On new fractional integral inequalities for p-convexity within interval-valued functions

This work mainly investigates a class of convex interval-valued functions via the Katugampola fractional integral operator. By considering the p-convexity of the interval-valued functions, we establish some integral inequalities of the Hermite–Hadamard type and Hermite–Hadamard–Fejér type as well as some product inequalities via the Katugampola fractional integral operator. In addition, we compare our results with the results given in the literature. Applications of the main results are illustrated by using examples. These results may open a new avenue for modeling, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables at the same time.


Introduction
Fractional calculus  is invariably important in almost all areas of mathematics and other natural sciences. Indeed, we can clearly realize that fractional operators have appeared in all fields of natural science and in fractional differential equations [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. In particular, it has been used in the study of waves in liquids, propagation of sound, gravitational attraction, and vibrations of strings. Numerous significant definitions and concepts have been established for the investigation of the fractional operators, for instance, Riemann, Liouville, Caputo, Hadamard, Katugampola, Atangana-Baleanu operators, and so on. Some well-known operators have been utilized for finding the existence of solutions to the boundary value problems, fractional integrodifferential equations or inclusions were elaborated [36][37][38][39][40].
Recently, the following Hermite-Hadamard inequality [94], one of the famous distinguished classical inequalities, has gained much consideration.
Let Q : I → R be a convex function. Then the double inequality holds for all e, f ∈ I with f = e. If Q is concave, then both the inequalities in (1.1) hold in the reverse direction. Many generalizations, modifications, applications, refinements, and variants for Hermite-Hadamard inequality (1.1) can be found in the literature [95,96].
The following weighted generalization of Hermite-Hadamard inequality (1.1) was derived by Fejér: Due to the modification among the ideas of convexity, the refinements for double inequality (1.2) have been widely investigated by many researchers. To meet the development trend of this research field, we delineate a new scheme and future plan in the present framework. We consider the p-convex function which assumes a dynamic job in portraying the idea of the interval-valued function just as establishing several generalizations by employing the Katugampola fractional integral operator.
On the other hand, a long history that can be followed back to Archimede's computation of the circumference of a circle has based on the theory of interval analysis. It fell into obscurity for a long time because of the dearth of utilities to different sciences. To the preeminence of our understanding, the substantial effort did not seem to this extent until the 1950s. In 1966, the first celebrated monograph concerned with interval analysis was written by Moore, who is famous as the founder of intervals, in order to compute the error bounds of the numerical solutions of a finite state machine. After his exploration, several researchers focused on studying the literature and applications of interval analysis in automatic error analysis, computer graphics, neural network output optimization, robotics, computational physics, and several other well-known areas in science and technology. Since then, several analysts have been broadly concentrated on and investigated the interval analysis and interval-valued functions in both mathematics and its applications.
The principal objective of this article is that we propose the notion of p-convex function for the interval-valued function. We also present the results concerning Hermite-Hadamard inequality, Fejér type inequality, and certain other related variants by employing p-convexity, which correlates with the Katugampola fractional integral operator. Finally, the repercussions of the employed technique depict the presentations for various existing outcomes. Results obtained by the novel approach disclose that the suggested scheme is very accurate, flexible, effective, and simple to use.

Preliminaries
For the basic notions and definitions on interval analysis, we use the literature [97].
Let M be the space of all intervals of R and D ∈ M be defined by Then D is said to be degenerate if d =d. If d > 0, then D is said to be positive, and ifd < 0, then D is said to be negative. We use M + and Mto symbolize the sets of all positive and negative intervals. Let η ∈ R and ηD be defined by Then the addition D 1 + D 2 and Minkowski difference D 1 -D 2 for D 1 , D 2 ∈ M are defined by and respectively. The inclusion relation "⊇" means that Let I ⊆ R be an interval and Q(z) = [Q(z),Q(z)] (z ∈ I). Then Q(z) is said to be Lebesgue integrable if Q(z) andQ(z) are measurable and Lebesgue integrable on I. Moreover, Now, we introduce the concept of Katugampola fractional integral operator for intervalvalued function.
Let q ≥ 1, c ∈ R, and χ q c (e, f ) be the set of all complex-valued Lebesgue integrable interval-valued functions Q on [e, f ] for which the norm Q χ q c is defined by is the Euler gamma function [99]. In [100], Zhang and Wan presented a definition of the p-convex function as follows.

Definition 2.2 ([100])
Let p ∈ R with p = 0 and I ⊆ R be a p-convex interval. Then the function Q : I → R is said to be a p-convex function if the inequality holds for all e, f ∈ I and η ∈ [0, 1].
From Definition 2.2 we clearly see that the p-convexity reduces to classical convexity and harmonic convexity if p = 1 and p = -1, respectively.
Next, we introduce a novel concept of interval p-convexity.

Definition 2.3
Let p ∈ R with p = 0 and I ⊆ R be a p-convex interval. Then the function for all e, f ∈ I and η ∈ [0, 1]. If the set inclusion (2.8) is reversed, then Q is said to be a p-concave interval-valued function.
Remark 2.4 From Definition 2.3 we clearly see that (1) If p = 1, then we get the definition given in [101].

Results and discussions
In this section, we establish several Hermite-Hadamard type inequalities for the p-convex interval-valued functions by employing the Katugampola fractional integral operator. In what follows, we denote by QC(I, M + ) the family of interval p-convex functions of the interval I.
(2) If p = 1 and q =q, then Theorem 3.1 reduces to the result given in [104].