Fractional inequalities of the Hermite–Hadamard type for m-polynomial convex and harmonically convex functions

Eze R. Nwaeze1, Muhammad Adil Khan2, Ali Ahmadian3,∗, Mohammad Nazir Ahmad3 and Ahmad Kamil Mahmood4 1 Department of Mathematics and Computer Science, Alabama State University, Montgomery, AL 36101, USA 2 Department of Mathematics, University of Peshawar, Peshawar, Pakistan 3 Institute of IR 4.0, The National University of Malaysia, 43600 Bangi, Selangor, Malaysia 4 High Performance Computing Centre, CISD, Universiti Teknologi Petronas, Seri Iskandar, Perak, Malaysia


Introduction
The sets T and S ⊆ R \ {0} are called convex and harmonically convex, respectively if            ςq + (1 − ς)z ∈ T for all q, z ∈ T and ς ∈ [0, 1]; qz ςq+(1−ς)z ∈ S for all q, z ∈ S and ς ∈ [0, 1]. Whenever used, we shall always consider T as a convex set and S as a harmonically convex set. Let m ∈ N. Recall that a function ϕ : T → R is said to be m-polynomial convex [31] on T if for all q, z ∈ S and ς ∈ [0, 1]. For this class of functions, Toplu et al. established the following double inequality of the Hermite-Hadamard type.
Theorem 1 ( [31] ). Let ϕ : T → R be an m-polynomial convex function. If ξ, δ ∈ T with ξ < δ, and ϕ is Lebesgue integrable on [ξ, δ], then the following Hermite-Hadamard type inequality holds: (1.1) The inequality (1.1) boils down to the classical Hermite-Hadamard inequality for convex functions if we take m = 1. Recently, Awan et al. [2] introduced the notion of m-polynomial harmonically convex functions as follows: a real valued function ϕ : S → R + : for all q, z ∈ S and ς ∈ [0, 1]. In the same paper, the authors established the following Hermite-Hadamard type inequality for this class of functions: Theorem 2 ( [2] ). Let ϕ : S → R + be an m-polynomial harmonically convex function. If ξ, δ ∈ S with 0 < ξ < δ, and ϕ is Lebesgue integrable on [ξ, δ], then the following Hermite-Hadamard type inequality holds: In the sequel, we will denote the sets of all m-polynomial convex and m-polynomial harmonically convex functions from A into B by XP m (A, B) and HXP m (A, B), respectively. The classical Hermite-Hadamard inequality has generated load of generalizations and extensions to other class of convexity. There are dozens of articles in this direction. We invite the interested reader to see the following articles [3-6, 8, 10-20, 22-30, 32-34] and the references cited therein. Now, recall that the left-and right-sided ζ-Riemann-Liouville fractional integral operators ζ J ξ + and ζ J δ − of order > 0, for a real valued continuous function ϕ(r), are defined as ( [21]): where ζ > 0, and Γ ζ is the ζ-gamma function given by with the properties Γ ζ (r + ζ) = rΓ ζ (r) and Γ ζ (ζ) = 1. If ζ = 1, we simply write The beta function B is defined by Another fractional integral operators of interest is the Caputo-Fabrizio operators [1]: let L 2 (ξ, δ) be the space of square integrable functions on the interval (ξ, δ) and H 1 (ξ, δ) := g | g ∈ L 2 (ξ, δ) and g ∈ L 2 (ξ, δ) .
Since the classes of convexity introduced here are new, much work have not been done in this sense. This work is geared towards further development around inequalities for these classes. In view of this, we aim to achieve the following objectives: 1. To establish new Hermite-Hadamard type inequalities for the class of m-polynomial convex functions involving the Caputo-Fabrizio integral operators. Our first result in this direction generalizes and extends Theorem 3. 2. To obtain inequalities of the Hermite-Hadamard type for functions that are m-polynomial harmonically convex functions via the ζ-Riemann-Liouville fractional integral operators. This, in turn, also complement and generalize some existing results in the literature.

Inequalities for m-polynomial convex functions
Inequalities of the Hermite-Hadamard type, for m-polynomial convex functions, are hereby presented. The results, presented herein, involve the Caputo-Fabrizio operators.
The required result follows.

Conclusion
Utilizing the Caputo-Fabrizio and generalized Riemann-Liouville fractional integral operators, we proved some inequalities of the Hermite-Hadamard kinds for m-polynomial convex and harmonically convex functions. Our results generalize, extend and complement results in [7,9,31].