Fractional inclusions of the Hermite–Hadamard type for m-polynomial convex interval-valued functions

The notion of m-polynomial convex interval-valued function Ψ=[ψ−,ψ+]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Psi =[\psi ^{-}, \psi ^{+}]$\end{document} is hereby proposed. We point out a relationship that exists between Ψ and its component real-valued functions ψ−\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi ^{-}$\end{document} and ψ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\psi ^{+}$\end{document}. For this class of functions, we establish loads of new set inclusions of the Hermite–Hadamard type involving the ρ-Riemann–Liouville fractional integral operators. In particular, we prove, among other things, that if a set-valued function Ψ defined on a convex set S is m-polynomial convex, ρ,ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rho,\epsilon >0$\end{document} and ζ,η∈S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\zeta,\eta \in {\mathbf{S}}$\end{document}, then mm+2−m−1Ψ(ζ+η2)⊇Γρ(ϵ+ρ)(η−ζ)ϵρ[ρJζ+ϵΨ(η)+ρJη−ϵΨ(ζ)]⊇Ψ(ζ)+Ψ(η)m∑p=1mSp(ϵ;ρ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \frac{m}{m+2^{-m}-1}\Psi \biggl(\frac{\zeta +\eta }{2} \biggr)& \supseteq \frac{\Gamma _{\rho }(\epsilon +\rho )}{(\eta -\zeta )^{\frac{\epsilon }{\rho }}} \bigl[{_{\rho }{\mathcal{J}}}_{\zeta ^{+}}^{\epsilon } \Psi (\eta )+_{ \rho }{\mathcal{J}}_{\eta ^{-}}^{\epsilon }\Psi (\zeta ) \bigr] \\ & \supseteq \frac{\Psi (\zeta )+\Psi (\eta )}{m}\sum_{p=1}^{m}S_{p}( \epsilon;\rho ), \end{aligned}$$ \end{document} where Ψ is Lebesgue integrable on [ζ,η]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[\zeta,\eta ]$\end{document}, Sp(ϵ;ρ)=2−ϵϵ+ρp−ϵρB(ϵρ,p+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S_{p}(\epsilon;\rho )=2-\frac{\epsilon }{\epsilon +\rho p}- \frac{\epsilon }{\rho }\mathcal{B} (\frac{\epsilon }{\rho }, p+1 )$\end{document} and B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{B}$\end{document} is the beta function. We extend, generalize, and complement existing results in the literature. By taking m≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m\geq 2$\end{document}, we derive loads of new and interesting inclusions. We hope that the idea and results obtained herein will be a catalyst towards further investigation.

for all w, z ∈ S and ξ ∈ [0, 1]. It is generally known that if ψ : [ζ , η] → R is convex, then Inequality (1) is today known as the Hermite-Hadamard inequality. It was named after two French mathematicians, Charles Hermite and Jacques Hadamard. The former [17] first established the result in 1883, and a decade later it was rediscovered by the latter [16].
There are loads of articles in the literature on generalizations and extensions of (1) for different kinds of convexities. Examples of such can be found in [1-5, 10, 11, 14, 15, 18-26, 33, 34, 38] and the references cited therein. Recently, Toplu et al. [39] proposed and defined an m-polynomial convex function as follows: a real-valued function ψ : S → R + : for all w, z ∈ S and ξ ∈ [0, 1]. In this paper, we shall denote the sets of all m-polynomial convex and m-polynomial concave functions from S into R + by XP m (S, R + ) and VP m (S, R + ), respectively. In the same paper, the authors established the following Hermite-Hadamard type inequality for this class of functions.
Now, recall that the left-and right-sided ρ-Riemann-Liouville fractional integral operators ρ J ζ + and ρ J η -of order > 0, for a real-valued continuous function ψ(w), are defined as follows: and where ρ > 0, and ρ is the ρ-gamma function given by with the properties ρ (w + ρ) = w ρ (w) and ρ (ρ) = 1. If ρ = 1, we simply write The beta function B is defined by Using these fractional integral operators, Sarikaya et al. [37] established the following fractional version of (1).
The theory of interval analysis [29] was initiated by the late American mathematician Ramon E. Moore in 1966. Since its advent, this field has received ample amount of attention from different researchers in the mathematical community. Experts have found applications of interval analysis in global optimization and constraint solution algorithms. It has since grown steadily in popularity over the past decades. Interval analysis has been found to be valuable to engineers and scientists interested in scientific computation, especially in reliability, effects of round-off error, and automatic verification of results, see [8,9,12,13]. With the birth of interval analysis, mathematicians, those who work in the field of mathematical inequalities, want to know if the inequalities in the above-mentioned results can be replaced with inclusions. In some cases, the answer to the question is in the affirmative. In this light, Sadowska (see also [28]) established the following result for a given interval-valued function.

Theorem 4 ([7]) Let be a convex interval-valued function defined on
This work is inspired by the above-mentioned articles. It is our purpose in this article to propose a new class of interval-valued functions called the m-polynomial convex functions and then obtain the interval-valued counterpart of (2). This result involves the ρ-Riemann-Liouville fractional integral operators and generalizes Theorem 4. In addition, we establish four more results in this direction. Our results complement and extend known results in [7] and others in the literature. The paper is arranged as follows: in Sect. 2, we present a quick overview of the theory of interval analysis. Section 3 contains our main results with detailed justifications. Interesting corollaries are also pointed out. A brief introduction follows thereafter.

Preliminaries
Interval analysis is roughly described as an analysis of interval-valued functions. It is an annex of numerical analysis where instead of real numbers intervals are used as its operating element. In this section, we collate some basic terms and essentials of the theory of interval analysis from the books [29][30][31]. In the sequel, let K c represent the class of all bounded closed nonempty intervals in R, i.e., The numbers ζand ζ + are called the left and right endpoints of [ζ -, ζ + ], respectively. The interval [ζ -, ζ + ] is called degenerated if ζ -= ζ + ; positive if ζ -> 0; and negative if ζ + < 0. We denote the sets of all negative intervals and positive intervals in R by Kc and K + c , respectively. That is, if and only if η -≤ ζand ζ + ≤ η + . The following arithmetic operations are defined thus: Interval addition is commutative, associative and 0 = [0, 0] is the identity element. Additive inverses do not exist, but the cancelation law holds. Also, interval multiplication is commutative, associative and 1 = [1, 1] is the identity element. Multiplicative inverses do not exist and the cancelation law does not hold either. The distributive rule is not valid in general. It is important to also note that the interval arithmetic is said to be inclusion isotonic (see [31, p. 34]). By this, we mean that if A, B, C, and D are intervals such that where stands for interval addition, subtraction, multiplication, or division. It follows therefore that if ζ ≤ η and C ⊆ D, It is generally known that (K c , d H ) is a complete metric space. The concept of a convergent sequence of intervals (A n ) n∈N , A n ∈ K c is considered in the complete metric space K c , endowed with the d H distance: We say that lim n→∞ A n = A if and only if for any real number > 0 there exists N ∈ N such that Next, we turn our attention to interval-valued functions.
Definition 5 An interval-valued function is defined to be any : We say that is Lebesgue integrable on [ζ , η] if the real-valued functions ψand ψ + are Lebesgue integrable on [ζ , η], and then we write For an interval function (w) = [ψ -(w), ψ + (w)], we define the ρ-Riemann-Liouville integral operators as follows:

Main results
We first introduce the notion of m-polynomial convex interval-valued function.
Definition 6 Let S be a convex set, : S → K + c be an interval-valued function, and m ∈ N. We say that is m-polynomial convex (concave) if and only if for all w, z ∈ S and ξ ∈ [0, 1]. In what follows, we shall denote the sets of all mpolynomial convex and m-polynomial concave interval-valued functions from S into K + c by XP m (S, K + c ) and VP m (S, K + c ), respectively.
Remark 7 If we take a particular value of m, then we get a corresponding set inclusion. Take, for instance: 1. If m = 1, then we get the definition of a convex interval-valued function for all w, z ∈ S and ξ ∈ [0, 1]; 2. For m = 2, we get the following inclusion for a 2-polynomial convex interval-valued function: for all w, z ∈ S and ξ ∈ [0, 1]; 3. For m = 3, we deduce the succeeding relation for a 3-polynomial convex interval-valued function: for all w, z ∈ S and ξ ∈ [0, 1].
We now present a theorem that gives a link between a given interval-valued function and its component real-valued functions ψand ψ + .
That completes the proof in both directions.
In a similar manner, one can prove the following result.

Theorem 9 Let
For the remaining part of this article, we shall assume that : S → K + c is always of the form (w) = [ψ -(w), ψ + (w)] ∈ K c and ψ -(w) ≤ ψ + (w) for all w ∈ [ζ , η]. We are now ready to formulate and prove some Hermite-Hadamard type results for m-polynomial convex (concave) interval-valued functions. Theorem 10 Let : S → K + c be an interval-valued function with ζ < η and ζ , η ∈ S, and Lebesgue integrable on [ζ , η]. If ∈ XP m (S, K + c ) and ρ, > 0, then and B is the beta function defined by (5). The inclusions are reversed if ∈ VP m (S, K + c ).
We get the desired inequality by applying Theorem 12.

Conclusion
Some new set inclusions of the Hermite-Hadamard types are established for the class of m-polynomial convex interval-valued functions. A relationship between a given m-