# On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications

Authors: Qi Li; Muhammad Shoaib Saleem; Peiyu Yan; Muhammad Sajid Zahoor; Muhammad Imran
Journal: Journal of Mathematics
 (2021)

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## Abstract

The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.


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## Body

## 1. Introduction

The theory of fractional calculus got rapid development, and it has brought the attention of many researchers from various disciplines [1–3]. In the last few years, it was observed that fractional calculus is very useful for modeling complicated problems of engineering, chemistry, mechanics, and many other branches. Various interesting notations of fractional calculus exist in the literature, for example, the Riemann–Liouville fractional integral and Caputo–Fabrizio fractional integral [4–14].Among these notions, Riemann–Liouville and Caputo involve the following singular kernal [11]:(1)Kζ,x=ζ−x−ςΓ1−ς,0<ς<1.However, it was observed by Caputo and Fabrizio in [8] that certain phenomena cannot be modelled by the already existing definition in the literature. That is why, they proposed a more general fractional derivative in [8] and named it as the Caputo–Fabrizio fractional integral operator. It mainly involves the following nonsingular kernal:(2)Kζ,x=e−ςζ−x/1−ς,0<ς<1.Nowadays, many researchers of applied sciences are using the Caputo–Fabrizio fractional integral operator to model their problem. For more details about the fractional integral with a nonsingular kernal, we refer [15–19] to the readers.The theory of inequalities also plays an important role in applied as well as in pure mathematics. The Hermite–Hadamard inequality is the most important inequality in the literature, and this inequality has been studied for different classes of convex functions, see [20–24]. The classical version of the Hermite–Hadamard inequality for convex functions is stated as follows:Ifϱ:I=a,b⊂ℝ⟶ℝ is an integrable and continuous convex function, then its mean value remains between the value of ϱ at a+b/2 of interval I=a,b and arithmetic mean value of ϱ at the endpoints a,b∈I=a,b. In other words, it means that(3)ϱa+b2≤1b−a∫abϱxx≤ϱa+ϱb2.Inequality (3), in the literature, is generalized by several fractional integral operators to meet the desired results, see, for instance, [25–28]. In this paper, we present the Hermite–Hadamard inequality for a strongly convex function in the setting of the Caputo–Fabrizio fractional integral operator. We also present some new inequalities for strongly convex functions in the setting of the Caputo–Fabrizio fractional integral operator. We also give some applications of the presented inequalities in special mean.


## 2. Preliminaries

In this section, we present some definitions from the literature.Definition 1.
A function is convex if(4)ϱζx+1−ζy≤ζϱx+1−ζϱy,for every x,y∈I and ζ∈0,1.Definition 2.
(see [29]). Assume λ≥0. A function ϱ:I⟶ℝ is strongly convex if(5)ϱζx+1−ζy≤ζϱx+1−ζϱy−λζ1−ζx−y2,for every x,y∈I and ζ∈0,1.Remark 1.
Settingλ=0 in inequality (5), we obtain convex function (4).Definition 3.
(see [8]). Let ϱ∈H1a,b,a<b,ς∈0,1; then, the left Caputo–Fabrizio fractional derivative is defined by(6)DςaCFϱζ=Bς1−ς∈ζaζϱ′xe−ςζ−xς/1−ςdx,and the left Caputo–Fabrizio fractional integral is defined by(7)IςaCFϱζ=1−ςBςϱζ+ςBς∈ζaζϱxdx,where Bς>0 is a normalization of function with B0=B1=1.Definition 4.
(see [8]). Let ϱ∈H1a,b,a<b,ς∈0,1; then, the right Caputo–Fabrizio fractional derivative is defined by(8)DbςCFϱζ=−Bς1−ς∈ζζbϱ′xe−ςx−ζς/1−ςdx,and the right Caputo–Fabrizio fractional integral is defined by(9)IbςCFϱζ=1−ςBςϱζ+ςBς∈ζζbϱxdx,where Bς>0 is a normalization of function with B0=B1=1.


## 3. Hermite–Hadamard-Type Inequalities via Caputo–Fabrizio Fractional Integrals for Strongly Convex Functions

Theorem 1.
Assumeϱ:I⟶ℝ to be a strongly convex function with modulus λ≥0 and ϱ∈L1a,b; then, the inequality(10)ϱa+b2+λ12b−a2≤Bςςb−aIςaCFϱζ+IbςCFϱζ−21−ςBςϱζ≤ϱa+ϱb2−λ6b−a2,holds, where Bς>0 is a normalization function, ς∈0,1, and ζ∈0,1.Proof.
Sinceϱ is strongly convex function, we have(11)ϱa+b2−λ12b−a2≤1b−a∫abϱxdx≤ϱa+ϱb2−λ6b−a2.
The left side of inequality (11) yields(12)2ϱa+b2−λ6b−a2≤2b−a∫abϱxdx,=2b−a∫aζϱxgxdx+∫ζbϱxgxdx.
Multiplyingςb−a/2Bς on both sides of the abovementioned inequality, adding 21−ς/Bςϱζgζ and rearranging the terms, we obtain(13)ϱa+b2+λ12b−a2≤Bςςb−aIςaCFϱζ+IbςCFϱζ−21−ςBςϱζ,which is the left side of Theorem 1.
Now, to prove the right side of Theorem1, we use the right side of (11), which is(14)2b−a∫abϱxdx≤ϱa+ϱb−λ3b−a2.
Applying the same operations on the abovementioned inequality as on (12) yields the right side of Theorem 1, which is(15)≤Bςςb−aIςaCFϱζ+IbςCFϱζ−21−ςBςϱζ≤ϱa+ϱb2−λ6b−a2.
The combination of (13) and (15) completes the proof.Theorem 2.
Assume thatϱ,g:I⟶ℝ are two strongly convex functions with modulus λ≥0 and f,g∈L1a,b; then, the inequality(16)2Bςςb−aIςaCFϱgζ+IbςCFϱgζ−21−ςBςϱζgζ≤23Pa,b+13Qa,b−λ3b−a2Ra,b−λ5b−a2,holds with normalization function Bς>0, ς∈0,1, and ζ∈0,1, where Pa,b=ϱaga+ϱbgb,  Qa,b=ϱagb+ϱbga, and Ra,b=ϱa+ga+ϱb+gb.Proof.
Sinceϱ and g are strongly convex functions defined on I, by definition, we have(17)ϱζa+1−ζb≤ζϱa+1−ζϱb−λζ1−ζb−a2,(18)gζa+1−ζb≤ζga+1−ζgb−λζ1−ζb−a2,for all a,b∈I and ζ∈0,1.
Multiplying (17) and (18), we have(19)ϱζa+1−ζbgζa+1−ζb≤ζ2ϱaga+1−ζ2ϱbgb+ζ1−ζϱagb+ϱbga−λζ1−ζ2b−a2ϱb+gb−λζ21−ζgab−a2−λζ21−ζb−a2ϱa+ga+λ2ζ21−ζ2b−a4.
Integrating the abovementioned inequality w.r.t “ζ” over 0,1, we obtain(20)2b−a∫abϱxgxdx≤23ϱaga+ϱbgb+13ϱagb+ϱbga−λ3b−a2ϱa+ϱb+ga+gb−λ5b−a2,2b−a∫abϱxgxdx≤23Pa,b+13Qa,b−λ3b−a2Ra,b−λ5b−a2.
Multiplyingςb−a/2Bς on both sides and adding 21−ς/Bςϱζgζ, we obtain(21)ςBς∫aζϱxgxdx+∫ζbϱxgxdx+21−ςBςϱζgζ≤ςb−a2Bς23Pa,b+13Qa,b−λ3b−a2Ra,b−λ5b−a2+21−ςBςϱζgζ.
Now, the use of (7) and (9) and rearrangements of the terms of abovementioned inequality complete the proof.Theorem 3.
Assume thatf,g:I⟶ℝ are two strongly convex functions with modulus λ≥0 and f,g∈L1a,b; then, the inequality(22)2ςBςϱa+b2ga+b2−1b−aIςaCFϱgζ+IbςCFϱgζ+21−ςBςb−aϱζgζ≤ς2Bς23Qa,b+13Pa,b−λ3b−a2Ra,b−λ5b−a2−b−a22−15holds with normalization function Bς>0, ς∈0,1, and ζ∈0,1, where Pa,b=ϱaga+ϱbgb,  Qa,b=ϱagb+ϱbga, and Ra,b=ϱa+ga+ϱb+gb.Proof.
Sinceϱ and g be the two strongly convex functions, so for ζ=1/2, we have(23)ϱa+b2≤ϱζa+1−ζb+ϱζa+1−ζa2−λ42ζ−1b−a2,(24)ga+b2≤gζa+1−ζb+gζa+1−ζa2−λ42ζ−1b−a2,for all a,b∈I and ζ∈0,1.
Multiplying (23) and (24), we obtain(25)ϱa+b2ga+b2≤14ϱζa+1−ζbgζa+1−ζb+ϱζa+1−ζagζa+1−ζa+ζ2+1−ζ2ϱagb+bga+2ζ1−ζϱaga+ϱbgb−λb−a2ζ21−ζ+ζ1−ζ2ϱa+gb+ϱb+ga+2λ2ζ21−ζ2b−a2+2λ2ζ1−ζ2ζ−12b−a4+λ22ζ−12b−a4−λ22ζ−12b−a2Ra,b.
Integrating the abovementioned inequality w.r.t “ζ” over 0,1 and using the technique of change of variable, we obtain(26)4ϱa+b2ga+b2≤2b−a∫abϱxgxdx+13Pa,b+23Qa,b−λ3b−a2Ra,b−λ5−b−a22−15.
Multiplyingςb−a/2Bς on both sides and subtracting 21−ς/Bςϱζgζ, we obtain(27)2ςb−a2Bςϱa+b2ga+b2≤ςBς∫aζϱxgxdx+∫ζbϱxgxdx−21−ςBςϱζgζ+ςb−a2Bς13Pa,b+23Qa,b−λ3b−a2Ra,b−λ5−b−a22−15−21−ςBςϱζgζ.
Now, the use of (7) and (9) and rearrangements of the terms of the abovementioned inequality complete the proof.


## 4. Some New Caputo–Fabrizio Fractional Integral Inequalities for Strongly Convex Functions

Lemma 1.
(see [28, 30]). Assume that ϱ:I⟶ℝ is a differentiable mapping on I°, where a,b∈I with a<b. If ϱ′∈L1a,b, then the inequality(28)b−a2∈ζ101−2ζϱ′ζa+1−ζbdζ−21−ςςb−aϱζ=ϱa+ϱb2−Bςςb−aIςaCFϱgζ+IbςCFϱgζholds, where Bς>0 is a normalization function, ς∈0,1, and ζ∈0,1.Theorem 4.
Assume thatϱ:I⟶ℝ is a differentiable positive mapping on I°, where a,b∈I with a<b. If ϱ′∈L1a,b and ϱ′ are two strongly convex functions, then the inequality(29)ϱa+ϱb2+21−ςςb−aϱζ−Bςςb−aIςaCFϱgζ+IbςCFϱgζ≤b−aϱ′a+ϱ′b8−λ32b−a3holds, where Bς>0 is a normalization function, ς∈0,1, and ζ∈0,1.Proof.
By using Lemma1, convexity of ϱ′, and the property of absolute value, we get(30)ϱa+ϱb2+21−ςςb−aϱζ−Bςςb−aIςaCFϱgζ+IbςCFϱgζ,=b−a2∈ζ011−2ζϱ′ζa+1−ζbdζ≤b−a2∈ζ011−2ζϱ′ζa+1−ζbdζ≤b−a2∈ζ011−2ζtϱ′a+1−ζϱ′b−λζ1−ζb−a2dζ,=b−aϱ′a+ϱ′b8−λ32b−a3.
This completes the proof.Theorem 5.
Assumeϱ:I⟶ℝ to be a differentiable positive mapping on I°, a,b∈I with a<b, and 1/p+1/q=1. If ϱ′∈L1a,b and ϱ′q is a strongly convex function, then the inequality(31)ϱa+ϱb2+21−ςςb−aϱζ−Bςςb−aIςaCFϱgζ+IbςCFϱgζ≤b−a2p+11/pϱ′ap/p−1+ϱ′bp/p−12−λ6b−a2p−1/pholds, where Bς>0 is a normalization function, ς∈0,1, and ζ∈0,1.Proof.
We start the proof by using Lemma1, convexity of ϱ′q, the property of absolute value, where 1/p+1/q=1, and Holder’s inequality to obtain(32)ϱa+ϱb2+21−ςςb−aϱζ−Bςςb−aIςaCFϱgζ+IbςCFϱgζ,=b−a2∈ζ011−2ζϱ′ζa+1−ζbdζ≤b−a2∈ζ011−2ζϱ′ζa+1−ζbdζ≤b−a2∈ζ011−2ζpdζ1/p∫01ϱ′ζa+1−ζbqdζ1/q≤b−a2∈ζ011−2ζpdζ1/p∫01tϱ′a+1−ζϱ′b−λζ1−ζb−a2dζ1/q,=b−a2p+11/pϱ′aq+ϱ′bq2−λ6b−a31/q,=b−a2p+11/pϱ′ap/p−1+ϱ′bp/p−12−λ6b−a3p−1/p,where ∫011−2ζpdζ=∫01/21−2ζpdζ+∫1/211−2ζpdζ=2∫01/21−2ζpdζ=1/p+1. This completes the proof.


## 5. Some Applications of Caputo–Fabrizio Fractional Integral Inequalities to Special Means

Means are important in applied and pure mathematics; especially, they are used frequently in numerical approximation. In the literature, they are ordered in the following way:(33)H≤G≤L≤I≤A.The special means of two numbersa and b in the order of b>a are known as arithmetic mean, geometric mean, harmonic mean, power mean, logarithmic mean, p-logarithmic mean, and identric mean. They are listed below from (34)–(40), respectively.(34)Aa,b=a+b2,(35)Ga,b=ab,(36)Ha,b=2aba+b,(37)Mpa,b=ap+bp21/p,p≠0,(38)La,b=b−alnb−lna,(39)Lpa,b=bp+1−ap+1p+1b−a1/p,p∈ℝ∖−1,0,(40)Ia,b=1ebpap1/b−a.There are several results connecting these means, see [31] for some new relations; however, very few results are known for arbitrary real numbers. For this, it is clear that we can extend some of the abovementioned means as follows:(41)Aa,b=a+b2,a,b∈ℝ,L¯a,b=b−alnb−lna,a,b∈ℝ∖0,Lna,b=bn+1−an+1n+1b−a1/n.n∈ℕ,n≥1,a,b∈ℝ,a<b.Now, we shall use the results of Sections3 and 4 to prove the following new inequalities connecting the abovementioned means for arbitrary real numbers.Proposition 1.
Supposea,b∈ℝ+,a<b and n∈ℕ,n≥2. Then, the following inequality holds:(42)Aan,bn−Lnna,b≤b−a8nAan−1,bn−1−λ4b−a2.Proof.
Insertion ofϱx=xn, where n∈ℕ,n≥2, with ς=1 and Bς=B1=1 in Theorem 4 completes the proof.Proposition 2.
Supposea,b∈ℝ+,a<b and n∈ℕ,n≥2. Then, the following inequality holds:(43)Aan,bn−Lnna,b≤nb−a2p+11/pnAan−1p/p−1,bn−1p/p−1−λb−a26np/p−1p−1/p.Proof.
Insertion ofϱx=xn, where n∈ℕ,n≥2, with ς=1 and Bς=B1=1 in Theorem 5 completes the proof.Proposition 3.
Supposea,b∈ℝ+,a<b and n∈ℕ,n≥2. Then, the following inequality holds:(44)A−1a,b+λ12b−a2≤L¯−1a,b≤Aa−1,b−1+λ6b−a2.Proof.
Insertion ofϱx=xn, where n∈ℕ,n≥2, with ς=1 and Bς=B1=1 in Theorem 1 completes the proof.Proposition 4.
Supposea,b∈ℝ+,a<b and n∈ℕ,n≥2. Then, the following inequality holds:(45)Aa−1,b−1−L¯−1a,b≤b−a4Aa−2,b−2−λ32b−a3.Proof.
Insertion ofϱx=x−1, where x∈a,b, with ς=1 and Bς=B1=1 in Theorem 4 completes the proof.Proposition 5.
Supposea,b∈ℝ+,a<b and n∈ℕ,n≥2. Then, the following inequality holds:(46)Aa−1,b−1−L¯−1a,b≤b−a2p+11/pAa−2p/p−1,|b|−2p/p−1−λ6b−a2p−1/p.Proof.
Insertion ofϱx=x−1, where x∈a,b, with ς=1 and Bς=B1=1 in Theorem 4 completes the proof.


## 6. Some Applications of Caputo–Fabrizio Fractional Integral Inequalities to the Trapezoidal Formula

Supposed is the division of interval a,b, d:a=x0<x1<⋯<xn−1<xn=b, and consider the trapezoidal formula(47)Tϱ,d=∑i=0i=1ϱxi+ϱxi+12xi+1−xi.It is well known that if the mappingϱ:I⟶ℝ is twice differentiable on a,b and M=maxx∈a,bϱ″x<∞, then(48)∫abϱxdx=Tϱ,d+Eϱ,d,where the approximation error Eϱ,d of the integral ∫abϱxdx by the trapezoidal formula Tϱ,d satisfies(49)Eϱ,d≤M12∑i=0n−1xi+1−xi3.It is clear that if the mappingf is not twice differentiable or the second derivative is not bounded on a,b, then (49) cannot be applied. In recent studies [30, 32–35], Dragomir and Wang showed that the remainder term Eϱ,d can be estimated in terms of the first derivative only. These estimates have a wider range of applications. Here, we shall propose some new estimates of the remainder term Eϱ,d which supplement, in a sense, those established in [30, 32–35].Proposition 6.
Assume thatϱ:I⟶ℝ is a differentiable positive mapping on I°, a,b∈I with a<b. If ϱ′∈L1a,b and ϱ′ is a strongly convex function, then for every division d of a,b, the following inequality holds:(50)Eϱ,d≤18∑n−1i=1xi+1−xi2ϱ′xi+ϱ′xi+1−λ4∑n−1i=1xi+1−xi2≤14∑n−1i=1xi+1−xi2max|ϱ′a|,ϱ′b−λ8∑n−1i=1xi+1−xi2.Proof.
Applying subintervalxi,xi+1,i=0,…,n−1, of the division d from Theorem 4, we obtain(51)ϱxi+ϱxi+12xi+1−xi−∫xixi+1ϱxdx≤xi+1−xi24ϱ′xi+ϱ′xi+12−λ16xi+1−xi2.
Summing overi=0,…,n−1 and taking that ϱ′ is a strongly convex function, then by using (47), (48), and triangular inequality, we complete the proof.Proposition 7.
Assume thatϱ:I⟶ℝ is a differentiable positive mapping on I°, a,b∈I with a<b and 1/p+1/q=1. If ϱ′∈L1a,b and ϱ′q is a strongly convex function, then for every division d of a,b, the following inequality holds:(52)Eϱ,d≤12p+11/p∑n−1i=1xi+1−xi2ϱ′xip/p−1+ϱ′xi+1p/p−12−λ6∑n−1i=1xi+1−xi2p/p−1≤12p+11/p∑n−1i=1xi+1−xi2max|ϱ′a|,ϱ′b−λ6∑n−1i=1xi+1−xi2.Proof.
Applying subintervalxi,xi+1,i=0,…,n−1, of the division d, we obtain from Theorem 5(53)ϱxi+ϱxi+12xi+1−xi−∫xixi+1ϱxdx≤xi+1−xi22p+11/pϱ′xip/p−1+ϱ′xi+1p/p−12−λ6xi+1−xi2p−1/p.
Summing overi=0,…,n−1 and taking that ϱ′q, where 1/p+1/q=1, is a strongly convex function, then by using (47), (48), and triangular inequality, we complete the proof.


## 7. Conclusions

The convex functions play an important role in approximation theory, and the fractional calculus has been found the best to model physical and engineering processes. Some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator have been studied in this paper. Precisely speaking, Hermite–Hadamard-type and some new inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator are proved, and applications of the proposed inequalities to special means are also presented in this paper.



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