# Inequalities for Estimations of Integrals Related to Higher-Order Stronglyn-Polynomial Preinvex Functions

Authors: Yongping Deng; Muhammad Uzair Awan; Sadia Talib; Muhammad Aslam Noor; Khalida Inayat Noor; Pshtiwan Othman Mohammed; Shanhe Wu
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, we establish an integral identity associated withm-times differentiable functions. The result is then used to derive some integral estimations for higher-order strongly n-polynomial preinvex functions. Finally, we apply the obtained inequalities to construct new inequalities involving special means.


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## Body

## 1. Introduction

The convexity properties of functions have been a powerful tool for dealing with various problems of pure and applied sciences. In recent decades, the concept of classical convexity has been extended and generalized in different directions, and the study relevant to this subject has stimulated an increased interest. In [1], Weir and Mond introduced the notion of preinvexity as follows.Definition 1.
A setX⊂ℝ is said to be invex with respect to ξ:ℝ×ℝ⟶ℝ if(1)e+τξf,e∈X,∀e,f∈X,τ∈0,1.Definition 2.
A functionΛ:X⟶ℝ is said to be preinvex with respect to ξ:ℝ×ℝ⟶ℝ if(2)Λe+τξf,e≤1−τΛe+τΛf,∀e,f∈X,τ∈0,1.
Forξf,e=f−e, the class of preinvex functions reduces to the class of convex functions. It had been shown that the preinvex functions may not be convex functions. For example, the function Λ=−x is not a convex function, but it is a preinvex function with respect to ξ provided that(3)ξf,e=f−e,iff≤0,e≤0andf≥0,e≥0,e−f,otherwise.
Mohan and Neogy [2] and Mititelu [3] discussed the properties of invex sets and preinvex functions. Noor [4] and Noor et al. [5] investigated integral inequalities of Hermite–Hadamard type for log-preinvex functions and h-preinvex functions, respectively. Wu et al. [6] considered harmonically p,h,m-preinvex functions and relevant inequalities. Peng et al. [7] established fractional Simpson’s inequalities with the aid of m,h1,h2-preinvexity. Du et al. [8] dealt with integral inequalities involving generalized m-convex functions on fractal sets. Saleh [9] defined ε,m-preinvex function and studied the properties of ε,m-preinvex functions. Mohammed [10] focused on the integral inequalities for preinvex functions via the generalized beta function.
Very recently, Toplu et al. [11] introduced and investigated the class of n-polynomial convex functions as follows.Definition 3.
Letn∈ℕ. A nonnegative function Λ:I⊂ℝ⟶ℝ is called an n-polynomial convex function if for any e,f∈I and τ∈0,1,(4)Λτe+1−τf≤1n∑s=1n1−1−τsΛe+1n∑s=1n1−τsΛf.
Note that if we taken=1 in the above inequality, then we have 1-polynomial convexity which is just the classical convexity. It is also necessary to mention here that every nonnegative convex function is also an n-polynomial convex function. For more properties on the n-polynomial convex function, see [11].
Kadakal [12] and Karamardian [13] put forward the class of strongly convex functions, respectively. Strong convexity is just the strengthening version of convexity, which reads as follows.Definition 4.
A functionΛ:I⊂ℝ⟶ℝ is said to be a strongly convex function with modulus μ>0 if(5)Λ1−τe+τf≤1−τΛe+τΛf−μτ1−τf−e2,∀e,f∈I,τ∈0,1.
In order to deal with the problems of mathematical programs with equilibrium constraints, Lin and Fukushima [14] proposed a generalization of the strongly convex function, which is called the higher-order strongly convex functions.Definition 5.
A functionΛ:I⊂ℝ⟶ℝ is said to be a strongly convex function with order σ>0 and modulus μ>0 if(6)Λ1−τe+τf≤1−τΛe+τΛf−μτ1−τf−eσ,∀e,f∈I,τ∈0,1.
The higher-order strongly convex functions would reduce to the strongly convex function in the special case whenσ=2.
As an extension of classical convex functions, the strongly convex functions have been widely used in establishing new inequalities and dealing with estimations of integrals. Angulo et al. [15] and Zhang et al. [16] presented some inequalities for strongly h-convex functions and strongly s,m-convex functions, respectively. In order to further generalize the strongly convex functions, recently, some researchers, such as Mishra and Sharma [17], Noor and Noor [18], and Mohsen et al. [19], began to study the higher-order strongly convex functions. Wu et al. [20, 21] gave some estimates of the upper bound for differentiable functions associated with k-fractional integrals and higher-order strongly convex functions. Khan et al. [22] and Ullah et al. [23] introduced the concept of coordinate and the technique of majorization into the study of strongly convex functions.
In this paper, we introduce the notion for a new class of strongly convex functions which are named as the stronglyn-polynomial preinvex functions of order σ>0.Definition 6.
Letn∈ℕ. A nonnegative function Λ:X⟶ℝ is said to be a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0 if(7)Λe+1−τξf,e≤1n∑s=1n1−1−τsΛe+1n∑s=1n1−τsΛf−μτσ1−τ+τ1−τσξf,eσholds for all e,f∈X and τ∈0,1.Remark 1.
Note that(i)
If we takeσ=2 in Definition 6, then we have the class of strongly n-polynomial preinvex functions(ii)
If we takeξf,e=f−e in Definition 6, then we have the class of strongly n-polynomial convex functions of order σ>0(iii)
If we takeξf,e=f−e and σ=2 in Definition 6, then we have the class of strongly n-polynomial convex functions(iv)
If we taken=1 in Definition 6, then we have the class of strongly preinvex functions of order σ>0
This shows that the class of stronglyn-polynomial preinvex functions of order σ>0 is quite unifying one as it relates several unrelated classes of convexity.
The main objective of this paper is to establish a new integral identity form-times differentiable functions. With the help of this identity, we deduce some inequalities for estimations of integrals associated with m-times differentiable strongly preinvex functions of order σ>0. As applications, we discuss some special cases of the main results and establish several new inequalities involving special means.


## 2. A Key Lemma

Here, we establish an auxiliary result which will play a significant role in the development of our main results.Lemma 1.
LetΛ:e,e+ξf,e⟶ℝ be an m-times differentiable function on e,e+ξf,e with ξf,e>0. If Λm∈Le,e+ξf,e, then the following integral equality holds for m∈ℕ+:(8)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!=ξmf,e2m!∫01τm+τ−1mΛme+1−τξf,edτ.Proof.
Integrating by parts, we have(9)ξm+1f,e2m!∫01τmΛme+1−τξf,edτ=−ξmf,e2m!Λm−1e+ξmf,e2m−1!∫01τm−1Λm−1e+1−τξf,edτ=−ξmf,e2m!Λm−1e−ξm−1f,e2m−1!Λm−2e+ξm−1f,e2m−2!∫01τm−2Λm−2e+1−τξf,edτ=−∑k=1m−1ξk+1f,eΛke2k+1!+ξ2f,e2∫01τΛ′e+1−τξf,edτ=−∑k=1mξkf,eΛk−1e2k!+12∫ee+ξf,eΛxdx.
Similarly, we deduce that(10)ξm+1f,e2m!∫01τ−1mΛme+1−τξf,edτ=ξmf,e2m!−1mΛm−1e+ξf,e+ξmf,e2m−1!∫01τ−1m−1Λm−1e+1−τξf,edτ=ξmf,e2m!−1mΛm−1e+ξf,e+ξm−1f,e2m−1!−1m−1Λm−2e+ξf,e+ξm−1f,e2m−2!∫01τ−1m−2Λm−2e+1−τξf,edτ=∑k=1m−1ξk+1f,e−1k+1Λke+ξf,e2k+1!+ξ2f,e2∫01τ−1Λ′e+1−τξf,edτ=∑k=1mξkf,e−1kΛk−1e+ξf,e2k!+12∫ee+ξf,eΛxdx.
Combining (9) and (10) completes the proof of Lemma 1.


## 3. Main Results

We are now in a position to state our main results.Theorem 1.
LetΛ:e,e+ξf,e⟶ℝ be an m-times differentiable function on e,e+ξf,e with ξf,e>0 and Λm∈Le,e+ξf,e. If Λm is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then(11)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!Λmen∑s=1nℋ1+Λmfn∑s=1nℋ1−μℋ2ξσf,e,where(12)ℋ1=m+2s+1m+1m+s+1−Bm+1,s+1,ℋ2=2m+σ+1m+σ+2+2Bσ+1,m+2,and Bx,y=∫01τx−11−τy−1dτ is the beta function.Proof.
Using Lemma1 and the fact that Λm is a strongly n-polynomial preinvex function of order σ>0, we have(13)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!∫01τm+1−τmΛme+1−τξf,edτ≤ξmf,e2m!∫01τm+1−τm1n∑s=1n1−1−τsΛme+1n∑s=1n1−τsΛmf−μξσf,eτσ1−τ+τ1−τσdτ=ξmf,e2m!Λmen∫01τm+1−τm∑s=1n1−1−τsdτ+Λmfn∫01τm+1−τm∑s=1n1−τsdτ−μξσf,e∫01τm+1−τmτσ1−τ+τ1−τσdτ=ξmf,e2m!Λmen∑s=1n∫01τm+1−τm1−1−τsdτ+Λmfn∑s=1n∫01τm+1−τm1−τsdτ−μξσf,e∫01τm+1−τmτσ1−τ+τ1−τσdτ,where(14)∫01τm+1−τm1−1−τsdτ=m+2s+1m+1m+s+1−Bm+1,s+1=ℋ1,∫01τm+1−τm1−τsdτ=m+2s+1m+1m+s+1−Bs+1,m+1=ℋ1,∫01τm+1−τmτσ1−τ+τ1−τσdτ=2m+σ+1m+σ+2+2Bσ+1,m+2=ℋ2.
The proof of Theorem1 is completed.
In the following, we discuss some special cases of Theorem1.(1)
If we take the limitμ⟶0 in Theorem 1, we get a result for the n-polynomial preinvex function.Corollary 1.
Under the assumptions of Theorem1, if Λm is an n-polynomial preinvex function, then(15)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!Λmen∑s=1nℋ1+Λmfn∑s=1nℋ1,where(16)ℋ1=m+2s+1m+1m+s+1−Bm+1,s+1.(2)
If we takeσ=2 in Theorem 1, we get a result for the strongly n-polynomial preinvex function.Corollary 2.
Under the assumptions of Theorem1, if Λm is a strongly n-polynomial preinvex function, then(17)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!Λmen∑s=1nℋ1+Λmfn∑s=1nℋ1−μℋ3ξ2f,e,where(18)ℋ1=m+2s+1m+1m+s+1−Bm+1,s+1,ℋ3=∫01τm+1−τmτ21−τ+τ1−τ2dτ=2m+2m+4.(3)
If we taken=1 in Theorem 1, we get a result for the higher-order strongly preinvex function.Corollary 3.
Under the assumptions of Theorem1, if Λm is a strongly preinvex function of order σ>0, then(19)1ξf,e∫ee+ξf,eΛx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!Λme+Λmfm+1−μℋ2ξσf,e.Theorem 2.
LetΛ:e,e+ξf,e⟶ℝ be an m-times differentiable function on e,e+ξf,e with ξf,e>0,1/p+1/q=1, p>1,q>1, and Λm∈Le,e+ξf,e. If Λmq is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then(20)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2pmp+11p2n∑s=1nss+1AΛmeq,Λmfq−2μξσf,eσ+1σ+21/q,where A.,. is the arithmetic mean.Proof.
Utilizing Lemma1, Hölder’s integral inequality, and the fact that Λmq is a strongly n-polynomial preinvex function of order σ>0, one has(21)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!∫01τm+1−τmΛme+1−τξf,edτ≤ξmf,e2m!∫01τm+1−τmpdτ1/p∫01Λme+1−τξf,eqdτ1/q≤ξmf,e2m!∫012p−1τmp+1−τmpdτ1/p∫01Λme+1−τξf,eqdτ1/q≤ξmf,e2m!2pmp+11/p∫011n∑s=1n1−1−τsΛmeqdτ+∫011n∑s=1n1−τsΛmfqdτ−μξσf,e∫01τσ1−τ+τ1−τσdτ1/q=ξmf,e2m!2pmp+11/pΛmeqn∫01∑s=1n1−1−τsdτ+Λmfqn∫01∑s=1n1−τsdτ−μξσf,e∫01τσ1−τ+τ1−τσdτ1/q=ξmf,e2m!2pmp+11/pΛmeqn∑s=1n∫011−1−τsdτ+Λmfqn∑s=1n∫011−τsdτ−μξσf,e∫01τσ1−τ+τ1−τσdτ=ξmf,e2m!2pmp+11/p2n∑s=1nss+1AΛmeq,Λmfq−2μξσf,eσ+1σ+21/q.
This completes the proof of Theorem2.
In the following, we discuss some special cases of Theorem2.(1)
Puttingμ⟶0 in Theorem 2, we obtain a result for the n-polynomial preinvex function.Corollary 4.
Under the assumptions of Theorem2, if Λmq is an n-polynomial preinvex function of order σ>0, then(22)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2pmp+11/p2n∑s=1nss+1AΛmeq,Λmfq1/q.(2)
Puttingσ=2 in Theorem 2, we obtain a result for the strongly n-polynomial preinvex function.Corollary 5.
Under the assumptions of Theorem2, if Λmq is a strongly n-polynomial preinvex function, then(23)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2pmp+11/p2n∑s=1nss+1AΛmeq,Λmfq−μξf,e261/q.(3)
Puttingn=1 in Theorem 2, we obtain a result for the higher-order strongly preinvex function.Corollary 6.
Under the assumptions of Theorem2, if Λmq is a strongly preinvex function of order σ>0, then(24)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2pmp+11/pAΛmeq,Λmfq−2μξσf,eσ+1σ+21/q.Theorem 3.
LetΛ:e,e+ξf,e⟶ℝ be an m-times differentiable function on e,e+ξf,e with ξf,e>0,q≥1, and Λm∈Le,e+ξf,e. If Λmq is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then(25)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2m+11−1/qΛmeqn∑s=1nℋ1+Λmfqn∑s=1nℋ1−μℋ2ξσf,e1/q,where ℋ1 and ℋ2 are the expressions as described in Theorem 1.Proof.
Applying Lemma1, power-mean integral inequality, and the fact that Λmq is a strongly n-polynomial preinvex function of order σ>0, we have(26)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!∫01τm+1−τmΛme+1−τξf,edτ≤ξmf,e2m!∫01τm+1−τmdτ1−1/q×∫01τm+1−τmΛme+1−τξf,eqdτ1/q≤ξmf,e2m!∫01τm+1−τmdτ1−1/q×∫01τm+1−τm1n∑s=1n1−1−τsΛmeq+1n∑s=1n1−τsΛmfq−μξσf,e×τσ1−τ+τ1−τσdτ1/q=ξmf,e2m!2m+11−1/qΛmeqn∫01τm+1−τm∑s=1n1−1−τsdτ+Λmfqn∫01τm+1−τm∑s=1n1−τsdτ−μξσf,e∫01τm+1−τmτσ1−τ+τ1−τσdτ1/q=ξmf,e2m!2m+11−1/qΛmeqn∑s=1n∫01τm+1−τm1−1−τsdτ+Λmfqn∑s=1n∫01τm+1−τm1−τsdτ−μξσf,e∫01τm+1−τmτσ1−τ+τ1−τσdτ1/q=ξmf,e2m!2m+11−1/qΛmeqn∑s=1nℋ1+Λmfqn∑s=1nℋ1−μℋ2ξσf,e1/q.
Theorem3 is proved.
Finally, we discuss some special cases of Theorem3.(1)
Lettingμ⟶0 in Theorem 3, we get a result for the n-polynomial preinvex function.Corollary 7.
Under the assumptions of Theorem3, if Λmq is an n-polynomial preinvex function of order σ>0, then(27)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2m+11−1/qΛmeqn∑s=1nℋ1+Λmfqn∑s=1nℋ11/q.(2)
Choosingσ=2 in Theorem 3, we get a result for the strongly n-polynomial preinvex function.Corollary 8.
Under the assumptions of Theorem3, if Λmq is a strongly n-polynomial preinvex function, then(28)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2m+11−1/qΛmeqn∑s=1nℋ1+Λmfqn∑s=1nℋ1−μℋ3ξf,e21/q.(3) Choosingn=1 in Theorem 3, we get a result for the higher-order strongly preinvex function.Corollary 9.
Under the assumptions of Theorem3, if Λmq is a strongly preinvex function of order σ>0, then(29)1ξf,e∫ee+ξf,eΛxdx−∑k=1mξk−1f,eΛk−1e+−1k−1Λk−1e+ξf,e2k!≤ξmf,e2m!2m+11−1/qΛmeq+Λmfqm+1−μℋ2ξσf,e1/q.


## 4. Application to Special Means

We consider the following special means.(1)
The arithmetic mean:(30)Ae,f=e+f2.(2)
The mean of the functionΨ on[e, f](31)AΨe,f=1f−e∫efΨedx.Proposition 1.
LetΨ:e,e+ξf,e→ℝ be m-times differentiable function on e,e+ξf,e with ξf,e>0 and Ψm∈Le,e+ξf,e. If Ψm is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then the following inequality holds:(32)AΨe,e+ξf,e−∑k=1mξk−1f,ek!AΨk−1e,−1k−1Ψk−1e+ξf,e≤ξmf,e2m!2n∑s=1nℋ1AΨme,Ψmf−μℋ2ξσf,e,where ℋ1 and ℋ2 are as in Theorem 1.Proof.
TakingΛx=Ψx, x∈e,e+ξf,e and using Theorem 1, we get the desired result (32).Proposition 2.
LetΨ:e,e+ξf,e→ℝ be m-times differentiable function on e,e+ξf,e with ξf,e>0, 1/p+1/q=1, p>1,q>1 and Ψm∈Le,e+ξf,e. If Ψmq is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then the following inequality holds:(33)AΨe,e+ξf,e−∑k=1mξk−1f,ek!AΨk−1e,−1k−1Ψk−1e+ξf,e≤ξmf,e2m!2pmp+11/p2n∑s=1nss+1AΨmeq,Ψmfq−2μξσf,eσ+1σ+21/q.Proof.
SettingΛx=Ψx, x∈e,e+ξf,e and applying Theorem 2, we get the desired result (33).Proposition 3.
LetΨ:e,e+ξf,e→ℝ be m-times differentiable function on e,e+ξf,e with ξf,e>0, q≥1 and Ψm∈Le,e+ξf,e. If Ψmq is a strongly n-polynomial preinvex function of order σ>0 with modulus μ>0, then the following inequality holds:(34)AΨe,e+ξf,e−∑k=1mξk−1f,ek!AΨk−1e,−1k−1Ψk−1e+ξf,e≤ξmf,e2m!2m+11−1/q2n∑s=1nℋ1AΨmeq,Ψmfq−μℋ2ξσf,e1/q,where ℋ1 and ℋ2 are as in Theorem 1.Proof.
TakingΛx=Ψx, x∈e,e+ξf,e and utilizing Theorem 3, we get the desired result (34).



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