Abstract
In interval analysis, the fuzzy inclusion relation and the fuzzy order relation are two different concepts. Under the inclusion connection, convexity and non-convexity form a substantial link with various types of inequalities. Moreover, convex fuzzy-interval-valued functions are well known in convex theory because they allow us to infer more exact inequalities than convex functions. Most likely, integral operators play significant roles to define different types of inequalities. In this paper, we have successfully introduced the Riemann–Liouville fractional integrals on coordinates via fuzzy-interval-valued functions (FIVFs). Then, with the help of these integrals, some fuzzy fractional Hermite–Hadamard-type integral inequalities are also derived for the introduced coordinated convex FIVFs via a fuzzy order relation (FOR). This FOR is defined by φ-cuts or level-wise by using the Kulish–Miranker order relation. Moreover, some related fuzzy fractional Hermite–Hadamard-type integral inequalities are also obtained for the product of two coordinated convex fuzzy-interval-valued functions. The main results of this paper are the generalization of several known results.
Highlights
The classical version of Hermite–Hadamard inequality can be put in the following manner: Let S : K → R be a convex function on a convex set K and ρ, ς ∈ K with ρ ≤ ς
Khan et al [34] introduced the new class of convexity in fuzzy-interval calculus, which consists of coordinated convex fuzzy-interval-valued functions, and with the support of these classes, some Hermite–Hadamard-type inequalities are obtained via newly defined fuzzy-interval double integrals
Inspired by ongoing research work, we provide a novel class of Hermite–Hadamardtype inequalities for coordinated convex fuzzy-interval-valued functions through fuzzy-interval Riemann–Liouville-type fractional integrals
Summary
The classical version of Hermite–Hadamard inequality can be put in the following manner: Let S : K → R be a convex function on a convex set K and ρ, ς ∈ K with ρ ≤ ς. For interval-valued functions, Liu et al [30] demonstrated Hermite–Hadamard inequality using interval Riemann–Liouville-type fractional integrals. Budak et al [33] introduced several novel Hermite–Hadamard inequalities and defined Riemann–Liouville-type fractional integrals for interval-valued coordinated functions. Khan et al [34] introduced the new class of convexity in fuzzy-interval calculus, which consists of coordinated convex fuzzy-interval-valued functions, and with the support of these classes, some Hermite–Hadamard-type inequalities are obtained via newly defined fuzzy-interval double integrals. Inspired by ongoing research work, we provide a novel class of Hermite–Hadamardtype inequalities for coordinated convex fuzzy-interval-valued functions through fuzzy-interval Riemann–Liouville-type fractional integrals. Motivated by the work of Khan et al [27,28,34] and Budak et al [33], we obtain Hermite–Hadamard-type inequalities for the products of two fuzzy-interval-valued coordinated functions
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