# On Caputo–Fabrizio Fractional Integral Inequalities of Hermite–Hadamard Type for Modifiedh-Convex Functions

Authors: Xiaobin Wang; Muhammad Shoaib Saleem; Kiran Naseem Aslam; Xingxing Wu; Tong Zhou
Journal: Journal of Mathematics
 (2020)

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## Abstract

The theory of convex functions plays an important role in engineering and applied mathematics. The Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. The aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting ofh-convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. The results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.


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## Body

## 1. Introduction and Preliminaries

The subject of fractional calculus got rapid development because of its diverse applications, not only in mathematics but also into many other fields of sciences. Nowadays, the researchers from biology (e.g., Cesarone et al. [1] and Caputo and Cametti [2]), economy (e.g., Caputo [3]), demography (e.g., Jumarie [4]), geophysics (e.g., Iaffaldano et al. [5]), medicine (e.g., El Sahede [6]), and bioengineering (e.g., Magin [7]) and signal processing are using fractional calculus as a key tool.Many researchers in the last three decades are studying fractional calculus [8–12]. Some researchers deduced that it is essential to define new fractional derivatives with different singular or nonsingular kernels in order to provide more sufficient area to model more real-world problems in different fields of science and engineering [13–19].In the present research, we will restrict ourselves to Caputo–Fabrizio fractional derivative. The features that make the operators different from each other comprise singularity and locality, while kernel expression of the operator is presented with functions such as the power law, the exponential function, or a Mittag–Leffler function. The unique feature of the Caputo–Fabrizio operator is that it has a nonsingular kernel. The main feature of the Caputo–Fabrizio operator can be described as a real power turned in to the integer by means of the Laplace transformation, and consequently, the exact solution can be easily found for several problems.Fractional calculus plays a very significant role in the development of inequality theory. To study convex functions and its generalizations, the Hermite–Hadamard-type inequality is considered as one of the fundamental inequality is given as.Theorem 1 (see [20]).
Letψ:J⊆ℝ⟶ℝ be a convex function and a1,b1∈J with a1<b1, then the following double inequality holds:(1)ψa1+b12≤1b1−a1∫a1b1ψxdx≤ψa1+ψb12.The Hermite–Hadamard inequality has been generalized by numerous fractional integral operators [21–23]. For the interesting readers, we refer [24–27] to study about Hermite–Hadamard inequalities.The paper is organized as follows: first of all, we give some definitions and preliminary material related to our work. In Section2, we will establish Hermite–Hadamard-type inequalities via Caputo–Fabrizio fractional integral operator for modified h-convex functions. Section 3 is devoted for some new inequalities via Caputo–Fabrizio fractional operator. At last, we give some application to special means and concluding remarks for our paper.Now, we start by some necessary definitions and preliminary results which will be used and in this paper.In [28], Toader gave the concept of modified h-convex functions as follows.Definition 1.
(see [28]). Let ψ,h:J⊂ℝ⟶ℝ be a nonnegative function, then ψ is called modified h-convex function, if(2)ψαa1+1−αb2≤hαψa1+1−hαψb2,for all a1,b2∈J and α∈0,1 holds.
In [8, 29, 30], the concept of Caputo–Fabrizio fractional operator has been given.Definition 2.
(see [8, 29, 30]). Let ψ∈H1a1,b1,a1<b1,σ∈0,1, then the left fractional derivative in the sense of Caputo and Fabrizio is given by(3)a1CFCDσψt=Bσ1−σ∫a1tψ′xe−σt−xσ/1−σdx,and the associated fractional integral is(4)a1CFIσψt=1−σBσψt+σBσ∫a1tψxdx,where Bσ>0 is a normalization function satisfying B0=B1=1.
The right fractional derivative is given as(5)CFCDb1σψt=−Bσ1−σ∫tb1ψ′xe−σx−tσ/1−σdx,and the associated fractional integral is(6)CFIb1σψt=1−σBσψt+σBσ∫tb1ψxdx.
The following lemma is proven by Dragomir and Agarwal in [31].Lemma 1 (see [31], Lemma 2.1).
Letψ:J=a1,b1⟶ℝ be a differentiable mapping on J∘,a1,b1∈J∘ with a1<b1. If ψ∈La1,b1, then the following equality holds:(7)ψa1+ψb12−1b1−a1∫a1b1ψxdx=b1−a12∫011−2αψ′αa1−1−αb1dα.Mustafa Gurbuz et al., in [32], generalized the kernal used in Lemma 1 with the help of Caputo–Fabrizio fractional integral operator.Lemma 2 (see [32], Lemma 2).
Letψ:J=a1,b1⟶ℝ be a differentiable mapping on J∘,a1,b1∈J∘ with a1<b1. If ψ∈L1a1,b1 and σ∈0,1, then the following equality holds”(8)b1−a12∫011−2αψ′αa1−1−αb1dα−21−σσb1−a1ψk=ψa1+ψb12−Bσσb1−a1a1CFIσψk+CFIb1σψk,where k∈a1,b1 and Bσ>0 is a normalization function.Iscan gave a refinement of Hölder integral inequality in [33], which is given in the following theorem.Theorem 2 (Hölder–Iscan integral inequality [33]).
Letp>1 and 1/p+1/q=1. If ψ and ϕ are real functions defined on interval a1,b1 and ψq,ϕq are integrable functions on a1,b1, then(9)∫a1b1ψxϕxdx≤1b1−a1∫a1b1b1−xψxpdx1/p∫a1b1b1−xϕxqdx1/q+∫a1b1x−a1ψxpdx1/p∫a1b1x−a1ϕxqdx1/q.
A refinement of power-mean integral inequality is given in the following theorem.Theorem 3 (improved power-mean integral inequality [34]).
Letq≥1. If ψ and ϕ are real functions defined on interval a1,b1 and ψ,ψgq are integrable functions on a1,b1, then(10)∫a1b1ψxϕxdx≤1b1−a1∫a1b1b1−xψxdx1−1/q∫a1b1b1−xψxϕxqdx1/q+∫a1b1x−a1ψxdx1−1/q∫a1b1x−a1ψxϕxqdx1/q.


## 2. Generalization of Hermite–Hadamard Inequality via the Caputo–Fabrizio Fractional Operator

The following theorem is a variant of Hermite–Hadamard inequality for modifiedh-convex functions.Theorem 4.
Suppose thatψ:J=a1,b1⟶ℝ is a modified h-convex function and ψ∈L1a1,b1. If σ∈0,1, then the following double inequality holds:(11)ψa1+b12≤Bσσb1−a1a1CFIσψk+CFIb1σψk−21−σBσψk≤ψa1+ψb1−ψa1∫01hαdα,where k∈a1,b1 and Bσ>0 is a normalization function.Proof.
Sinceψ is modified h-convex function on a1,b1, we can write(12)2ψa1+b12≤2b1−a1∫a1b1ψxdx=2b1−a1∫a1kψxdx+∫kb1ψxdx.
Multiplying both sides of (12) by σb1−a1/2Bσ and adding 21−σ/Bσψk, we get(13)21−σBσψk+σb1−a1Bσψa1+b12≤21−σBσψk+σBσ∫a1kψxdx+∫kb1ψxdx=1−σBσψk+σBσ∫a1kψxdx+1−σBσψk+σBσ∫kb1ψxdx=a1CFIσψk+CFIb1σψk.
After suitable rearrangement of (13), we get the required left-hand side of (11).
For the right-hand side, we will use the right-hand side of Hermite–Hadamard inequality for modifiedh-convex functions:(14)2b1−a1∫a1b1ψxdx≤2ψa1+2ψb1−ψa1∫01hαdα.
By using the same operator with (12) in (14), we have(15)a1CFIσψk+CFIb1σψk≤21−σBσψk+σb1−a1Bσψa1+ψb1−ψa1∫01hαdα.
After suitable rearrangement of (15), we get the required right-hand side of (11), which completes the proof.Remark 1.
If we takehα=α, then inequality (11) reduces to the Hermite–Hadamard inequality for convex functions via Caputo–Fabrizio fractional operator [32].Theorem 5.
Letψ and ϕ are modified h-convex functions on J. If ψϕ∈La1,b1, then we have the following inequality:(16)2Bσσb1−a1a1CFIσψϕk+CFIb1σψϕk−21−σBσψkϕk≤2Ma1,b1−Na1,b1∫01hα2dα+2ψb1ϕb1+2Na1,b1−2ψb1ϕb1∫01hαdα,where(17)Ma1,b1=ψa1ϕa1+ψb1ϕb1,Na1,b1=ψa1ϕb1+ψb1ϕa1,k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Sinceψ and ϕ are convex on a1,b1, we have(18)ψαa1+1−αb1≤hαψa1+1−hαψb1,∀α∈0,1,(19)ϕαa1+1−αb1≤hαϕa1+1−hαϕb1,∀α∈0,1.
Multiplying both sides of (18) and (19), we have(20)ψαa1+1−αb1ϕαa1+1−αb1≤hα2ψa1ϕa1+1−hα2ψb1ϕb1+hα1−hαψa1ϕb1+ψb1ϕa1.
Integrating (20) with “α” over 0,1, and using the change of variable technique, we obtain(21)1b1−a1∫a1b1ψxϕxdx≤∫01hα2ψa1ϕa1+1−hα2ψb1ϕb1dα+∫01hα1−hαψa1ϕb1+ψb1ϕa1dα≤Ma1,b1∫01hα2dα+ψb1ϕb1∫011−2hαdα+Na1,b1∫01hα1−hαdα.
So,(22)2b1−a1∫a1b1ψxϕxdx≤2Ma1,b1−Na1,b1∫01hα2dα+2ψb1ϕb1+2Na1,b1−2ψb1ϕb1∫01hαdα.
Multiplying both sides of (22) by σb1−a1/2Bσ and adding 21−σ/Bσψkϕk, we get(23)21−σBσψkϕk+σBσ∫a1kψxϕxdx+∫kb1ψxϕxdx≤21−σBσψkϕk+σb1−a12Bσ2Ma1,b1−Na1,b1∫01hα2dα+2ψb1ϕb1+2Na1,b1−2ψb1ϕb1∫01hαdα.
So,(24)1−σBσψkϕk+σBσ∫a1kψxϕxdx+1−σBσψkϕk+σBσ∫kb1ψxϕxdx≤21−σBσψkϕk+σb1−a12Bσ2Ma1,b1−Na1,b1∫01hα2dα+2ψb1ϕb1+2Na1,b1−2ψb1ϕb1∫01hαdα.
Thus,(25)a1CFIσψϕk+CFIb1σψϕk≤21−σBσψkϕk+σb1−a12Bσ2Ma1,b1−Na1,b1∫01hα2dα+2ψb1ϕb1+2Na1,b1−2ψb1ϕb1∫01hαdα,with suitable rearrangements, and the proof is completed.Remark 2.
If we takehα=α in Theorem 5, we obtain ([32], Theorem 3).Theorem 6.
Letψ and ϕ are modified h-convex functions on J. If ψϕ∈La1,b1, the set of integrable functions and h1/21−h1/2≠0, then we have the following inequality:(26)1h1/21−h1/2ψa1+b12ϕa1+b12−Bσh1/22+1−h1/22σb1−a1h1/2.1−h1/2⋅a1CFIσψϕk+CFIb1σψϕk+21−σh1/22+1−h1/22σb1−a1h1/2.1−h1/2ψkϕk≤2Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα,where(27)Ma1,b1=ψa1ϕa1+ψb1ϕb1,Na1,b1=ψa1ϕb1+ψb1ϕa1,and k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Sinceψ and ϕ are modified h-convex functions on J, then for α=1/2, we have(28)ψa1+b12≤h12ψ1−αa1+αb1+1−h12ψαa1+1−αb1,ϕa1+b12≤h12ϕ1−αa1+αb1+1−h12ϕαa1+1−αb1.
Multiplying the above inequalities at both sides, we have(29)ψa1+b12ϕa1+b12≤h122ψ1−αa1+αb1ϕ1−αa1+αb1+1−h122ψαa1+1−αb1ϕαa1+1−αb1+h12.1−h12ψ1−αa1+αb1ϕαa1+1−αb1+ψαa1+1−αb1ϕ1−αa1+αb1≤h122ψ1−αa1+αb1ϕ1−αa1+αb1+1−h122ψαa1+1−αb1ϕαa1+1−αb1+h12.1−h121−hαψa1+hαψb1hαϕa1+1−hαϕb1+hαψa1+1−hαψb11−hαϕa1+hαϕb1≤h122ψ1−αa1+αb1ϕ1−αa1+αb1+1−h122ψαa1+1−αb1ϕαa1+1−αb1+h12.1−h12hα1−hαψa1ϕa1+ψb1ψb1+hα2ψb1ϕa1+1−hα2ψa1ϕb1+hα1−hαψa1ϕa1+ψb1ψb1+hα2ψa1ϕb1+1−hα2ψb1ϕa1≤h122ψ1−αa1+αb1ϕ1−αa1+αb1+1−h122ψαa1+1−αb1ϕαa1+1−αb1+h12.1−h12hα1−hα2ψa1ϕa1+2ψb1ψb1+1−hα2+hα2ψa1ϕb1+ψb1ϕa1.
Integrating (29) with respect to α over 0,1 and using change of variable technique, we obtain(30)2ψa1+b12ϕa1+b12≤h122+1−h1222b1−a1∫a1b1ψxϕxdx+2h12.1−h122Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα.
So,(31)2h1/22+1−h1/22ψa1+b12ϕa1+b12≤2b1−a1∫a1b1ψxϕxdx+2h1/2.1−h1/2h1/22+1−h1/222Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα.
Multiplying both sides of (31) by σb1−a1/2Bσ and adding 21−σ/Bσψkϕk, we get(32)σb1−a1Bσh1/22+1−h1/22ψa1+b12ϕa1+b12+21−σBσψkϕk≤σBσ∫a1kψxϕxdx+∫kb1ψxϕxdx+σb1−a1h1/2.1−h1/2Bσh1/22+1−h1/222Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα+21−σBσψkϕk≤1−σBσψkϕk+σBσ∫a1kψxϕxdx+1−σBσψkϕk+σBσ∫kb1ψxϕxdx+σb1−a1h1/2.1−h1/2Bσh1/22+1−h1/222Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα.
Thus,(33)σb1−a1Bσh1/22+1−h1/22ψa1+b12ϕa1+b12+21−σBσψkϕk≤a1CFIσψϕk+CFIb1σψϕk+σb1−a1h1/2.1−h1/2Bσh1/22+1−h1/222Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα.
This implies that(34)σb1−a1Bσh1/22+1−h1/22ψa1+b12ϕa1+b12−a1CFIσψϕk+CFIb1σψϕk+21−σBσψkϕk≤σb1−a1h1/2.1−h1/2Bσh1/22+1−h1/222Ma1,b1∫01hα1−hαdα+Na1,b1∫01hα2+1−hα2dα.
Multiplying both sides of the above inequality byBσh1/22+1−h1/22/σb1−a1h1/2.1−h1/2, we obtain our required result.Remark 3.
If we takehα=α in Theorem 6, we obtain ([32], Theorem 4).


## 3. Some New Results Related with Caputo–Fabrizio Fractional Operator

In this section, we establish some new inequalities for modifiedh-convex functions via Caputo–Fabrizio fractional operator.Theorem 7.
Letψ:J⟶ℝ be a differentiable function on J∘. If ψ′ is a modified h-convex function on interval a1,b1, where a1,b1∈J with a1<b1, ψ′∈L1a1,b1 and σ∈0,1, then the following inequality holds:(35)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12ψ′b12+ψ′a1−ψ′b1∫011−2αhαdα,where k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Using Lemma2 and the definition of modified h-convexity of ψ′, we get(36)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12∫011−2αψ′αa1−1−αb1dα≤b1−a12∫011−2αhαψ′a1+1−hαψ′b1dα≤b1−a12ψ′a1∫011−2αhαdα+ψ′b1∫011−2α1−hαdα≤b1−a12ψ′b1∫011−2αdα+ψ′a1−ψ′b1∫011−2αhαdα≤b1−a12ψ′b12+ψ′a1−ψ′b1∫011−2αhαdα.which completes the proof.Remark 4.
If we takehα=α in Theorem 7, we obtain ([32], Theorem 5).Theorem 8.
Letψ:J⟶ℝ be a positive differentiable function on J∘. If ψ′q is a modified h-convex function on interval a1,b1, a1,b1∈J∘ with a1<b1,q>1,1/p+1/q=1, where a1,b1∈J with a1<b1, ψ′∈L1a1,b1 and σ∈0,1, then the following inequality holds:(37)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a121p+11/q×ψ′b1q+ψ′a1q−ψ′b1q∫01hαdα1/q,where k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Using Lemma2, Hölder’s integral inequality and modified h-convexity of ψ′q, we get(38)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12∫011−2αψ′αa1−1−αb1dα≤b1−a12×∫011−2αpdα1/p∫01ψ′αa1−1−αb1qdα1/q≤b1−a121p+11/p×∫01ψ′αa1−1−αb1qdα1/q≤b1−a121p+11/p×∫01hαψ′a1q+1−hαψ′b1qdα1/q≤b1−a121p+11/p×ψ′b1q+ψ′a1q−ψ′b1q∫01hαdα1/q.which completes the proof.Remark 5.
If we takehα=α in Theorem 8, we obtain ([32], Theorem 6).Theorem 9.
Letψ:J⟶ℝ be a positive differentiable function on J∘. If ψ′q is a modified h-convex function on interval a1,b1, a1,b1∈J∘ with a1<b1,q≥1, where a1,b1∈J with a1<b1, ψ′∈L1a1,b1 and σ∈0,1, then the following inequality holds:(39)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12121−1/q×ψ′b1q2+ψ′a1q−ψ′b1q∫011−2αhαdα1/q,where k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Assumingq>1, using Lemma 2 and the power mean inequality and modified h-convexity of ψ′q, we get(40)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12∫011−2αψ′αa1−1−αb1dα≤b1−a12×∫011−2αdα1−1/q∫011−2αψ′αa1−1−αb1qdα1/q≤b1−a12121−1/q×∫011−2αψ′αa1−1−αb1qdα1/q≤b1−a12121−1/q×∫011−2αhαψ′a1q+1−hαψ′b1qdα1/q≤b1−a12121−1/q×ψ′a1q∫011−2αhαdα+ψ′b1q∫011−2α1−hαdα1/q≤b1−a12121−1/q×ψ′b1q2+ψ′a1q−ψ′b1q∫011−2αhαdα1/q.
Forq=1, we use the estimates of Theorem 7, which also follows step by step the above estimates. This completes the proof of theorem.Remark 6.
(1)
Under the assumptions of Theorem9 with q=1, we get conclusion of Theorem 7(2)
If we takehα=α and q=1 in Theorem 9, we obtain ([32], Theorem 5)
Now, we will prove Theorems8 and 9 by Hölder–Iscan and improved power mean integral inequality, respectively. Then, we will show that the results we have obtained in these theorems gives better approximation of Theorems 8 and 9, respectively.Theorem 10.
Letψ:J⟶ℝ be a positive differentiable function on J∘. If ψ′q is a modified h-convex function on interval a1,b1, a1,b1∈J∘ with a1<b1,q>1,1/p+1/q=1, where a1,b1∈J with a1<b1, ψ′∈L1a1,b1 and σ∈0,1, then the following inequality holds:(41)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫011−αhαdα1/q+b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫01αhαdα1/q,where k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Using Lemma2, Hölder–Iscan integral inequality and modified h-convexity of ψ′q, we get(42)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12∫011−2αψ′αa1−1−αb1dα≤b1−a12×∫011−α1−2αpdα1/p∫011−αψ′αa1−1−αb1qdα1/q+b1−a12×∫01α1−2αpdα1/p∫01αψ′αa1−1−αb1qdα1/q≤b1−a12×12p+11/p∫011−αψ′αa1−1−αb1qdα1/q+b1−a12×12p+11/p∫01αψ′αa1−1−αb1qdα1/q≤b1−a12×12p+11/p∫011−αhαψ′a1q+1−hαψ′b1qdα1/q+b1−a12×12p+11/p∫01αhαψ′a1q+1−hαψ′b1qdα1/q≤b1−a1212p+11/p×ψ′a1q∫011−αhαdα+ψ′b1q∫011−α1−hαdα1/q+b1−a1212p+11/p×ψ′a1q∫01αhαdα+ψ′b1q∫01α1−hαdα1/q≤b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫011−αhαdα1/q+b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫01αhαdα1/q.which completes the proof.Corollary 1.
If we takehα=α in inequality (41), we get the following inequality:(43)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a141p+11/p×ψ′a1q+2ψb1q31/q+2ψ′a1q+ψb1q31/q.Remark 7.
The inequality (41) gives better results than inequality[35], and we have the following inequality:(44)b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫011−αhαdα1/q+b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫01αhαdα1/q≤b1−a121p+11/q×ψ′b1q+ψ′a1q−ψ′b1q∫01hαdα1/q.Proof.
Using concavity ofμ:0,∞⟶ℝ,μx=xλ,0<λ≤1, we get(45)b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫011−αhαdα1/q+b1−a1212p+11/p×ψ′b1q2+ψ′a1q−ψ′b1q∫01αhαdα1/q≤b1−a1212p+11/p×2ψ′b1q2+ψ′a1q−ψ′b1q2∫01hαdα1/q=b1−a121p+11/q×ψ′a1q+ψ′a1q−ψ′b1q∫01hαdα1/q.which completes the proof.Theorem 11.
Letψ:J⟶ℝ be a positive differentiable function on J∘. If ψ′q is a modified h-convex function on interval a1,b1, a1,b1∈J∘ with a1<b1,q≥1, where a1,b1∈J with a1<b1. If ψ′∈L1a1,b1 and σ∈0,1, then the following inequality holds:(46)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫011−α1−2αhαdα1/q+b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫01α1−2αhαdα1/q,where k∈a1,b1, and Bσ>0 is a normalization function.Proof.
Assumingq>1 and using Lemma 2, improved power-mean integral inequality and modified h-convexity of ψ′q, we get(47)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a12∫011−2αψ′αa1−1−αb1dα≤b1−a12×∫011−α1−2αdα1−1/q∫011−α1−2αψ′αa1−1−αb1qdα1/q+b1−a12×∫01α1−2αdα1−1/q∫01α1−2αψ′αa1−1−αb1qdα1/q≤b1−a12×141−1/q∫011−α1−2αψ′αa1−1−αb1qdα1/q+b1−a12×141−1/q∫01α1−2αψ′αa1−1−αb1qdα1/q≤b1−a12141−1/q×∫011−α1−2αhαψ′a1q+1−hαψ′b1qdα1/q+b1−a12141−1/q×∫01α1−2αhαψ′a1q+1−hαψ′b1qdα1/q≤b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫011−α1−2αhαdα1/q+b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫01α1−2αhαdα1/q.
Forq=1, we use the estimates of Theorem 7 which also follows step by step the above estimates. This completes the proof of theorem.Corollary 2.
If we takehα=α in inequality (41), we get the following inequality:(48)ψa1+ψb12+21−σσb1−a1ψk−Bσσb1−a1a1CFIσψk+CFIb1σψk≤b1−a18ψ′a1q+3ψb1q41/q+3ψ′a1q+ψb1q41/q.Remark 8.
Inequality (46) gives better results than inequality (39), and we have the following inequality:(49)b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫011−α1−2αhαdα1/q+b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫01α1−2αhαdα1/q≤b1−a12121−1/q×ψ′b1q2+ψ′a1q−ψ′b1q∫011−2αhαdα1/q.Proof.
Using concavity ofψ:0,∞⟶ℝ,ψx=xλ,0<λ≤1, we get(50)b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫011−α1−2αhαdα1/q+b1−a12141−1/q×ψ′b1q4+ψ′a1q−ψ′b1q∫01α1−2αhαdα1/q≤b1−a12141−1/q×2ψ′b1q4+ψ′a1q−ψ′b1q2∫011−2αhαdα1/q=b1−a12121−1/q×ψ′b1q2+ψ′a1q−ψ′b1q∫011−2αhαdα1/q.which completes the proof.


## 4. Application to Means

For two positive numbersa1>0 and b1>0, define(51)Aa1,b1=a1+b12,Lp=Lpa1,b1=b1p+1−a1p+1p+1b1−a11/p,a1≠b1,p∈ℝ,−1,0,a1,a1=b2.These means are, respectively, called the arithmetic andp-logarithmic means of two positive numbers a1 and b1.Proposition 1.
Leta1,b1∈0,∞ with a1<b1, then the following inequality holds:(52)Ana1,b1−Lnna1,b1≤nb1−a12b1n−12+a1n−1−b1n−1∫011−2αhαdα.Proof.
In Theorem7, if we set ψx=xn, where n is an even number with σ=1 and Bσ=B1=1, we obtain the required result.Proposition 2.
Leta1,b1∈0,∞ with a1<b1, then the following inequality holds:(53)A2a1,b1−L22a1,b1≤b1−a12b1+2a1−b1∫011−2αhαdα.Proof.
In Theorem7, if we set ψx=x2, with σ=1 and Bσ=B1=1, we obtain the required result.


## 5. Conclusion

Hermite–Hadamard-type inequalities for modifiedh-convex functions via Caputo–Fabrizio integral operator are derived. Some new and interesting integral inequalities involving Caputo–Fabrizio fractional integral operator are also obtained for modified h-convex functions. Many existing results in literature become the particular cases for these results as mentioned in remarks.



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