Hermite–Hadamard and Fractional Integral Inequalities for Interval-Valued Generalized 
 p
 -Convex Function

In the present paper, the new interval-valued generalized 
 
 p
 
 convex functions are introduced. By using the notion of interval-valued generalized 
 
 p
 
 convex functions and some auxiliary results of interval analysis, new Hermite–Hadamard and Fejér type inequalities are proved. The established results are more generalized than existing results in the literature. Moreover, fractional integral inequality for this generalization is also established.


Introduction
e theory of interval analysis introduced in numerical analysis by Moore in [1] had rapid development in last few decades. In computational problems, one can program a computer to find interval that contains the exact answer to the problems. Also, interval analysis provides rigorous enclosure of solution to the model equation. Moreover, the interval analysis is widely used in chemical and structured engineering, economics, control circuitry design, robotics, beam physics, behavioral ecology, constraint satisfaction, computer graphics, signal processing, asteroid orbits and global optimization [2], neural network output optimization [3], and many others. For interesting fundamental results, we refer [2,[4][5][6][7][8] to the readers.
Since the convexity play a vital role not only in convex analysis but also in almost all branches of mathematics. e famous inequalities in convex analysis are Jensen type, Hermite-Hadamard type, Fejér type, Ostrowski type, etc. For deeper insight about these inequalities, we refer [9][10][11][12][13][14][15][16] and references therein.
In order to introduce the main definition of this paper, let us recall few generalizations of convexity present in the literature.
Definition 1 (see [17]). An interval I 1 is p-convex set, if for any x 1 , x 2 ∈ I 1 , α 1 ∈ [0, 1], we have Definition 3 (see [29]). e mapping f defined from I 1 to R is said to be η-convex if holds with respect to η: Definition 4 (see [29]). A mapping is nonnegatively ho- Definition 5 (see [30]). A mapping f defined from a p-convex set I 1 to R is said to be generalized p convex function, if holds for η: B 1 × B 1 ⟶ B 2 be a bifunction for appropriate B 1 , B 2 ⊆ R and for each x 1 , x 2 ∈ I 1 and α 1 ∈ [0, 1]. Now, we present the concept of interval-valued generalized p convex function.
is article is in the direction of the concepts and some results discussed in [30], but now we use interval-valued generalized p-convex function instead of generalized p convex function. After this introduction, in Section 2, we develop some basic properties of interval-valued generalized p convex functions. InSection 3, we make some new inequalities like Hermite-Hadamard's and Fejér type for interval-valued generalized p convex functions.

Basic Results
Here, we derive some operations which preserves intervalvalued generalized p convex function. (1) If η is additive, then f 1 + f 2 is interval-valued generalized p convex (2) If η is nonnegatively homogeneous, then λf 1 is interval-valued generalized p convex for any λ ≥ 0.
Proof. e proof is similar to that of eorem 1.
Proof. Take u p � tξ By definition of interval-valued generalized p convex functions, we have Now, by the definition of interval we have It follows that (1/p) , Integrating (17) with respect to "x" on [0, 1], we get which implies
Put z � x p and simplify, we get Journal of Mathematics Combining (32), (34), and (35), we get

Fejér-Type Inequality for Interval-Valued Generalized p Convex Function
Now, we develop Fejér type inequality for interval-valued generalized p convex functions.
Theorem 4. Let f and g be nonnegative interval-valued generalized p convex functions ξ 1 , ξ 2 ∈ I ξ 1 < ξ 2 such that fg ∈ L 1 [ξ 1 , ξ 2 ], then where Proof. Since f and g are interval-valued generalized p convex functions, we have f tξ for all t ∈ [ξ 1 , ξ 2 ]. Since f and g are nonnegative, By the definition of interval, we have f tξ It follows f tξ Integrating (42) over (0, 1), we obtain the following inequality: Journal of Mathematics respectively, where Γ(α) is the Gamma function defined as Reimann integral is reduced as classical integral for α � 1. (1/p) ) and holds for all x, y ∈ I � [a, b].
Following lemma will help us in obtaining our fractional integrals inequalities which can be found in [36].
Now, we are ready to develop the Fractional Hermite-Hadamard-type inequalities for interval-valued generalized p convex functions. en, following fractional integral inequality holds, if p ∈ (R/(0)) and p > 0: Proof. Let ϕ be a generalized p convex function with p ≥ 0 and η is bounded above by M η . Take x � (ka p + (1 − k)b p ) (1/p) and y � (kb p + (1 − k)a p ) (1/p) . Since Multiplying both sides of (54) by k α− 1 and then integrating the resulting inequality with respect to k over [0, (1/2)], we obtain By definition of RiemannLiouville integrable function with g(x) � x (1/p) , we obtain which is the left-hand side of theorem (56). 8 Journal of Mathematics To prove the right-hand side, we take x � (ka p + (1 − k)b p ) (1/p) and y � (kb p + (1 − k)a p ) (1/p) : kη(ϕ(b), ϕ(a)). (58) Adding the (57) and (58) and multiplying the resulting inequality with 2k α− 1 and integrating with respect to k over By definition of RiemannLiouville integrable function, we get Rearranging the above inequality, we get the right-hand side: Γ(α + 1) b p − a p α 2 1−α J α a p +b p /2 ( is completes the proof.

Conclusions
e convex functions and fractional calculus play an important role in applied sciences [38][39][40][41][42][43]. Here, the new interval-valued generalized convex functions are introduced. By using the notion of interval-valued generalized p convex functions and some auxiliary results of interval analysis, some new Hermite-Hadamard-and Fejér-type inequalities are presented. Our results can be considered as generalization of many existing results. Moreover, fractional integral inequality for this generalization is also established.

Data Availability
e data used to support the article are available within the article.

Conflicts of Interest
e authors declare that do not have any conflicts of interest.

Authors' Contributions
All the authors contributed equally to this paper.