# Hermite–Hadamard-Type Inequalities for Generalized Convex Functions via the Caputo-Fabrizio Fractional Integral Operator

Authors: Dong Zhang; Muhammad Shoaib Saleem; Thongchai Botmart; M. S. Zahoor; R. Bano
Journal: Journal of Function Spaces
 (2021)

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## Abstract

Due to applications in almost every area of mathematics, the theory of convex and nonconvex functions becomes a hot area of research for many mathematicians. In the present research, we generalize the Hermite–Hadamard-type inequalities forp,h-convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for p,h-convex functions. Finally, the applications of our main findings are also given.


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## Body

## 1. Introduction

In the last few decades, the subject of fractional calculus got attention of many researchers of different fields of pure and applied mathematics like mechanics, convex analysis, and relativity [1–3]. Nowadays, the researches in convex analysis cannot ignore the deep connectivity of both inequalities in convex analysis and fractional integral operator. Niels Henrik Abel gave birth to fractional calculus. The applications of fractional calculus can be seen in [4–9]. The first appearance of fractional derivative had been seen in a letter. The letter was written to Guillaume de lHopital by Gottfried Wilhelm Leibniz in 1695.The fractional calculus techniques can be seen in many branches of science and engineering. Geometric and physical interpretation of fractional integration and fractional differentiation can be viewed in [10]. There are different fractional integral operators in which we use the integral inequalities (see, for example, [11–16]). The well-known inequality given by Hermite in 1881 can be stated as follows.Theorem 1.
Letζ:L⟶ℝ be a p,h-convex function defined on the interval L of real numbers and c,d∈L with c<d. Then, the following inequality holds:
(1)12h1/2ζcp+dp21/p≤pdp−cp∫cdxp−1ζxdx≤ζc+ζd∫01htdt.The fractional Hermite–Hadamard and Hermite–Hadamard inequalities via fractional integral can be seen in [17, 18]. For the history of Hermite–Hadamard-type inequalities, we refer to the readers [19, 20]. The outstanding applications of fractional calculus and fractional derivatives and integrals are given in [21]. Moreover, we refer the readers for a detailed study [22–27].In the present article, we generalize the Hermite–Hadamard-type inequalities forp,h-convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for p,h-convex functions. Finally, the applications of our main findings are also given.This paper is organized as follows: in Section2, some preliminaries are given. In Section 3, we generalized Hermite–Hadamard via Caputo-Fabrizio for p,h-convex functions. In Section 4, we give some results related to Caputo-Fabrizio, and in Section 5, some applications to special means are given.


## 2. Preliminaries

We will start with some basic definitions related to our work.Definition 2 (convex function) [28].
Letζ:L⟶ℝ be an extended real-valued function defined on a convex set L⊆ℝn. Then, the function ζ is convex on L if
(2)ζtc+1−td≤tζc+1−tζd,for all c,d∈L and t∈0,1.Definition 3 (p-convex function) [29].
A functionζ:L=c,d⊆ℝ is said to be a p-convex function if
(3)ζtcp+1−tdp1/p≤tζc+1−tζd∈L,∀c,d∈L,t∈0,1,p≠0.Definition 4 (h-convex function) [30].
Leth:K⟶ℝ be a nonnegative function. We say that ζ:L⟶ℝ is an h-convex function or that ζ∈SXh,L, if ζ is nonnegative, and for all ∀c,d∈L,t∈0,1, we have
(4)ζtc+1−td≤htζc+h1−tζd.If inequality (4) is reversed, then ζ is said to be h-convex, i.e., ζ∈SVh,L.Remark 5.
(1)
If we takeht=t, then (4) reduces to (2)(2)
If the functionh has the property: ht≥t for all t∈0,1, then any nonnegative convex function ζ belongs to the class SXh,L(3)
If the functionh has the property: ht≤t for all t∈0,1, then any nonnegative convex function ζ belongs to the class SVh,LDefinition 6 (p,h-convex function) [31].
A functionζ:L=c,d⟶ℝ is called a p,h-convex function if
(5)ζtcp+1−tpd1/p≤htζc+h1−tζd,∀c,d∈L,t∈0,1.Definition 7 (Caputo-Fabrizio fractional time derivative) [32].
The usual Caputo fractional time derivativeUFDt of order γ is given by
(6)Dtγζt=1α1−γ∫ctζ′xt−xγdx,with γ∈0,1 and d∈−∞,t,ζ∈H1c,d,c<d. By changing the Kernal t−x−γ with the function exp−γt−xγ/1−γ and 1/α1−γ with Bγ/1−γ, we obtain the new definition of fractional time derivative:
(7)Dtγζt=Bγ1−γ∫ctζ′xexp−γt−xγ1−γdx.Definition 8 (Caputo-Fabrizio fractional integral) [32].
Letζ∈H1c,d,c<d,γ∈0,1; then, the definition of the left fractional derivative in the sense of Caputo and Fabrizio becomes
(8)cCFCDγζt=Bγ1−γ∫ctζ′xexp−γt−xγ1−γdx,and the associated fractional integral is
(9)IγcCFζt=1−γBγζt+γBγ∫ctζxdx,where Bγ>0 is a normalization function satisfying Bγ=B1=1.
In the right case, we have(10)cCFCDγζt=Bγ1−γ∫tdζ′xexp−γt−xγ1−γdx,and the associated fractional integral is
(11)cCFIγζt=1−γBγζt+γBγ∫tdζxdx.Lemma 9.
Letζ:L⟶ℝ be a differentiable mapping on L,c,d∈L with c<d. If ζ′∈L1c,d, then the following inequality holds:
(12)ζc+ζd2−1dp−cp∫cdζxx1−pdx=dp−cp2p∫01Mp−1c,d;t1−2tζ′Mpc,d;tdt,where Mp−1c,d;t=tcp+1−tdp1/p−1.


## 3. A Generalized Hermite–Hadamard-Type Inequality via the Caputo-Fabrizio Fractional Operator for ap,h-Convex Function

The double inequality named as Hermite–Hadamard inequality is considered one of the fundamental inequalities for convex functions.Theorem 10.
Let a functionζ:c,d⊆ℝ⟶ℝ be a p,h-convex function on c,d and ζ∈L1c,d if γ∈0,1; then, the following double inequality holds:
(13)12h1/2ζcp+dp21/p≤2pBγγdp−cpcCFIγηk+CFIdγηk−21−γBγηk≤ζc+ζd∫01htdt,where ηx=x1−pζx.Proof.
Letζ be a p,h-convex function; then, Hermite–Hadamard for a p,h-convex function is as follows:
(14)12h1/2ζcp+dp21/p≤pdp−cp∫cdxp−1ζxdx≤ζc+ζd∫01htdt.
Sinceζ is a p,h-convex function on c,d, we can write
(15)12h1/2ζcp+dp21/p≤pdp−cp∫cdxp−1ζxdx22h1/2ζcp+dp21/p≤2pdp−cp∫cbxp−1ζxdx12h1/2ζcp+dp21/p≤2pdp−cp∫ckxp−1ζxdx+∫kdxp−1ζxdx.
Multiplying both sides of (15) with γdp−cp/2pBγ and adding kp−121−γ/Bγζk, we have
(16)kp−121−γBγζk+γdp−cp2pBγh1/2ζcp+dp21/p≤kp−121−γBγζk+γBγ∫ckxp−1ζxdx+∫kdxp−1ζxdx21−γBγηk+γdp−cp2pBγh1/2ζcp+dp21/p≤21−γBγηk+γBγ∫ckηxdx+∫kdηxdx=1−γBγηk+γBγ∫ckηxdx+1−γBγηk+γBγ∫kdηxdx=cCFIγηk+CFIdγηkζcp+dp21/p≤2pBγh1/2γdp−cpcCFIγηk+CFIdγηk21−γBγηk.
For the proof of the right hand side of the Hermite–Hadamard-type inequality, we have(17)2pdp−cp∫cdxp−1ζxdx≤2ζc+ζd∫01htdt.
Multiplying both sides of (17) with γdp−cp/2pBγ and adding kp−121−γ/Bγζk, we get
(18)cCFIγηk+CFIdγηk≤21−γBγηk+γdp−cpγBγζc+ζd∫01htdt.
By recognizing (18), the proof of the right hand side of (13) is completed. This completes the proof.Remark 11.
If we putp=1 and ht=t, then we will get the Hermite-Hadamard inequality for convex function.Theorem 12.
Letζ,θ:L⊆ℝ⟶ℝ be a p,h-convex function. If ζθ∈L1c,d, then we have the following inequality:
(19)2pBγγdp−cpcCFIγηβk+CFIdγηβk−21−γBγηkβk≤2Mc,d∫01h1th2tdt+Nc,d∫01h1th21−tdt,with ηx=x1−pζx and βx=x1−pθx, where Mc,d=ζcθc+ζdθd, Nc,d=ζcθd+ζdθc, and k∈c,d,Bγ>0 is a normalization function.Proof.
Sinceζ and θ are p,h-convex functions on c,d, we have
(20)ζtcp+1−tdp1/p≤h1tζc+h11−tζd,θtcp+1−tdp1/p≤h2tθc+h21−tθd.
Multiplying both sides of the above inequalities, we have(21)ζtcp+1−tdp1/pθtcp+1−tdp1/p≤h1th2tζcθc+h11−th21−tζdθd+h1th21−tζcθb+h11−th2tζdθc.
Integrating with respect tot over 0,1 and making the change of variable, we obtain
(22)pdp−cp∫cbxp−1ζxθxdx≤∫01h1th2tdtζcθc+ζdθd+∫01h1th21−tdtζcθd+ζdθc,which implies
(23)2pdp−cp∫ckxp−1ζxθxdx+∫ckxp−1ζxθxdx≤2Mc,d∫01h1th2tdt+Nc,d∫01h1th21−tdt.
By multiplying both sides withγcp−dp/2PBγ and adding kp−121−γ/BγζKgk, we have
(24)γBγ∫ckηxβxdx+∫kdηxβxdx+21−γBγηkβk≤γcp−dp2pBγ2Mc,d∫01h1th2tdt+Nc,d∫01h1th21−tdt+21−γBγηkβk.
Thus,(25)cCFIγηβk+CFIdγηβk≤γcp−dp2pBγ2Mc,d∫01h1th2tdt+2Nc,d∫01h1th21−tdt+21−γBγηkβk,and with suitable rearrangements, the proof is completed.Remark 13.
If we putp=1 and ht=t in the above theorem, we get the results for the classical convex function.Theorem 14.
Letζ,θ:L⊆ℝ⟶ℝ be a p,h-convex function. If ζθ∈L1c,d, then we have the following inequality:
(26)1h11/2h21/2ζcp+dp21/pθcp+dp21/p−2pγdp−cpcCFIγηβk+CFIdγηβk+4p1−γγdp−cpηkβk≤2Mc,d∫01h1th2tdt+2Nc,d∫01h1th21−tdt,with ηx=x1−pζx and βx=x1−pθx, where Mc,d=ζcθc+ζdθd, Nc,d=ζcθd+ζdθc, and k∈c,d,Bγ>0 is a normalization function.Proof.
Sincecp+dp/2=tcp+1−tdp/2+1−tcp+tdp/2 for t=1/2, we have
(27)ζ1−tcp+tdp+tcp+1−tdp21/p≤h112ζ1−tcp+tdp1/p+h112ζtcp+1−tdp1/p,(28)θ1−tcp+tdp+tcp+1−tdp21/p≤h212θ1−tcp+tdp1/p+h212θtcp+1−tdp1/p.
Multiplying (2) and (28), we have
(29)ζcp+dp21/pθcp+dp21/p≤h112h212ζ1−tcp+tdp1/pθ1−tcp+tdp1/p+h112h212ζtcp+1−tdp1/pθtcp+1−tdp1/p+h112h212h1tζc+h11−tζdh2tθc+h21−tθd+h112h212h11−tζc+h1tζdh21−tθc+h2tθd=h112h212ζ1−tcp+tdp1/pθ1−tcp+tdp1/p+h112h212ζtcp+1−tdp1/pθtcp+1−tdp1/p+h112h212h1t+h21−t+h11−th2tMc,d+h112h212h1th2t+h11−th21−tNc,d.
Integrating the above inequality with respect tot over 0,1 and making the change of variable, one obtains
(30)1h11/2h21/2ζcp+dp21/pθcp+dp21/p−pdp−cp∫cdxp−1ζxθxdx≤2Mc,d∫01h1th2tdt+2Nc,d∫01h1th21−tdt.
By multiplying both sides withγdp−cp/2pBγ and subtracting Kp−121−γ/Bγζkθk, we have
(31)2γdp−cppBγζcp+dp21/pθcp+dp21/p−γBγ∫ckFxGxdx+∫kdFxGxdx−21−γBγζkgk≤γdp−cp2pBγ2Mc,d∫01h1th2tdt+2Nc,d∫01h1th21−tdt−21−γBγFkGk,2γdp−cppBγζcp+dp21/pθcp+dp21/p−γBγcCFIγηβk+CFIdγηβk≤γdp−cp2pBγ2Mc,d∫01h1th2tdt+Nc,d∫01h1th21−tdt−21−γBγηkβk.
By multiplying both sides of the above inequality by2pBγ/γdp−cp, we get the required inequality (28).Remark 15.
If we putp=1 and ht=t in the above theorem, then we will get the result for the convex function.


## 4. Some New Results Related to the Caputo-Fabrizio Fractional Operator

In this section, firstly we generalize a lemma; then, we prove our main theorem with the help of this lemma.Lemma 16.
Letζ:L⊆ℝ⟶ℝ be a differentiable mapping on L,c,d∈L with c<d. If ζ′∈L1c,d and γ∈0,1, the following equality holds:
(32)dp−cp2∫01M−1c,d;t1−2tζ′Mpc,d;tdt−21−γγdp−cpηk=ζc+ζd2−pBγγdp−cpcCFIγηk+CFIdγηk,where k∈c,d and Bγ>0 is a normalization function with ηx=ζx/x1−p.Proof.
From Lemma9, we can observe that
(33)∫01M−1c,d;t1−2tζ′Mpc,d;tdt=ζc+ζddp−cp−2pdp−cp2∫ckζxx1−pdx+∫kdζxx1−pdx.
By multiplyingγdp−cp2/2pBγ and subtracting 21−γ/x1−pBγζk, we have
(34)γdp−cp22pBγ∫01M−1c,d;t1−2tζ′Mpc,d;tdt−21−γx1−pBγζk=γdp−cpζc+ζd2pBγ+21−γx1−pBγζk−γBγ∫ckζxx1−pdx+∫kdζxx1−pdx=γdp−cpζc+ζd2pBγ+21−γBγηk−γBγ∫ckηxdx+∫kdηxdx=γdp−cpζc+ζd2pBγ−1−γBγηk+γBγ∫ckηxdx−1−γBγηk+γBγ∫ckηxdx=γdp−cpζc+ζd2pBγ−cCFIγηk+CFIdγηk.
This completes the proof.Theorem 17.
Letζ:L⊆ℝ⟶ℝ be a differential mapping on L and ζ′ be p,h-convex on c,d where c,d∈L with c<d,p>0. If ζ′∈L1c,d and γ∈0,1, the following inequality holds:
(35)ζc+ζd2+21−γγdp−cpζk−pBγγdp−cpcCFIγζk+CFIdγζk≤dp−cp2pd1c,d,pζ′c+d2c,d,pζ′d,where
(36)d1c,d,p=∫01/21−2thttcp+1−tdp1/p−1dt+∫1/212t−1httcp+1−tdp1/p−1dt,d2c,d,p=∫01/21−2th1−ttcp+1−tdp1/p−1dt+∫1/212t−1h1−ttcp+1−tdp1/p−1dt,where k∈c,d and Bγ>0 is a normalization function.Proof.
(37)ζc+ζd2+21−γγdp−cpζk−pBγγdp−cpcCFIγζk+CFIdγζk≤dp−cp2p∫01M−1c,d;t1−2tζ′Mpc,d;tdt≤dp−cp2p∫011−2tζ′tcp1−tdp1/ptcp1−tdp1/p−1dt≤dp−cp2p∫011−2thtζ′c+h1−tζ′dtcp1−tdp1/p−1dt=dp−cp2p∫01/21−2thtζ′ctcp+1−tdp1/p−1dt+∫1/212t−1htζ′ctcp+1−tdp1/p−1dt+dp−cp2p∫01/21−2th1−tζ′dtcp+1−tdp1/p−1dt+∫1/212t−1h1−tζ′dtap+1−tdp1/p−1dt≤dp−ap2pd1c,d,pζ′c+d2c,d,pζ′d.
This completes the proof.Theorem 18.
Letζ:L⊆ℝ⟶ℝ be a differential mapping on L and ζ′ be p,h-convex on c,d where c,d∈L with c<d, p>1, and 1/r+1/q=1. If ζ′∈L1c,d and γ∈0,1, the following inequality holds:
(38)ζc+ζd2+21−γγdp−cpζk−pBγγdp−cpcCFIγζk+CFIdγζk≤dp−ap2pd1c,d,p1/rd2c,d,pζ′cq+d3c,d,pζ′dq1/q,where
(39)d1c,d,p=∫011−2trtcp1−tdp1/p−1dt,d2c,d,p=∫01httcp1−tdp1/p−1dt,d3c,d,p=∫01h1−ttcp1−tdp1/p−1dt,where k∈c,d and Bγ>0 is a normalization function.Proof.
By a similar argument to the proof of the previous theorem, by using lemma, the Hölder inequality, and convexity ofζ′, we get
(40)ζc+ζd2+21−γγdp−cpfk−pBγγdp−cpcCFIγζk+CFIdγζk≤dp−cp2p∫01M−1c,d;t1−2tζ′Mpc,d;tdt≤dp−cp2p∫011−2trtcp1−tdp1/p−1dt1/r∫01ζ′tcp1−tdpq/ptcp1−tdp1/p−1dt1/q≤dp−cp2p∫011−2trtcp1−tdp1/p−1dt1/r∫01htζ′cq+h1−tζ′dqtcp1−tdp1/p−1dt1/q≤dp−cp2p∫011−2trtcp1−tdp1/p−1dt1/r∫01htζ′cqtcp1−tdp1/p−1dt+∫01h1−tζ′dqtcp1−tdp1/p−1dt1/q≤dp−cp2pd1c,d,p1/rd2c,d,pζ′aq+d3c,d,pζ′dq1/q.
This completes the proof.


## 5. Application to Special Means

The applications to special means are also used to confirm the accuracy of the findings for real numbersc,d such that c≠d.The arithmetic mean of two numbersc and d is defined as
(41)A=Ac,d=c+d2,c,d∈ℝ.The generalized logarithmic mean is defined as(42)L=Lrr=ct+1−dt+1r+1d−cr∈ℝ−−1,0,t∈ℝc,d∈ℝ,c≠d.Now, using the results in Section4, we have some applications to the special means of real numbers.Proposition 19.
Letc,d∈ℝ,c<d; then,
(43)Ac2,d2−pL22c,d≤dp−cp2pd12c+d22d.Proof.
In the inequality proven in Theorem17, if we set ζz=z2 with γ=1 and Bγ=B1=1, then we obtain the result immediately.Remark 20.
If we putht=t and p=1 in this proposition, we will obtain this result for the convex function.Proposition 21.
Letc,d∈ℝ,c<d; then,
(44)Acn,dn−pLnnc,d≤ndp−cp2pd12cn−1+d22dn−1.Proof.
In the inequality proven in Theorem17, if we set ζz=zn where n is an even number with γ=1 and Bγ=B1=1, then we obtain the result immediately.Remark 22.
If we putht=t and p=1 in this preposition, we will obtain this result for the convex function.


## 6. Conclusion

Convexity is very important for solving optimization problems. Fractional calculus together with convexity plays an important role in solving real-life problems. In this paper, we established several Hermite–Hadamard-type inequalities in the setting of a fractional integral operator forp,h-convex functions. We also presented some applications in means. Our results generalized several existing results.



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