Hermite-Hadamard- and Jensen-Type Inequalities for Interval (h1, h2) Nonconvex Function

Since its inception five decades ago, the theory of fuzzy sets has advanced in a variety of ways. Application of the theory of fuzzy sets covered many areas like artificial intelligence, decision theory, computer science, logic operational research, and robotics [1–3]. Initially good books like probability theory by Dubois and Prade in 1988, Behavioral and social science by Smithsons in 1987, and Fuzzy Control by sugeno 1985 and pedrycz 1989 and others have been published. We refer [4, 5] recent developments in this field. For some other results and application of interval analysis theory, we refer the readers [4, 6–13]. Due to vast application of fuzzy sets, many integral inequalities have been derived by different authors [1–3, 14–17] Costa [18] there is a new fuzzy version of Jensen-type integral inequality for fuzzy interval valued function. Also in [19], Zhao et al. develop new Harmite-Hadamard-type inequality for h-convex interval valued function. For more about Hermite-Hadamard inequalities, refer [20–24]. We will introduce the interval (h1, h2) non-convex function. .e second objective of this article is to develop Hermite-Hadamardand Jensen-type inequality for above said generalization.


Introduction
Since its inception five decades ago, the theory of fuzzy sets has advanced in a variety of ways. Application of the theory of fuzzy sets covered many areas like artificial intelligence, decision theory, computer science, logic operational research, and robotics [1][2][3]. Initially good books like probability theory by Dubois and Prade in 1988, Behavioral and social science by Smithsons in 1987, and Fuzzy Control by sugeno 1985 and pedrycz 1989 and others have been published. We refer [4,5] recent developments in this field.

Preliminaries
In this section, we define some basic definitions, properties, results, and notations on interval analysis, which are used throughout the paper [17,25]. Here, R I and R + I denote the family of all intervals and positive interval and it is equipped with the algebraic operations "+" and "." given, respectively, and if (3) roughout the paper, IR-integrable means interval Riemann integrable. e concept of IR-integrable is given in [19], Definition 2.2, is equivalent to IR integral given in [10], Definition 9.1. [26]). e real interval I is known as p-convex set if for all x, y ∈ I and α ∈ [0, 1], implies that where p � 2k + 1 or p � (n 1 /n 2 ), n 1 � 2r + 1, n 2 � 2t + 1, and k, r, t ∈ N. [26]). For a p-convex set I, the mapping f: I ⟶ R: is called p-convex function, for all x, y ∈ I, and λ ∈ [0, 1]. [27]). e nonnegative function f: ∀x, y ∈ [a, b] and λ ∈ (0, 1) and h ≠ 0, h is nonnegative real-valued function or f belongs to the class SX(h; [a, b]).
If the inequality (7) is reversed, then f is said to be h-concave, i.e., f ∈ SV(h, I).
We say that f: J ⟶ R + I is an interval (h 1 , h 2 )-convex function or that f belongs to the class SX ((h 1 , h 2 ); J, R + I ), if f is nonnegative and for all x, y ∈ J and λ ∈ (0, 1), we have If the inequality (9) is reversed, then f is said to be interval (h 1 , h 2 )-concave, i.e., f ∈ SV ((h 1 , h 2 ), J, R).
We wind up the current section by introducing the new concept of interval (h 1 , h 2 ) nonconvexity. is idea is inspired by An et al. [29]. roughout the paper, for interval [x, x] and [y, y], x, x ⊆ y, y ⇒ y ≤ x and x ≤ y.

Main Result
Proof. By assumption, we have Integrating above w.r.t. "x" on [0, 1], we get it follows that 2p is implies that Now by def. of interval nonconvex function, we have integrated with respect to "x" on [0, 1], we get Journal of Mathematics it follows that Combining (19) and (22) we get (27).
Putting z � λ p in (24) and simplifying, we get