Uncertain Fuzzy Hermite-Hadamard Type Inequalities for MT(m,φ)-Preinvex Functions

In this paper, a new class of MT (m,φ) -preinvex functions is introduced and some uncertain fuzzy Hermite-Hadamard type inequalities for MT (m,φ) -preinvex functions via Riemann-Liouville fractional integrals are established. At the end, some applications to special means are given.


Introduction and Preliminaries
In this paper, we denote R F the set of all fuzzy numbers on R. Denote L F [a, b] the space of fuzzy Lebesgue integrable functions on [a, b] and C F [a, b] the space of fuzzy continuous functions on [a, b]. Also, we use I α a+ f and I α b− f for fuzzy fractional left and right Riemann-Liouville operators, where 0 < α ≤ 1.
Fractional calculus (see [3]) and the references cited therein, was introduced by Riemann and Liouville. Fuzzy Riemann integrals were introduced by Wu (see [5]). Let r, s ∈ R F and λ ∈ R. Define It is easy to show that D is a metric on R F and (R F , D) is a complete metric space with the following properties: where0 ∈ R F is defined0(x) = 0 for all x ∈ R. Let u, v ∈ R F . If there exists a w ∈ R F such that u = v ⊕ w, then we call w the H-difference on u and v and it is denoted by w = u v.
The aim of this paper is to applied the notion of MT (m,ϕ) -preinvex function for establish some uncertain fuzzy Hermite-Hadamard type inequalities via Riemann-Liouville fractional integrals. Holder's inequality and the well-known power mean inequality will be used to find new bounds for uncertain fuzzy Hermite-Hadamard inequalities. At the end, some applications to special means are given.

Fractional uncertain fuzzy Hermite-Hadamard for M T (m,ϕ) -preinvex functions
In order to prove in this section our main results regarding uncertain fuzzy Hermite-Hadamard type inequalities for M T (m,ϕ) -preinvex functions we need the following new lemma: Proof. Denote By integration by parts and using properties of fuzzy numbers the lemma follows.
By using Lemma 2.1, we have the following interesting results.