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

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 for ðp, hÞ -convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for ðp, hÞ -convex functions. Finally, the applications of our main findings are also given.


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][2][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][5][6][7][8][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][12][13][14][15][16]). The well-known inequality given by Hermite in 1881 can be stated as follows. Theorem 1. Let ζ : L ⟶ ℝ be a ðp, hÞ-convex function defined on the interval L of real numbers and c, d ∈ L with c < d. Then, the following inequality holds: 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][23][24][25][26][27].
In the present article, we generalize the Hermite-Hadamard-type inequalities for ðp, hÞ-convex functions. Moreover, we establish some new inequalities via the Caputo-Fabrizio fractional integral operator for ðp, hÞ-convex functions. Finally, the applications of our main findings are also given. This paper is organized as follows: in Section 2, some preliminaries are given. In Section 3, we generalized Hermite-Hadamard via Caputo-Fabrizio for ðp, hÞ-convex functions. In Section 4, we give some results related to Caputo-Fabrizio, and in Section 5, some applications to special means are given.

Preliminaries
We will start with some basic definitions related to our work.
Definition 2 (convex function) [28]. Let ζ : L ⟶ ℝ be an extended real-valued function defined on a convex set L ⊆ ℝ n . Then, the function ζ is convex on L if for all c, d ∈ L and t ∈ ð0, 1Þ.

A Generalized Hermite-Hadamard-Type Inequality via the Caputo-Fabrizio Fractional Operator for a ðp, hÞ-Convex Function
The double inequality named as Hermite-Hadamard inequality is considered one of the fundamental inequalities for convex functions.
Remark 11. If we put p = 1 and hðtÞ = t, then we will get the Hermite-Hadamard inequality for convex function.
Proof. Since ζ and θ are ðp, hÞ-convex functions on ½c, d, we have Multiplying both sides of the above inequalities, we have Integrating with respect to t over ½0, 1 and making the change of variable, we obtain which implies By multiplying both sides with ðγc p − d p Þ/2PBðγÞ and adding ðk p−1 2ð1 − γÞ/BðγÞÞζðKÞgðkÞ, we have Thus, CF c I γ ηβ and with suitable rearrangements, the proof is completed.
Remark 13. If we put p = 1 and hðtÞ = t in the above theorem, we get the results for the classical convex function.
Remark 15. If we put p = 1 and hðtÞ = t in the above theorem, then we will get the result for the convex function.

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.