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

,e theory of convex functions plays an important role in the study of optimization problems. ,e fractional calculus has been found the best to model physical and engineering processes. ,e 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.


Introduction
e theory of fractional calculus got rapid development, and it has brought the attention of many researchers from various disciplines [1][2][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][5][6][7][8][9][10][11][12][13][14].
Among these notions, Riemann-Liouville and Caputo involve the following singular kernal [11]: However, it was observed by Caputo and Fabrizio in [8] that certain phenomena cannot be modelled by the already existing definition in the literature. at 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: (1−ς) , 0 < ς < 1. (2) 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][16][17][18][19] to the readers. e theory of inequalities also plays an important role in applied as well as in pure mathematics.
e 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][21][22][23][24]. e classical version of the Hermite-Hadamard inequality for convex functions is stated as follows: If ϱ: 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 Inequality (3), in the literature, is generalized by several fractional integral operators to meet the desired results, see, for instance, [25][26][27][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.

Preliminaries
In this section, we present some definitions from the literature.

Hermite-Hadamard-Type Inequalities via Caputo-Fabrizio Fractional Integrals for Strongly Convex Functions
Theorem 1. Assume ϱ: I ⟶ R to be a strongly convex function with modulus λ ≥ 0 and ϱ ∈ L 1 [a, b]; then, the inequality Proof. Since ϱ is strongly convex function, we have e left side of inequality (11) yields Multiplying ς(b − a)/2B(ς) on both sides of the abovementioned inequality, adding (2(1 − ς)/B(ς))ϱ(ζ) g(ζ) and rearranging the terms, we obtain which is the left side of eorem 1. Now, to prove the right side of eorem 1, we use the right side of (11), which is Applying the same operations on the abovementioned inequality as on (12) yields the right side of eorem 1, which is e combination of (13) and (15) completes the proof. □ Theorem 2. Assume that ϱ, g: I ⟶ R are two strongly convex functions with modulus λ ≥ 0 and f, g ∈ L 1 [a, b]; then, the inequality Proof. Since ϱ and g are strongly convex functions defined on I, by definition, we have for all a, b ∈ I and ζ ∈ [0, 1].
Multiplying (17) and (18), we have Integrating the abovementioned inequality w.r.t "ζ" over Multiplying ς(b − a)/2B(ς) on both sides and adding Now, the use of (7) and (9) and rearrangements of the terms of abovementioned inequality complete the proof. □ Theorem 3. Assume that f, g: I ⟶ R are two strongly convex functions with modulus λ ≥ 0 and f, g ∈ L 1 [a, b]; then, the inequality Proof. Since ϱ and g be the two strongly convex functions, so for ζ � 1/2, we have for all a, b ∈ I and ζ ∈ [0, 1].
Proof. We start the proof by using Lemma 1, convexity of |ϱ ′ | q , the property of absolute value, where (1/p) + (1/q) � 1, and Holder's inequality to obtain is completes the proof.

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: e special means of two numbers a 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. ey are listed below from (34)-(40), respectively. Journal of Mathematics ere 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: Now, we shall use the results of Sections 3 and 4 to prove the following new inequalities connecting the abovementioned means for arbitrary real numbers. Proposition 1. Suppose a, b ∈ R + , a < b and n ∈ N, n ≥ 2.

Some Applications of Caputo-Fabrizio Fractional Integral Inequalities to the Trapezoidal Formula
Suppose d is the division of interval [a, b], d: a � x 0 < x 1 < · · · < x n−1 < x n � b, and consider the trapezoidal formula It is well known that if the mapping ϱ: where the approximation error E(ϱ, d) of the integral b a ϱ(x)dx by the trapezoidal formula T(ϱ, d) satisfies It is clear that if the mapping f is not twice differentiable or the second derivative is not bounded on (a, b), then (49) Journal of Mathematics cannot be applied. In recent studies [30,[32][33][34][35], Dragomir and Wang showed that the remainder term E(ϱ, d) can be estimated in terms of the first derivative only. ese estimates have a wider range of applications. Here, we shall propose some new estimates of the remainder term E(ϱ, d) which supplement, in a sense, those established in [30,[32][33][34][35]. Proposition 6. Assume that ϱ: I ⟶ R is a differentiable positive mapping on I°, a, b ∈ I with a < b. If ϱ ′ ∈ L 1 [a, b] and |ϱ ′ | is a strongly convex function, then for every division d of [a, b], the following inequality holds: Proof. Applying subinterval [x i , x i+1 ], i � 0, . . . , n − 1, of the division d from eorem 4, we obtain Summing over i � 0, . . . , n − 1 and taking that |ϱ ′ | is a strongly convex function, then by using (47), (48), and triangular inequality, we complete the proof.

Conclusions
e 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. 8 Journal of Mathematics

Data Availability
All data required for this research are available within this paper.

Conflicts of Interest
e authors declare that they have no conflicts of interest.