(19)I4=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1rdx1/r∫01Λ′1−xi+1aϱ+i+xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1rdx1/r∫011n∑s=1n1−1−1−xi+1sΛ′aϱq+1n∑s=1n1−1−xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q,where 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I3 and I4 in (17), we get the required result.Theorem 3.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′q is a higher-order n-polynomial convex function of order σ>0 where q>1, then(20)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21/r1ii+2∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q.Proof.
Using Lemma1, Power mean inequality, and because Λ′q is a higher-order n-polynomial convex function of order σ>0, we have (21)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I5+I6.
Note that(22)I5=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+11n∑s=1n1−1−i+xi+1sΛ′aϱq+1n∑s=1n1−i+xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21/r1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q.
Similarly,(23)I6=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+11n∑s=1n1−1−1−xi+1sΛ′aϱq+1n∑s=1n1−1−xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21/r1ii+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q,where we have used the fact that 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I5 and I6 in (21), we get the required result.


## 2.1. A Key Lemma

We now derive the main auxiliary result of the paper.Lemma 1.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ, then(8)1i+1i+2Λaϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1+Λbϱ−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1=ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt−∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt.Proof.
Consider(9)I=ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt−∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdtI=ϱbϱ−aϱi+1I1−I2.Integrating by parts I1, we have(10)I1=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=−i+1ϱbϱ−aϱ1i+2−tαϱi+1Λi+tϱi+1aϱ+1−tϱi+1bϱ10−αbϱ−aϱ∫01tαϱ−1Λi+tϱi+1aϱ+1−tϱi+1bϱdt,=Λaϱϱbϱ−aϱi+2+i+1ϱi+2bϱ−aϱΛiaϱ+bϱi+1−i+1αϱα−1Γα+1bϱ−aϱα+1ϱIa+αΛiaϱ+bϱi+1.
Similarly,(11)I2=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=−Λbϱϱbϱ−aϱi+2−i+1ϱi+2bϱ−aϱΛaϱ+ibϱi+1+i+1αϱα−1Γα+1bϱ−aϱα+1ϱIb−αΛaϱ+ibϱi+1.
By substituting the values ofI1 and I2 in I, we get the required result.
This completes the proof.


## 2.2. Results and Discussions

We now derive the main results using Lemma1.Theorem 1.
Letα,ϱ>0 and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′ is higher-order n-polynomial convex function of order σ>0, then(12)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21n∑s=1ni+1s2s−i+1+is+1−1i+1ss+1Λ′aϱ+Λ′bϱ−4μσ+1σ+2bϱ−aϱσ.Proof.
By using Lemma1 and because Λ′ is higher-order n-polynomial convex function of order σ>0, we have(13)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I1+I2.
Note that(14)I1=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫01Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫011n∑s=1n1−1−i+xi+1sΛ′aϱ+1n∑s=1n1−i+xi+1sΛ′bϱ−μxσ1−x+x1−xσbϱ−aϱσdx=1ϱi+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱ+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱ−2μσ+1σ+2bϱ−aϱσ1/q.
Similarly,(15)I2=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫01Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx∫011n∑s=1n1−1−1−xi+1sΛ′aϱ+1n∑s=1n1−1−xi+1sΛ′bϱ−μxσ1−x+x1−xσbϱ−aϱσdx=1ϱi+21n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱ+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱ−2μσ+1σ+2bϱ−aϱσ,where we have used the fact that 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I1 and I2 in (13), we get the required result.Theorem 2.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′q is higher-order n-polynomial convex function of order σ>0, where r−1+q−1=1 and q>1, then(16)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q.Proof.
Using Lemma1, Holder’s inequality and because Λ′q is a higher-order n-polynomial convex function of order σ>0, we have(17)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I3+I4.
Note that(18)I3=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1rdx1/r∫01Λ′i+xi+1aϱ+1−xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1rdx1/r∫011n∑s=1n1−1−i+xi+1sΛ′aϱq+1n∑s=1n1−i+xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21n∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1n∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q.
Similarly,(19)I4=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1rdx1/r∫01Λ′1−xi+1aϱ+i+xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1rdx1/r∫011n∑s=1n1−1−1−xi+1sΛ′aϱq+1n∑s=1n1−1−xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21n∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1n∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μσ+1σ+2bϱ−aϱσ1/q,where 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I3 and I4 in (17), we get the required result.Theorem 3.
Letα,ϱ>0, and let Λ:aϱ,bϱ⟶ℝ be a differentiable function on aϱ,bϱ, with 0≤α<b such that Λ′∈L1aϱ,bϱ. If Λ′q is a higher-order n-polynomial convex function of order σ>0 where q>1, then(20)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤bϱ−aϱi+1i+21/r1ii+2∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q.Proof.
Using Lemma1, Power mean inequality, and because Λ′q is a higher-order n-polynomial convex function of order σ>0, we have (21)1i+1i+2Λaϱ+Λbϱ+i+1Λiaϱ+bϱi+1+Λaϱ+ibϱi+1−i+1α−1ϱαΓα+1bϱ−aϱαIa+αϱΛiaϱ+bϱi+1+Ib−αϱΛaϱ+ibϱi+1≤ϱbϱ−aϱi+1∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt+∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=ϱbϱ−aϱi+1I5+I6.
Note that(22)I5=∫011i+2−tαϱi+1tϱ−1Λ′i+tϱi+1aϱ+1−tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+1Λ′i+xi+1aϱ+1−xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+11n∑s=1n1−1−i+xi+1sΛ′aϱq+1n∑s=1n1−i+xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21/r1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q.
Similarly,(23)I6=∫011i+2−tαϱi+1tϱ−1Λ′1−tϱi+1aϱ+i+tϱi+1bϱdt=1ϱ∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱdx≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+1Λ′1−xi+1aϱ+i+xi+1bϱqdx1/q≤1ϱ∫011i+2−xαi+1dx1/r∫011i+2−xαi+11n∑s=1n1−1−1−xi+1sΛ′aϱq+1n∑s=1n1−1−xi+1sΛ′bϱq−μxσ1−x+x1−xσbϱ−aϱσdx1/q=1ϱi+21/r1ii+2∑s=1ni+1ss−i+is+1i+1s1+sΛ′aϱq+1ni+2∑s=1ni+1s1+s−1i+1s1+sΛ′bϱq−2μi+2σ+1σ+2bϱ−aϱσ1/q,where we have used the fact that 1/i+2−xα/i+1≤1/i+2 for all x∈0,1. Using I5 and I6 in (21), we get the required result.



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