FRACTIONAL OSTROWSKI-SUGENO FUZZY UNIVARIATE INEQUALITIES

. Here we present fractional univariate Ostrowski-Sugeno Fuzzy type inequalities. These are of Ostrowski-like inequalities in the setting of Sugeno fuzzy integral and its special-particular properties. In a fractional environment, they give tight upper bounds to the deviation of a function from its Sugeno- fuzzy averages. The fractional derivatives we use are of Canavati and Caputo types. This work is greatly inspired by [8], [1] and [2].

1. Introduction. The famous Ostrowski ([8]) inequality motivates this work and has as follows: where f ∈ C ([a, b]), x ∈ [a, b], and it is a sharp inequality. Another motivation is author's next fractional result, see [2], Another great source for inspiration is [4].
Here first we give a complete survey about Sugeno fuzzy integral and its basic properties. Then we derive a series of Ostrowski-Sugeno type fractional inequalities in the univariate case to all directions in the context of Sugeno integral and its basic important particular properties. We employ right and left fractional derivatives of Canavati and Caputo types. At the end we present a reverse fractional Polya-Sugeno inequality.
2. Background -I. In this section, some definitions and basic important properties of the Sugeno integral which will be used in the next section are presented. Also a preparation for the main results in Section 5 is given. [10,12]) Let Σ be a σ-algebra of subsets of X, and let µ : Σ → [0, +∞] be a non-negative extended real-valued set function. We say that µ is a fuzzy measure iff: Let (X, Σ, µ) be a fuzzy measure space and f be a non-negative real-valued function on X. We denote by F + the set of all non-negative real valued measurable functions, and by L α f the set: Definition 2.2. Let (X, Σ, µ) be a fuzzy measure space. If f ∈ F + and A ∈ Σ, then the Sugeno integral (fuzzy integral) [11] of f on A with respect to the fuzzy measure µ is defined by where ∨ and ∧ denote the sup and inf on [0, ∞], respectively.
The basic properties of Sugeno integral follow: 9,12]) Let (X, Σ, µ) be a fuzzy measure space with A, B ∈ Σ and f, g ∈ F + . Then In general we have see [12], p. 137.
We need We mention for all measurable functions f, g : We have in particular it holds for any A ∈ Σ.
A very important property of Sugeno integral follows.
From now on in this article we work on the fuzzy measure space ( , and µ is a finite fuzzy measure on B. Typically we take it to be subadditive.
The functions f we deal with here are continuous from [a, b] into R + . We make We notice that