Abstract
In this paper, we introduce the non-convex interval-valued functions for fuzzy-interval-valued functions, which are called -convex fuzzy-interval-valued functions, by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation given on the interval space. By using the -convexity concept, we present fuzzy-interval Hermite–Hadamard inequalities for fuzzy-interval-valued functions. Several exceptional cases are debated, which can be viewed as useful applications. Interesting examples that verify the applicability of the theory developed in this study are presented. The results of this paper can be considered as extensions of previously established results.
Highlights
The following integral inequality is known in the literature as the Hermite–Hadamard inequality [16, 17]: F u+θ ≤ 1 θ F (x) dx ≤ F (u) + F (θ), (1)θ–u u where F : K → R is a convex function on the interval K = [u, θ] with u
1 Introduction The following integral inequality is known in the literature as the Hermite–Hadamard inequality [16, 17]: F u+θ ≤ 1 θ
Theorem 5 Let F : [c, d] ⊂ R → FC(R) be a fuzzy-interval-valued function, whose γ levels define the family of interval-valued functions Fγ : [c, d] ⊂ R → KC given by Fγ (x) = [F∗(x, γ ), F ∗(x, γ )] for all x ∈ [c, d] and for all γ ∈ [0, 1]
Summary
Theorem 5 Let F : [c, d] ⊂ R → FC(R) be a fuzzy-interval-valued function, whose γ levels define the family of interval-valued functions Fγ : [c, d] ⊂ R → KC given by Fγ (x) = [F∗(x, γ ), F ∗(x, γ )] for all x ∈ [c, d] and for all γ ∈ [0, 1]. Theorem 6 Let K be a convex set, h1, h2 : [0, 1] ⊆ K → R+ such that h1, h2 ≡ 0, and let F : K → FC(R) be a fuzzy-interval-valued function whose γ -levels define the family of interval valued functions Fγ : [c, d] ⊂ R → KC+ given by
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