Hermite-Hadamard and Schur-Type Inequalities for Strongly h -Convex Fuzzy Interval Valued Functions

The concept of fuzzy theory was developed in 1965 and becomes an acknowledged research subject in both pure and applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications. In the present study, we introduced the de ﬁ nition of fuzzy interval valued strongly h -convex function and investigated some of its properties. We established Hermite-Hadamard and Schur-type inequalities for the class of fuzzy interval valued strongly h -convex function.


Introduction
The theory of interval analysis is one of the top areas of research nowadays because of its enormous application in various fields especially in automatic error analysis [1], computer graphics [2], and neural network output optimization [3]. In [4], Moore et al. give the first monograph on interval analysis, and since then, a huge amount of work have been done in interval calculus; for example, in [5], Chalco Cano et al. studied the interval-valued functions using generalized Hukuhara derivative and presented applications of interval valued calculus. An efficient method for solving fuzzy optimization problems using interval valued calculus is presented in [6]. Costa et al. [7] calculated the possible conformations arising from uncertainty in the molecular distance geometry problem using constraint interval analysis. Interval vector spaces and interval optimizations are given in [8], while optimality conditions for generalized differentiable interval valued functions are presented in [9]. Several classical inequalities for interval valued functions have been established in [10].
To address the modern problems, the convexity has been generalized in number of ways and some interesting generalizations are strongly ðh, sÞ-convex functions [11], strongly generalized convex functions [12], p-convex function [13], and many more. The authors introduce h-convex function in [14] as follows; A function ϕ : for all s, t ∈ I = ½μ 1 , μ 2 and β ∈ ½0, 1.
In recent years, mathematical inequalities for interval valued convex and nonconvex function got attention of many mathematician [17]. Bai et al. [10] established Hermite-Hadamard and Jensen-type inequalities for the class of interval valued nonconvex functions. In 2017, Costa [18] presented Jensen-type inequality for the class of fuzzy interval valued functions, and in the same year, Costa and Román-Flores [19] presented some integral inequalities for the same class of functions. Some Opial-type inequalities were studied in [20]. For other remarkable results, we refer [21] and references therein.
In this report, we proposed the definition of fuzzy interval valued strongly h-convex function. We investigated some of its properties and established Hermite-Hadamard and Schur-type inequalities for the proposed definition.
The paper is organized as follows: In Section 2, we will give some preliminaries and basic definitions, and we will established some basic properties. However, Section 3 is devoted for the establishment of main results like Hermite Hadamard and Schur-type inequalities.

Some Basic Properties of Interval Calculus
Let ℝ + I and ℝ I be all positive intervals and all intervals. The algebraic operations "+" and "." are defined as [22]  For intervals ½s, s and ½t, t, the distance dð½s, s, ½t, tÞ = max fjs − tj, j s − tjg is the Hausdorff distance. Then ðℝ I , dÞ is complete.
The tagged partition P of interval ½s, t is the set of num- Moreover, letting Δs i = s i − s i−1 and Δs i ≤ δ for every i, the partition is said to be δ-fine. The family of the all such δ-fine partitions of ½s, t is represented as Pðδ, ½s, tÞ [17].
We end this section of preliminaries by introducing the new concept of interval valued strongly h-convex function. Note that for interval ½s, s and ½t, t, the inclusion ′′ ⊆ ′′ is defined by Definition 4. Let X be a normed space and D ⊆ X is convex. h : ð0, 1Þ ⟶ ℝ is a function. We call a function ϕ : I = ½μ 1 , for all s, t ∈ I = ½μ 1 , μ 2 and β ∈ ½0, 1.
The notion SXðh ; ðμ 1 , μ 2 , ℝ + I ÞÞ is reserved for the class of interval valued strongly h-convex functions.
Proof. The proof is similar to that of Theorem 6.
But it is not interval valued h-convex function. Because by taking β = 1/2, hðβÞ = 1, s = −1, t = 1, and ϑ = 1, the inequality (16) is not satisfied. So the set inclusion (9) is holds, and we say that it is not interval valued h-convex function.

Journal of Function Spaces
Substituting the values of u and v in (21), we get integrating (23) with respect toβoverð0, 1Þ; we obtain Similarly substituting the values of u and v in (22), we get Combining (24) and (25), we obtain which gives the proof of left hand side. For the proof of right hand side, take Using the definition of interval valued strongly h-convex function, we obtain which gives right hand side.