On modified convex interval valued functions and related inclusions via the interval valued generalized fractional integrals in extended interval space

In this paper, we propose a new family of interval valued (IV) convex functions termed as generalized modified (p, h)-convex IV functions. We obtain the counterpart of Hermite-Hadamard H · H inequality by extending the IV fractional integral to the IV ψk-Riemann-Liouville (ψk − RL) fractional integrals. Also, several inequalities using extended operations on the newly defined class of convex IV functions are given.


Introduction
The function H : T → R (T is a finite interval of R) will be convex if the inequality: holds for all a 1 , a 2 ∈ T , µ ∈ [0, 1]. The beauty of mentioned functions was revealed when, Hermite and Hadamard [1] discovered the inequality called H·H inequality, which ranked among the best-established inequalities. Due to a nice geometrical interpretation, various applications has been found, see for instance [2,3]. For a detailed study about the generalization of this inequality, we refer to [4][5][6][7][8][9][10][11][12][13].
Interval analysis is a special case of set-valued analysis. The subject emerged as an attempt to handle the interval uncertainty which appears in different computational or mathematical models of some deterministic real-world problems. One of the oldest example of an interval enclosure is Archimedes method which is related to computation of the circumference of a circle. In 1966, Moore [14] wrote the first manuscript on to the interval analysis. After his book, scientists began to explore the theory and application of interval arithmetic.
The interval analysis has its applications in robotics, computer graphics, chemical and structured engineering, economics, behavioral ecology, constraint satisfaction, signal processing and global optimization, neural network output optimization and many others [15][16][17].
Motivated by these papers, we develop some new H · H inequalities by introducing a new class of IV functions termed as modified generalized (p, h)-convex IV function. Furthermore we explore the counter part of H · H inequality for the new and larger class of IV convex functions. We also extend the concept from interval arithmetic to extended interval arithmetic for the deduction of the including sub-classes of modified generalized (p, h)-convex IV functions. We establish the H · H inequality via IV ψ k − RL fractional integrals, which is the extension of IV Riemann-Liouville fractional integral introduced in [27].

Preliminaries
In this section, we recall some useful definitions and results which will be helpful for the further study. Most of the literature has been taken from [28][29][30].

Interval arithmetic; S = (S, +, ×, ⊆)
In this subsection we recall some basic concepts of interval-arithmetic. An interval [u, v], u ≤ v is a compact subset of R, the real line, defined by [u, v] So, for U ∈ T R the symbol u r (or U r ), with r ∈ {+, −}, represents the left or right end-point of U, depending on the value of r. Define the product rs for r, s ∈ {+, −} by setting −− = + = ++, +− = −+ = −, so that u −− = u ++ = u + etc. Let us denote the set of intervals containing zero by Z = {U ∈ T R| 0 ∈ U} = {U ∈ T R| u − ≤ 0, u + ≥ 0} and the set of intervals which do not contain zero is The interval arithmetic G = (T R, +, ., /, ⊆) consists of the set T R together with a relation for inclusion ⊆ and the basic operations addition + : In particular, if U is a degenerate interval of the form U = [u, u] = u, then The operation of subtraction U − V and division U/V are defined in G as composite operations respectively by Note that the operation inversion 1/V in G can not be composed by means of " + " and "." and thus should be assumed as basic. One can observe that the operations +, −, ., / in G defined by (2.3)-(2.8) satisfy: U V = {a b|a ∈ U, v ∈ V} , ∈ {+, −, ., /}. The properties of G = (T R, +, ., /, ⊆) are well established. Also note that: G 1 . The operations " + " and "." satisfy the following associative laws: The distributive law is valid if V.W > 0 (see [14]).

Interval integration
This section deals with reloading the notion of integral for IV functions. Throughout the study, we will denote set of positive proper intervals by T R + , negative T R − and all proper intervals by T R and the set of extended interval set by H, positive extended intervals by H + and negative by H − .
An IV function F of t on [a 1 , a 2 ] assigns a nonempty interval to each t ∈ [a 1 , Let P be a partition of [a 1 , a 2 ] having the form P : a 1 = t 0 < t 1 < . . . < t n = a 2 and mesh(P) = max t j t j−1 : j = 1, 2, . . . , n be the mesh of the partition P. Also, the set of all partitions of [a 1 , a 2 ] is denote by P([a 1 , a 2 ]). Let P(δ, [a 1 , a 2 ]) be the set of all P in P([a 1 , a 2 ]) such that mesh(P) < δ. Let us choose an arbitrary point ζ j in [t j−1 , t j ], (i = 1, 2, . . . , n) and let us define the sum Then, we say that S (F, P, δ) is a Riemann sum of F corresponding to P ∈ P(δ, [a 1 , a 2 ]). For any Riemann sum S of F corresponding to P ∈ P(δ, [a 1 , a 2 ]) and independent from choice of ζ j ∈ [t j−1 , t j ] for all 1 ≤ j ≤ n, we call M as the (I )-integral of F on [a 1 , a 2 ] and denoted by In recent past, researchers have focused interval analysis and established different inequalities, for detail see [18-26, 34, 35]. In [36], Varosanec introduced the class of S X functions as follows.
In [37], Zhao et al. introduced the notion of h-convex IV function as follows.
Lupulescu [27] derived the following IV definition for the left-sided RL fractional integral.
and H ∈ I ([a 1 ,a 2 ]) . Then, the IV left-sided RL fractional integral of function H is given by Budak et al. [39] defined the IV right-sided RL fractional integrals in the similar passion as following.
Budak et al. [39] establish the following H · H inequalities for convex IV functions.
Interested reader may see also the recent papers on inclusions of H · H type [40][41][42]. The purpose of this paper is to derive some new counterparts of H · H inequality via generalized (p, h)-convex IV functions in terms of ψ k − RL fractional integrals in interval form. In order to obtain the main results of the paper we first introduce the notion of (p, h)-convex IV functions on S and ψ k − RL fractional integrals in interval form. Then we obtain some subclasses of (p, h)-convex IV functions by considering the interval family K. We obtain a number of inequalities for including classes for the extended class. Finally, the validity of our findings is checked by taking some particular examples.
where Γ k (·) is the k-gamma function.

27)
Proof. The desired result can be obtained by making use of Theorem 3.1.
Interval analysis have uncertainty while solving equations of the form U + [1, 2] = [4,6] are not solvable by interval arithmetic. To remove this kind of uncertainty, extended interval arithmetic is introduced. The main objective of this paper is to discover H · H for general classes where extended arithmetic can be use successfully. For the desirable results, we introduce the notion of generalized modified (p, h)-convex IV functions. We have develop several inequalities using extended operations on the newly defined class of convex IV functions.

Main results
This section deals with introducing the notion of generalized modified (p, h)-convex IV function.
Throughout the study, we consider an IV bi-function η : ∆ T × ∆ T → ∆ T for suitable ∆ T ⊆ H.
for all a 1 , a 2 ∈ K 1 , µ ∈ [0, 1]. Now we present some special cases by taking into account the algebraic properties of intervals.  which is convex IV function, see [39]. (f) If p = −1, then (4.6) gives H(a 2 ) + h(µ)η (H(a 1 ), H(a 2 )) ⊆ H a 1 a 2 (1 − µ)a 1 + µa 2 , (4.10) we say H is a generalized modified harmonically h-convex IV function. We have following cases for harmonic convexity subject to the different choices of h and η as following. we call it harmonically s-convex IV function of first kind.
We start with the following double inequality of H · H type.