Some integral inequalities for fuzzy-interval-valued functions

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Some integral inequalities for fuzzy-interval-valued functions T.M. Costa, H. Roman-Flores ´ PII: DOI: Reference:

S0020-0255(17)30896-4 10.1016/j.ins.2017.08.055 INS 13059

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Please cite this article as: T.M. Costa, H. Roman-Flores, Some integral inequalities for fuzzy-interval´ valued functions, Information Sciences (2017), doi: 10.1016/j.ins.2017.08.055

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Some integral inequalities for fuzzy-interval-valued functions T. M. Costaa,∗, H. Rom´ an-Floresb a Instituto

de Ciˆ encias Exatas e Naturais, Universidade Federal do Par´ a, Bel´ em, Par´ a, Brasil de Alta Investigaci´ on, Universidad de Tarapac´ a, Casilla 7D, Arica, Chile

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b Instituto

Abstract

This study presents new fuzzy versions of Minkowski and Beckenbach’s integral inequalities without making use of the Sugeno integral. These new inequalities generalize the interval versions of the Minkowski and Beckenbach inequalities recently published and are obtained by introducing the

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p−F A−integrability concept for fuzzy-interval-valued functions by means of the Kaleva integral and a fuzzy order relation. This fuzzy order relation is defined level-wise through the Kulisch-Miranker order relation given on the interval space. Numerical examples that illustrate the applicability of the theory developed in this study are presented.

Keywords: Fuzzy interval space, Fuzzy integral, Fuzzy Minkowski’s inequality, Fuzzy

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Beckenbach’s inequality.

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1. Introduction

It is known that the classical Minkowski integral inequality [32] plays an important role in mathematical analysis and in many other areas of mathematics (see [46]). This fact has driven

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the development of its extensions into different mathematical spaces, including those that do not satisfy the axioms of a vector space such in the case of intervals [29, 45] and fuzzy intervals [2, 3, 12, 18, 31, 37, 39]. These inequalities applied to interval and fuzzy interval spaces provide tools for

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modeling problems under uncertainties. In general, the fuzzy Minkowski inequalities [2, 3, 12, 18, 37] as well as other fuzzy integral

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inequalities [4, 7, 13, 14, 20, 21, 30, 39, 43, 44, 48, 50] use the Sugeno integral [47] and, consequently, from a semantical viewpoint, the fuzzy uncertainties contained in those inequalities are generated ∗ Corresponding

author: T.M. Costa Email addresses: [email protected] (T. M. Costa), [email protected] (H. Rom´ an-Flores)

Preprint submitted to Journal of LATEX Templates

August 16, 2017

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by the fuzzy measures of integration and not by the real-valued functions integrands. On the other hand, the use of the class of fuzzy-interval-valued functions in such inequalities makes it possible to consider uncertainties in the prediction processes that are semantically different from those previously cited. More precisely, by using fuzzy-interval-valued functions to deal with fuzzy integral inequalities, the uncertainties are generated by the integrand and not by the measures of

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integration such as occurs when using the Sugeno integral. There are some fuzzy integral inequalities where the integrands are fuzzy-interval-valued functions. For example, Osuna-G´ omez et al. [35] presented a Hadamard’s integral inequality, and Costa [17] presented a Jensen’s integral inequality for fuzzy-interval-valued functions. These studies show that this approach is interesting both from a theoretical point of view as well as from a practical point of view since it allows the transformation of fuzzy integral inequalities to real integral inequalities. Thus, it is possible to obtain tools to deal

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with classes of non-deterministic problems different from those previously studied, using tools from the classical real analysis.

The main goal of this study is to present the Minkowski and Beckenbach’s integral inequalities, where the integrands are fuzzy-interval-valued functions. These fuzzy integral inequalities are obtained as generalization of their respective interval versions given by Rom´ an-Flores et al. [45] by

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means of the Aumann integral [10] and the Kulisch-Miranker’s order relation defined on the set of all closed and bounded intervals of real numbers. The generalizations arise from the Kaleva integral [24] obtained level-wise through the Aumann integral, from the fuzzy order relation defined level-

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wise by means of the Kulisch-Miranker’s order relation given on the set of all closed and bounded intervals of real numbers, and from the new concept presented here of p − F A−integrability for fuzzy-interval-valued functions given by means of the Kaleva integral.

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This study is organized as follows: Section 2 presents preliminary concepts and results in the interval space and in the space of fuzzy intervals. Moreover, Section 2 recalls the Minkowski and

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Beckenbach’s integral inequalities presented in [45]. Section 3 presents the p − F A−integrability concept for fuzzy-interval-valued functions and the Minkowski and Beckenbach’s integral inequalities for fuzzy-interval-valued functions. These are the main results of this study. Section 4 recalls known

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fuzzy inequalities given by other types of fuzzy integrals as well as the fuzzy Hadamard’s integral inequality and the recently published fuzzy Jensen’s integral inequality both generated by the same fuzzy integral concept used in this article. Section 5 presents numerical examples in order to illustrate the theory developed in this study. Finally, Section 6 presents our final considerations.

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2. Preliminaries 2.1. Interval-valued functions This research uses the symbol KC to denote the family of all closed and bounded intervals of

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real numbers equipped with the algebraic operations “+”, “·”, “×”, and “/” given, respectively, by   [λa, λa] if 0 ≤ λ , [a, a] + [b, b] = [a + b, a + b], λ · [a, a] =  [λa, λa] if λ < 0   [a, a] × [b, b] = min{ab, ab, ab, ab}, max{ab, ab, ab, ab} for all [a, a], [b, b] ∈ KC and for all λ ∈ R,

and

     [a, a] a a a a a a a a = min , , , , max , , , for all [a, a], [b, b] ∈ KC such that 0 6∈ [b, b]. b b b b b b b b [b, b]

The family KC is called the interval space.

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That is, KC denotes the set {[x, x] : x, x ∈ R, and x ≤ x} equipped with “+”, “·”, “×”, and “/”. Given A and B nonempty subsets of R, the Hausdorff separation of B from A is defined by dH ∗ (B, A) = sup{d(b, A)}, b∈B

where d(b, A) = inf |b − a|. By using the Hausdorff separation, the Pompeiu-Hausdorff distance a∈A

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between the non-empty subsets A and B, it is defined by dH (A, B) = max{dH ∗ (B, A), dH ∗ (A, B)}. It is known that the Pompeiu-Hausdorff distance restricted to KC is the metric dH : KC ×KC −→ R metric space [8, 9, 19].

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given by dH ([a, a], [b, b]) = max{|a − b|, |a − b|} for all [a, a], [b, b] ∈ KC , and (KC , dH ) is a complete A map F : U ⊆ R → KC given by F (t) = [f (t), f (t)], wheref , f : U → R are real functions,

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with f (t) ≤ f (t) for all t ∈ U , it is called an interval-valued function. The functions f and f are called the lower and the upper (endpoint) functions of F , respectively. Definition 2.1. ([8, 9]) Let F : U ⊆ R −→ KC be an interval-valued function. Then L ∈ KC is

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called a limit of F at t0 ∈ U if for every  > 0 there exists δ(, t0 ) = δ > 0 such that dH (F (t), L) <  for all t ∈ U with 0 < |t − t0 | < δ. This is denoted by lim F (t) = L. t→t0

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Theorem 2.1. ([8, 9]) Let F : U ⊆ R −→ KC be the interval-valued function given by F (t) = [f (t), f (t)] for all t ∈ U . Then L = [L, L] ∈ KC is a limit of F at t0 ∈ U if and only if L is the

limit of f at t0 and L is the limit of f at t0 . Moreover, if L is the limit of F at t0 , then   lim F (t) = lim f (t), lim f (t) . t→t0

t→t0

3

t→t0

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Definition 2.2. ([8, 9]) An interval-valued function F : U ⊆ R −→ KC is called dH −continuous at t0 ∈ U if lim F (t) = F (t0 ). If F is dH −continuous at every t ∈ U , then we say that F is t→t0

dH −continuous. Theorem 2.2. ([8, 9]) Let F : U ⊆ R −→ KC be the interval-valued function given by F (t) =

t→t0

t→t0

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[f (t), f (t)] for all t ∈ U . Then F is dH −continuous at t0 ∈ U if and only if f and f are continuous at     t0 . Moreover, if F is dH −continuous at t0 , then lim F (t) = lim f (t), lim f (t) = f (t0 ), f (t0 ) . t→t0

An interval-valued function F : [a, b] −→ KC is said to be measurable if and only if

{(t, x) : x ∈ F (t)} ∈ A × B, where A denotes the σ−algebra composed of the Lebesgue-measurable

subsets of R and B denotes the σ−algebra composed of the Borel-measurable subsets of R.

F is said to be integrably bounded on [a, b] if and only if there exists a Lebesgue-integrable

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function h : [a, b] −→ [0, +∞) such that |x| ≤ h(t) for all x and t such that x ∈ F (t).

Given an interval-valued function F : [a, b] −→ KC , the Aumann integral (IA− integral, for short) of F over [a, b] is defined [10] by (Z Z (IA) F (t)dt = [a,b]

)

f (t)dt : f ∈ S(F ) ,

[a,b]

(1)

where S(F ) := {f : [a, b] → R : f is integrable and f (t) ∈ F (t) for all t ∈ [a, b]}. We say that the

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IA-integral of F over [a, b] exists (or that F is IA-integrable over [a, b]) if S(F ) 6= ∅. Theorem 2.3. (see, e.g., [19, 24]) If an interval-valued function F : [a, b] −→ KC is measurable

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and integrably bounded on [a, b], then F is IA−integrable over [a, b]. Theorem 2.4. (see, e.g., [19, 24]) Let F : [a, b] → KC be the interval-valued function given by

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F (t) = [f (t), f (t)] for all t ∈ [a, b].

Then F is IA−integrable over [a, b] if and only if

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f , f ∈ S(F ). Moreover, if F is IA−integrable over [a, b], then (IA)

Z

[a,b]

F (t)dt =

"Z

f (t)dt,

[a,b]

Z

#

f (t)dt .

[a,b]

(2)

Corollary 2.1. Let F, G : [a, b] −→ KC be the interval-valued functions given, respectively, by

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F (t) = [f (t), f (t)] and G(t) = [g(t), g(t)] for all t ∈ [a, b]. Then if F and G are IA−integrable over [a, b], it follows that the interval-valued function (F + G) : [a, b] → KC defined by (F + G)(t) = F (t) + G(t) for all t ∈ [a, b], it is IA−integrable over [a, b] and Z Z Z (IA) (F (t) + G(t))dt = (IA) F (t)dt + (IA) [a,b]

[a,b]

4

[a,b]

G(t)dt.

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Proof. If F and G are IA−integrable over [a, b], from Theorem 2.1, it follows that f , f , g, g : [a, b] → R are integrable over [a, b], and consequently, (f + g), (f + g) : [a, b] → R are integrable over [a, b]. Since (F + G)(t) = [(f + g)(t), (f + g)(t)] = [f (t) + g(t), f (t) + g(t)] for

[a,b]

[a,b]

=

[a,b]

"Z

f (t)dt +

=

f (t)dt,

[a,b]

=

(IA)

Z

Z

F (t)dt + (IA)

(F (t) + G(t))dt = (IA)

[a,b]

Z

f (t)dt +

[a,b]

"Z

Z

g(t)dt

[a,b]

g(t)dt,

[a,b]

#

Z

[a,b]

#

g(t)dt

G(t)dt.

[a,b]

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(IA)

Z

#

Z

f (t)dt +

[a,b]

[a,b]

Therefore,

g(t)dt,

[a,b]

[a,b]

"Z

Z

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all t ∈ [a, b], from Theorem 2.1, it follows that (F + G) is IA−integrable over [a, b]. Moreover, "Z # Z Z (IA) (F (t) + G(t))dt = (f (t) + g(t))dt, (f (t) + g(t))dt

Z

[a,b]

F (t)dt + (IA)

Z

G(t)dt.

[a,b]

then F is IA−integrable over [a, b].

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Remark 2.1. From Theorem 2.1 and from Theorem 2.2, it is easy to see that, if F is dH −continuous,

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The relation ≤I defined on KC by

[a, a] ≤I [b, b] if and only if a ≤ b and a ≤ b

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for all [a, a], [b, b] ∈ KC , it is an order relation [27]. Given [a, a], [b, b] ∈ KC , we say that [a, a]
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way, the interval uncertainties contained in the input data of a given real-valued function ensuring that the result obtained, which is an interval, contains all possibles values that such a real-valued

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function could attain on such interval uncertainties. For example, by considering a real-valued function f , whose input data x are values expressing the measurements of a given experiment, since a measurement is the process of experimentally determining numerical magnitude value for a characteristic that can be assigned to an object or event, then x may not represent an accurate value. On the other hand, usually this kind of process provides the lower and upper bounds of 5

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x, i.e., the values x, x ∈ R such that x ≤ x ≤ x. Thus, by representing the interval uncertainty over x by [x, x], from the interval analysis viewpoint it is desirable to explicitly represent the set f ([x, x]) = {f (x) : x ∈ [x, x]} in terms of intervals. Although such a representation is not always possible since, in general, it is only possible to provide an interval estimate for f ([x, x]) for some

is obtained naturally as illustrated in the following examples.

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functions such as the monotonic and continuous real-valued functions, this process of representation

Example 2.1. ([33]) Given n ∈ N, let f : R → R be given by f (x) = xn for all x ∈ R. Given [x, x] ∈ KC , it follows that f ([x, x]) = {xn : x ∈ [x, x]} can be expressed in terms of intervals as    [xn , xn ] , if 0 ≤ x or n is odd,   f ([x, x]) = [xn , xn ] , if x < 0 and n is even,     [0, max{xn , xn }] , if x < 0 < x and n is even.

(3)

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follows

n

In this presentation the set f ([x, x]) is denoted by [x, x] . By considering the particular case, n = 2, it follows that

     x2 , x2 , if 0 ≤ x,     f ([x, x]) = x2 , x2 , if x < 0,     0, max{x2 , x2 } , if x < 0 < x.

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(4)

Remark 2.2. From Fundamental theorem of interval analysis (see [33]), it is easy to see that 2

[x, x] ⊆ [x, x] × [x, x] for all [x, x] ∈ KC . In order to obtain a case where the strict inclusion 2

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holds, it is sufficient to consider [x, x] = [−1, 1]. Thus, [x, x] = [0, 1] and [x, x] × [x, x] = [−1, 1]. The overestimation given by [x, x] × [x, x] occurs because the operation “×” does not capture the dependence between the interval factors involved in the operation. That is, by using “×”, the factor

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X = [−1, 1] is considered to be generated by a different semantics from the one that generated the factor Y = [−1, 1], and consequently, X is considered to be semantically different from Y (about 2

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semantics, to see [28]). On the other hand, by using (4) to generate “[x, x] ”, the dependence is completely captured such as occurs in the real space. However, for all [x, x] ∈ KC such that x > 0, 2

since x2 ≤ x · x ≤ x2 , it follows that [x, x] = [x, x] × [x, x]. Actually, for all n ∈ N and for all n

n

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[x, x] ∈ KC such that x > 0, it follows that [x, x] = [x, x] × · · · × [x, x]. That is, [x, x] is given by the multiplication of the n−factors [x, x] ∈ KC .


2 Example 2.2. ([33]) Let f : R → R be given by f (x) = 41 − x − 12 for all x ∈ R. Given n o
2 [x, x] ∈ KC , it follows that f ([x, x]) = 14 − x − 21 : x ∈ [x, x] can be expressed in terms of 6

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intervals as follows  h

i 1 2 1 1 2 1  x − , − , if 12 ≤ x, − − x  2 4 2   h4 i

2 1 1 2 1 f ([x, x]) = , 4 − x − 12 , if x ≤ 12 , 4 − x− 2  h i n o    1 − max x − 1
2 , x − 1
2 , 1 , if x < 4

2

2

4

(5) 1 2

< x.

the set f ([x, x]) can be expressed straightforward by

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It is known (see [33]) that if f is a monotonic and continuous real function on [x, x] ∈ KC , then

f ([x, x]) = [{min{f (x), f (x)}, max{f (x), f (x)}] .

Example 2.3. ([33]) Given r > 0, let g : [0, +∞) → R be given by g(x) = xr for all x ∈ [0, +∞). Then g is continuous and increasing. Given [x, x] ∈ KC such that x ≥ 0, it follows that the set

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g([x, x]) = {xr : x ∈ [x, x]} can be expressed by

g([x, x]) = [xr , xr ],

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and analogously to Example 2.1, for all r > 0 and for all [x, x] ∈ KC such that x ≥ 0, this presentation uses [x, x]r to denote the set g([x, x]).

Rom´an-Flores et al. [45] presented the following interval versions of the Minkowski and Becken-

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bach’s integral inequalities, which are based on Example 2.3 and based on the concept of intervalvalued proportional functions (we recall that, given F, G : [a, b] → KC , then F and G are said to

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be proportional if there is a non-negative real constant k such that F (t) = k · G(t) for all t ∈ [a, b]). Theorem 2.5. ([45])(Interval Minkowski’s inequality) Given p ∈ (0, +∞), let F, G : [a, b] → KC be given, respectively, by F (t) = [f (t), f (t)] and G(t) = [g(t), g(t)] for all t ∈ [a, b], with f (t) ≥ 0

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and g(t) ≥ 0 for all t ∈ [a, b]. Then

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(i) if F and G are IA-integrable over [a, b] and p ≥ 1, it follows that ! p1 ! p1 Z Z Z p p ≤I (IA) (F (t)) dt + (IA) (IA) (F (t) + G(t)) dt [a,b]

[a,b]

[a,b]

p

! p1

(G(t)) dt

.

(7)

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The equality holds if F and G are proportional;

(ii) if F and G are IA-integrable over [a, b] and 0 < p < 1, it follows that ! p1 ! p1 ! p1 Z Z Z p p p (IA) (F (t)) dt + (IA) (G(t)) dt ≤I (IA) (F (t) + G(t)) dt . [a,b]

[a,b]

[a,b]

(8)

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The equality holds if F and G are proportional. Theorem 2.6. ([45])(Interval Beckenbach’s inequality) Let F, G : [a, b] → KC be given, respectively, by F (t) = [f (t), f (t)] and G(t) = [g(t), g(t)] for all t ∈ [a, b], with f (t) > 0 and g(t) > 0 for all t ∈ [a, b]. Then if F and G are IA-integrable over [a, b] and 0 < p < 1, it follows that Z

p+1

(F (t) + G(t))

dt

[a,b]

(IA)

Z

p

(F (t) + G(t)) dt

(IA) ≤I

[a,b]

Z

(F (t))

p+1

dt

[a,b]

(IA)

Z

p

(F (t)) dt

[a,b]

Z

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(IA)

(IA)

+

(G(t))

p+1

dt

[a,b]

(IA)

Z

p

.

(9)

(G(t)) dt

[a,b]

Definition 2.3. Given p ∈ (0, +∞), let F : [a, b] −→ KC be the interval-valued function given by

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F (t) = [f (t), f (t)] for all t ∈ [a, b], with f (t) ≥ 0 for all t ∈ [a, b]. We say that F is p-IA-integrable

over [a, b] if F p : [a, b] −→ KC given by F p (t) = (F (t))p for all t ∈ [a, b], it is IA−integrable over [a, b].

Proposition 2.1. Given p ∈ (0, +∞), let F : [a, b] −→ KC be the interval-valued function given by F (t) = [f (t), f (t)] for all t ∈ [a, b], with f (t) ≥ 0 for all t ∈ [a, b]. Then F is p − IA−integrable over

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[a, b] if and only if f and f are p−integrable over [a, b] (we recalls that, a real function f : [a, b] −→ R

is said to be p−integrable over [a, b] if the real function f p : [a, b] −→ R given by f p (t) = (f (t))p is

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integrable over [a, b]).

Proof. This result follows directly from Theorem 2.1.

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Remark 2.3. The assumptions of Theorem 2.6, which were first given in [45], are not sufficient to ensure Theorem 2.6. However, if the conditions of p − IA−integrability and of (p + 1) − IA−integrability for the interval-valued functions are added in the assumptions of Theorem 2.6,

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then this theorem holds and the proof is exactly that given by Rom´ an-Flores at el. [45]. A similar remark must to be considered with regard to Theorem 2.5.

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2.2. Fuzzy-interval-valued functions This section recalls concepts and results from fuzzy literature used in this presentation.

Definition 2.4. (see e.g.,[11, 19]) A fuzzy subset A of R is characterized by a function u ˜ : R → [0, 1] called the membership function of A. In general, in order to simplify the notation, a fuzzy subset A 8

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of R is presented as being its membership function u ˜. That is, a fuzzy subset A of R is a function u ˜ : R → [0, 1]. In this study this representation is adopted. Moreover, the family of all fuzzy subset

of R is denoted by F(R).

(i) u ˜ is normal, i.e., there exists x ¯ ∈ R such that u ˜(¯ x) = 1;

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A fuzzy subset u ˜ of R is called a real fuzzy interval if it has the following properties:

(ii) u ˜ is fuzzy convex, i.e., min{˜ u(x1 ), u ˜(x2 )} ≤ u ˜(λx1 + (1 − λ)x2 ) for all x1 , x2 ∈ R and for all λ ∈ [0, 1];

(iii) u ˜ is upper semicontinuous on R, i.e., given x ¯ ∈ R, for every  > 0 there exists δ > 0 such that u ˜(x) − u ˜(¯ x) <  for all x ∈ R with |x − x ¯| < δ;

(iv) u ˜ is compactly supported, i.e., cl{x ∈ R : 0 < u ˜(x)} is compact, where cl(A) denotes the

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closure of a classical set A. The family of all real fuzzy intervals is denoted by FC (R).

Definition 2.5. (see e.g.,[11, 19]) Given u ˜ ∈ FC (R), the level sets of u ˜ are given by

[˜ u]α = {x ∈ R : α ≤ u ˜(x)} for all α ∈ (0, 1] and by [˜ u]0 = cl{x ∈ R : 0 < u ˜(x)}. These sets are called the α−level sets of u ˜ for all α ∈ [0, 1].

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Theorem 2.7. ([34]) If u ˜ ∈ FC (R) and [˜ u]α are its α−level sets, then: (i) [˜ u]α is a closed interval [˜ u]α = [uα , uα ], for all α ∈ [0, 1].

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(ii) If 0 ≤ α1 ≤ α2 ≤ 1, then [˜ u]α2 ⊆ [˜ u]α1 .

(iii) For any sequence (αn )n∈N which converges from below to α ∈ (0, 1], we have ∩∞ u]αn = [˜ u]α . n=1 [˜

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(iv) For any sequence (αn )n∈N which converges from above to 0, we have cl (∪∞ u]αn ) = [˜ u]0 . n=1 [˜

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It is well-known that, given λ ∈ R and u ˜, v˜ ∈ FC (R), then the addition u ˜ ⊕ v˜, the multiplication u ˜ can be characterized level-wise, by scalar λ u ˜, the multiplication u ˜ ⊗ v˜, and the division v˜ respectively, by  α u ˜ [˜ u]α [˜ u ⊕ v˜]α = [˜ u]α + [˜ v ]α , [λ u ˜]α = λ · [˜ u]α , [˜ u ⊗ v˜]α = [˜ u]α × [˜ v ]α , and if 0 6∈ [˜ v ]α then = α v˜ [˜ v] for all α ∈ [0, 1]. Another well-known fact is that, u ˜ = v˜ if and only if [˜ u]α = [˜ v ]α for all α ∈ [0, 1]. Next two results provide necessary and sufficient conditions to characterize a real fuzzy interval

via real-valued functions.

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Theorem 2.8. ([11, 22]) Let u ˜ ∈ FC (R) be the fuzzy interval, whose α−level sets are given by

[˜ u]α = [uα , uα ] for all α ∈ [0, 1]. Then the functions u, u : [0, 1] −→ R defined by u(α) = uα and u(α) = uα for all α ∈ [0, 1], respectively, have the following properties: (i) u is bounded, non-decreasing, left-continuous on (0, 1], and it is right-continuous at 0.

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(ii) u is bounded, non-increasing, left-continuous on (0, 1], and it is right-continuous at 0. (iii) u(1) ≤ u(1).

Theorem 2.9. ([11, 22]) If u, u : [0, 1] −→ R are real functions that have the following properties: (i) u is bounded, non-decreasing, left-continuous on (0, 1] and it is right-continuous at 0; (ii) u is bounded, non-increasing, left-continuous on (0, 1] and it is right-continuous at 0;

AN US

(iii) u(1) ≤ u(1),

then there is a fuzzy interval u ˜ ∈ FC (R) such that [˜ u]α = [u(α), u(α)] for all α ∈ [0, 1]. Proposition 2.2. The relation ≤F given on FC (R) by

u ˜ ≤F v˜ if and only if [˜ u]α ≤I [˜ v ]α for all α ∈ [0, 1],

M

it is a partial order relation.

(10)

Proof. This result follows directly from the partial interval order relation ≤I defined on KC .

ED

Theorem 2.10. (see, e.g, [19, 40]) The space FC (R) equipped with the supremum metric, i.e., d∞ (˜ u, v˜) = sup dH ([˜ u]α , [˜ v ]α ) ,

PT

α∈[0,1]

it is a complete metric space.

CE

A fuzzy-interval-valued map F˜ : U ⊆ R −→ FC (R) is called a fuzzy-interval-valued function. Definition 2.6. (see, e.g., [11, 19]) A fuzzy-interval-valued function F˜ : U ⊆ R −→ FC (R) is said

AC

to be continuous at t0 ∈ U if for any  > 0, there exists δ(, t0 ) = δ > 0 such that

for all t ∈ U with |t − t0 | < δ.

  d∞ F˜ (t), F˜ (t0 ) < ,

10

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Remark 2.4. ([17]) Given a fuzzy-interval-valued function F˜ : U ⊆ R −→ FC (R), from (i) of Theorem 2.7, it follows that for each α ∈ [0, 1], we can define the interval-valued function Fα : U −→ KC

by Fα (t) = [F˜ (t)]α for all t ∈ U . On the other hand, for each given interval-valued function

Fα : U ⊆ R −→ KC , two real-valued functions f α , f α : U −→ R can be defined such that

Fα (t) = [f α (t), f α (t)] for each t ∈ U . Then from definition of d∞ and from Definition 2.6, it

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follows that F˜ is continuous at t0 ∈ U if and only if Fα is dH −continuous at t0 ∈ U for all

α ∈ [0, 1] and, from Theorem 2.2 , this is equivalent to f α and f α being continuous at t0 ∈ U for all α ∈ [0, 1].

Given a fuzzy-interval-valued function F˜ : U −→ FC (R), the interval-valued function

Fα : U −→ KC given by Fα (t) = [F˜ (t)]α for all t ∈ U is called the α−level of F˜ for all α ∈ [0, 1]. A

AN US

fuzzy-interval-valued function F˜ : [a, b] −→ FC (R) is said to be integrably bounded if there exists a Lebesgue-integrable function h : [a, b] −→ [0, +∞) such that |x| ≤ h(t) for all x and t such that x ∈ F0 (t). A fuzzy-interval-valued function F˜ : [a, b] −→ FC (R) is said to be strongly measurable

[24] if and only if its α−levels Fα : [a, b] −→ KC are measurable for all α ∈ [0, 1]. This concept is equivalent to the concept of measurability given in [40] and such equivalence can be obtained through Theorem III-2 and Theorem III-30 given in [15].

[a,b]

(F A)

Z

#α

ED

"

M

Definition 2.7. ([24]) Let F˜ : [a, b] −→ FC (R) be a fuzzy-interval-valued function. The integral of Z b Z F˜ (t)dt or (F A) F˜ (t)dt, it is defined level-wise by F˜ over [a, b], denoted by (F A) F˜ (t)dt

[a,b]

= (IA)

Z

a

Fα (t)dt =

[a,b]

(Z

[a,b]

PT

for all α ∈ [0, 1]. F˜ is F A−integrable over [a, b] if (F A)

Z

[a,b]

)

f (t)dt : f ∈ S (Fα )

F˜ (t)dt ∈ FC (R).

Theorem 2.11. ([40]) Given a fuzzy-interval-valued function F˜ : [a, b] −→ FC (R), if F˜ is strongly

CE

measurable and integrably bounded, then F˜ is F A−integrable over [a, b].

Given the fuzzy-interval-valued function F˜ : [a, b] → FC (R), whose α−levels are given by

AC

Fα : [a, b] → KC for all α ∈ [0, 1], if Fα is dH −continuous for all α ∈ [0, 1], then from Definition 2.7, Remark 2.4, and from Remark 2.1 is follows that F˜ is F A−integrable over [a, b].

Theorem 2.12. (see, e.g., [19, 24]) Let F˜ : [a, b] → FC (R) be the fuzzy-interval-valued function, h i whose α−levels Fα : [a, b] → KC are given by Fα (t) = f α (t), f α (t) for all t ∈ [a, b] and for all 11

ACCEPTED MANUSCRIPT

α ∈ [0, 1], where f α , f α : [a, b] → R are real functions. Then F˜ is integrable over [a, b] if and only if f α , f α ∈ S(Fα ) for all α ∈ [0, 1]. Moreover, if F˜ is F A−integrable over [a, b], then (F A)

Z

[a,b]

#α "Z ˜ F (t)dt =

[a,b]

f α (t)dt,

Z

[a,b]

for all α ∈ [0, 1].

#

f α (t)dt

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"

3. Minkowski and Beckenbach’s integral inequalities for fuzzy-interval-valued functions This section presents the Minkowski and Beckenbach’s integral inequalities for fuzzy-intervalvalued functions. These integral inequalities are the main results of this study. In order to present

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these inequalities, the following results are required.

Definition 3.1. Given r > 0, let u ˜ ∈ FC (R) be the fuzzy interval, whose α−level sets are given by α

˜r level-wise by [˜ ur ] := [uα , uα ]r = [urα , urα ] for [uα , uα ] for all α ∈ [0, 1], and u0 ≥ 0. We define u

all α ∈ [0, 1].

Theorem 3.1. Given r > 0, let u ˜ ∈ FC (R) be the fuzzy interval, whose α−level sets are given by

M

[˜ u]α = [uα , uα ] for all α ∈ [0, 1], and u0 ≥ 0. Then u ˜r ∈ FC (R).

Proof. Given u ˜ ∈ FC (R) such that its α−level sets are given by [uα , uα ] for all α ∈ [0, 1], from

Theorem 2.8, it follows that the functions u, u : [0, 1] −→ R, defined by u(α) = uα and u(α) = uα

ED

for all α ∈ [0, 1], respectively, they have the following properties: (i) u is bounded, non-decreasing, left-continuous on (0, 1], and it is right-continuous at 0,

PT

(ii) u is bounded, non-increasing, left-continuous on (0, 1], and it is right-continuous at 0. Considering r > 0, let g : [0, +∞) → R be given by g(x) = xr for all x ∈ [0, +∞). Then g is left-

CE

continuous on (0, +∞) and right-continuous at 0, and increasing (in particular it is non-decreasing). Thus, from (i) and (ii), it follows that (iii) the function (g ◦ u) : [0, 1] → R given by (g ◦ u)(α) = urα , is bounded, non-decreasing,

AC

left-continuous on (0, 1], and right-continuous at 0,

(iv) the function (g ◦ u) : [0, 1] → R given by (g ◦ u)(α) = urα , is bounded, non-increasing, leftcontinuous on (0, 1], and right-continuous at 0.

Therefore, from (iii) and (iv), and from Theorem 2.9, it follows that u ˜r ∈ FC (R). 12

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Corollary 3.1. Given u ˜ ∈ FC (R) such that its α−level sets are given by [uα , uα ] for all α ∈ [0, 1], and u0 ≥ 0. Then

u ˜n = u ˜⊗u ˜ ⊗ ··· ⊗ u ˜. | {z } n−factors

(11)

That is, u ˜n coincides with the multiplication of n−factors u ˜ for every n ∈ N.

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Proof. The proof of this result is given by the mathematical induction principle. (i) For n = 1, it is trivial that u ˜1 = u ˜;

(ii) Consider n = 2. Since u0 ≥ 0, from item (ii) of Theorem 2.7, it follows that uα ≥ 0 and uα ≥ 0 for all α ∈ [0, 1], and consequently, from definition of “⊗” and from definition of u ˜2 , it follows that α

α

α

[˜ u] ⊗ [˜ u] = [min{uα uα , uα uα , uα uα , uα uα }, max{uα uα , uα uα , uα uα , uα uα }]      2 α = min{u2α , uα uα , u2α }, max{u2α , uα uα , u2α } = u2α , u2α = u ˜ for all α ∈ [0, 1].

=

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[˜ u⊗u ˜]

Therefore, u ˜2 = u ˜⊗u ˜.

=

  α α α ˜⊗u ˜ ⊗ ··· ⊗ u ˜ = [˜ u ⊗ (˜ u)p ] = [˜ u] ⊗ [˜ up ] u | {z }

p−factors [min{uα upα , uα upα , uα upα , uα upα }, max{uα upα , uα upα , uα upα , uα upα }]

ED

  ˜⊗u ˜ ⊗ ··· ⊗ u ˜ u | {z } (p+1)−factors

M

(iii) Given p ∈ N such that p > 2, suppose that (11) holds for p. That is, u ˜p = u ˜⊗u ˜ ⊗ ··· ⊗ u ˜. {z } | p−factors Then   α α =

=

PT

=

h i (p+1) (p+1) (p+1) min{uα }, max{uα } , uα upα , uα upα , u(p+1) , uα upα , uα upα , uα α h i h iα , u(p+1) u(p+1) = u ˜(p+1) for all α ∈ [0, 1]. α α

CE

That is, u ˜(p+1) = u ˜⊗u ˜ ⊗ ··· ⊗ u ˜. {z } | (p+1)−factors

Therefore, from (i)-(iii) and from the mathematical induction principle, it follows that (11) holds

AC

for every n ∈ N.

Definition 3.2. Given p ∈ (0, +∞), we say that the fuzzy-interval-valued function F˜ : [a, b] → FC (R), whose α−levels Fα : [a, b] −→ KC are given by Fα (t) = [f α (t), f α (t)] for all t ∈ [a, b] and for all α ∈ [0, 1], is p-FA-integrable over [a, b] if and only if F˜ p : [a, b] −→ KC given by F˜ p (t) = (F˜ (t))p for all t ∈ [a, b], is F A−integrable over [a, b]. If p = 1 we say that F is F A−integrable over [a, b]. 13

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Theorem 3.2. Given p ∈ (0, +∞). Let F˜ : [a, b] → FC (R) be the fuzzy-interval-valued function, whose α−levels Fα : [a, b] → KC are given by Fα (t) = [f α (t), f α (t)] for all t ∈ [a, b] and for all

α ∈ [0, 1], with f 0 (t) ≥ 0 for all t ∈ [a, b]. Then F˜ is p − F A−integrable over [a, b] if and only if

f α and f α are p−integrable over [a, b] for all α ∈ [0, 1]. Moreover, if F˜ is p − F A−integrable over

[a,b]

#α "Z  p F˜ (t) dt =

Z

p

[a,b]

(f α (t)) dt,

#

p

[a,b]

(f α (t)) dt = (IA)

for all α ∈ [0, 1].

Z

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[a, b], then " Z (F A)

p

(Fα (t)) dt

(12)

[a,b]

Proof. From definition of p − F A−integrability, it follows that F˜ is p − F A−integrable over [a, b]

if and only if F˜ p is F A−integrable over [a, b]. On the other hand, from Definition 2.7, it follows

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that F˜ p is F A−integrable over [a, b] if and only if [(F˜ (·))p ]α = Fαp is IA−integrable over [a, b] for all α ∈ [0, 1]. However, since [(F˜ (t))p ]α = (Fα (t))p = [(f α (t))p , (f α (t))p ] for all t ∈ [a, b] and for all

α ∈ [0, 1], then from Definition 2.7 and from Theorem 2.12, it follows that F˜ p is F A−integrable

over [a, b] if and only if f α and f α are p−integrable over [a, b] for all α ∈ [0, 1], and if F˜ is

p

(f α (t)) dt,

M

p − F A−integrable over [a, b], then " #α "Z Z  p F˜ (t) dt = (F A)

[a,b]

[a,b]

Z

#

p

[a,b]

(f α (t)) dt = (IA)

Z

p

(Fα (t)) dt

[a,b]

ED

for all α ∈ [0, 1]. Therefore F˜ is p − F A−integrable over [a, b] if and only if f α and f α are

p−integrable over [a, b] for all α ∈ [0, 1], and if F˜ is p − F A−integrable over [a, b], then (12) holds for all α ∈ [0, 1].

PT

Corollary 3.2. Given p, q ∈ (0, +∞), let F˜ : [a, b] → FC (R) be the fuzzy-interval-valued function, whose α−levels Fα : [a, b] → KC are given by Fα (t) = [f α (t), f α (t)] for all t ∈ [a, b] and for all

CE

α ∈ [0, 1], with f 0 (t) ≥ 0 for all t ∈ [a, b]. If F˜ is p − F A−integrable over [a, b], then

AC

"

(F A)

Z

[a,b]



!q #α p ˜ F (t) dt

=

" Z

p

[a,b]

=

(IA)

(f α (t)) dt

Z

[a,b]

for all α ∈ [0, 1].

14

!q p

,

Z

!q

(Fα (t)) dt

[a,b]

p

!q #

(f α (t)) dt

ACCEPTED MANUSCRIPT

Proof. If F˜ is p − F A−integrable over [a, b], from Theorem 3.2, it follows that # " #α "Z Z Z Z  p p p (f α (t)) dt = (IA) (F A) F˜ (t) dt = (f α (t)) dt, [a,b]

[a,b]

Since f 0 (t) ≥ 0 for all t ∈ [a, b], it follows that "

(F A)

Z

[a,b]



!q #α p ˜ F (t) dt

=

Z

[a,b]

(f 0 (t))p dt ≥ 0, and consequently, it implies that

" Z

p

[a,b]

=

p

(Fα (t)) dt.

[a,b]

(IA)

!q

(f α (t)) dt

Z

p

,

Z

!q

p

[a,b]

(f α (t)) dt

(Fα (t)) dt

[a,b]

for all α ∈ [0, 1].

!q #

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[a,b]

AN US

˜ : [a, b] → FC (R) be the fuzzy-interval-valued Corollary 3.3. Given p, q ∈ (0, +∞). Let F˜ , G h i functions, whose α−levels Fα , Gα : [a, b] → KC are given, respectively, by Fα (t) = f α (t), f α (t) h i and Gα (t) = g α (t), g α (t) for all t ∈ [a, b] and for all α ∈ [0, 1], with f 0 (t) ≥ 0 and g 0 (t) ≥ 0 for ˜ are p − F A−integrable over [a, b], then all t ∈ [a, b]. If F˜ and G " !q #α " Z !q Z Z  p p ˜ F˜ (t) ⊕ G(t) (F A) dt = (f α (t) + g α (t)) dt , [a,b]

M

[a,b]

for all α ∈ [0, 1].

(IA)

[a,b]

p

!q #

(f α (t) + g α (t)) dt

(Fα (t) + Gα (t)) dt

ED

=

Z

[a,b] !q

p

˜ are p − F A−integrable over [a, b], from Theorem 3.2, it follows that f , f α , g , Proof. If F˜ and G α α

PT

and g α are p−integrable over [a, b] for all α ∈ [0, 1], and consequently, from classical real analysis, it follows that (f α + g α ) and (f α + g α ) are p−integrable over [a, b] for all α ∈ [0, 1]. Since i  iα h iα h ˜ (t) = F˜ (t) ⊕ G(t) ˜ F˜ ⊕ G = (Fα (t) + Gα (t)) = f α (t) + g α (t), f α (t) + g α (t)

CE

h

(13)

for all t ∈ [a, b] and for all α ∈ [0, 1], with f 0 (t) + g 0 (t) ≥ 0 for all t ∈ [0, 1], and (f α + g α )

AC

and (f α + g α ) are p−integrable over [a, b] for all α ∈ [0, 1], from Theorem 3.2, it follows that   ˜ : [a, b] → FC (R) is p − F A−integrable over [a, b] and, from (13) and from Corollary 3.2, F˜ ⊕ G

15

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it follows that " !q #α Z  p ˜ ˜ (F A) F (t) ⊕ G(t) dt

=

[a,b]

" Z

p

[a,b]

=

(IA)

!q

(f α (t) + g α (t)) dt

Z

Z

,

p

[a,b] !q

p

!q #

(f α (t) + g α (t)) dt

(Fα (t) + Gα (t)) dt

[a,b]

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for all α ∈ [0, 1].

˜ : [a, b] → FC (R) be the fuzzy-interval-valued functions we say that F˜ and Definition 3.3. Let F˜ , G

˜ are proportional if there is a non-negative real scalar k such that F˜ = k G. ˜ G

˜ : [a, b] → FC (R) be the fuzzy-interval-valued functions, whose α−levels Proposition 3.1. Let F˜ , G Fα , Gα : [a, b] → KC are given, respectively, by Fα (t) = [f α (t), f α (t)] and Gα (t) = [g α (t), g α (t)] for

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˜ are all t ∈ [a, b] and for all α ∈ [0, 1], with f 0 (t) ≥ 0 and g 0 (t) ≥ 0 for all t ∈ [a, b]. Then F˜ and G proportional if and only if Fα and Gα are proportional for all α ∈ [0, 1].

Proof. Given k ≥ 0, from definition of and from definition of equality of fuzzy intervals, it follows h iα h iα h iα ˜ ˜ ˜ that F˜ (t) = k G(t) for all t ∈ [a, b] if and only if F˜ (t) = k G(t) = k · G(t) for all

˜ for all t ∈ [a, b] if and only if Fα (t) = k ·Gα (t) t ∈ [a, b] and for all α ∈ [0, 1]. That is, F˜ (t) = k G(t) are proportional for all α ∈ [0, 1].

M

˜ are proportional if and only if Fα and Gα for all t ∈ [a, b] and for all α ∈ [0, 1]. Therefore, F˜ and G

Theorem 3.3. (Minkowski’s inequality for fuzzy-interval-valued functions) Given p ∈ (0, +∞). Let

ED

˜ : [a, b] → FC (R) be the fuzzy-interval-valued functions, whose α−levels Fα , Gα : [a, b] → KC F˜ , G

are given, respectively, by Fα (t) = [f α (t), f α (t)] and Gα (t) = [g α (t), g α (t)] for all t ∈ [a, b] and for

PT

all α ∈ [0, 1], with f 0 (t) ≥ 0 and g 0 (t) ≥ 0 for all t ∈ [a, b]. Then

CE

˜ are p − F A-integrable over [a, b] with p ≥ 1, it follows that (i) if F˜ and G ! p1 ! p1 Z Z Z  p  p ˜ (F A) F˜ (t) ⊕ G(t) dt ≤F (F A) F˜ (t) dt ⊕ (F A) [a,b]

[a,b]

[a,b]

! p1  p ˜ G(t) dt . (14)

AC

˜ are proportional; The equality holds if F˜ and G

˜ are p − F A-integrable over [a, b] and 0 < p < 1, it follows that (ii) if F˜ and G ! p1 ! p1 ! p1 Z Z Z  p  p  p ˜ ˜ ˜ ˜ (F A) F (t) dt ⊕ (F A) G(t) dt ≤F (F A) F (t) ⊕ G(t) dt . [a,b]

[a,b]

[a,b]

(15)

16

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˜ are proportional. The equality holds if F˜ and G Proof. (i) From definition of ≤FC , it follows that (14) holds if and only if   ! p1 α ! p1 Z Z Z   p p ˜ ˜ ˜    (F A) ≤I ⊕ (F A) (F A) F (t) ⊕ G(t) dt F (t) dt [a,b]

[a,b]

˜ G(t)

p

! p1 α

dt

(16)

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[a,b]



for all α ∈ [0, 1]. However, from definition of ⊕, from Corollary 3.3, and from Theorem 3.2, it follows that (16) is equivalent to (IA)

Z

p

(Fα (t) + Gα (t)) dt

[a,b]

! p1

≤I

(IA)

Z

p

! p1

(Fα (t)) dt

[a,b]

+

(IA)

Z

p

! p1

(Gα (t)) dt

[a,b]

(17)

AN US

for all α ∈ [0, 1]. Therefore, (14) holds if and only if (17) holds for all α ∈ [0, 1]. On the other hand, from assumptions, it follows that the interval-valued functions Fα , Gα : [0, 1] → KC given, respectively, by Fα (t) = [f α (t), f α (t)] and Gα (t) = [g α (t), g α (t)] for all t ∈ [a, b] and for all α ∈ [0, 1], they are p − IA− integrable over [a, b] for all α ∈ [0, 1], with f α (t) ≥ 0 and g α (t) ≥ 0 for all α ∈ [0, 1]. Then from item (i) of Theorem 2.5 and from Remark 2.3, it follows that (17) holds for all α ∈ [0, 1],

˜ are proportional, from and consequently, it implies that (14) also holds. Moreover, if F˜ and G

M

Proposition 3.1, it follows that Fα and Gα are proportional for all α ∈ [0, 1] and then, by using again the item (i) of Theorem 2.5 and Remark 2.3, it follows that the equality in (17) holds for all ˜ are proportional. and G

ED

α ∈ [0, 1], and consequently, the equality in (14) also holds. Thus, the equality in (14) holds if F˜

PT

(ii) Using similar arguments to those used in the proof of (i) the proof of (ii) is obtained. Theorem

3.4.

(Beckenbach’s

inequality

for

fuzzy-interval-valued

functions)

Let

CE

˜ : [a, b] → FC (R) be the fuzzy-interval-valued functions, whose α−levels Fα , Gα : [a, b] → KC F˜ , G are given, respectively, by Fα (t) = [f α (t), f α (t)] and Gα (t) = [g α (t), g α (t)] for all t ∈ [a, b] and for

˜ are F A−integrable all α ∈ [0, 1], with f 0 (t) > 0 and g 0 (t) > 0 for all t ∈ [a, b]. Then if F˜ and G

AC

and p − F A−integrable over [a, b], with 0 < p < 1, it follows that Z Z Z  p+1  p+1  p+1 ˜ ˜ (F A) F˜ (t) ⊕ G(t) dt (F A) F˜ (t) dt (F A) G(t) dt [a,b] [a,b] [a,b] Z Z Z ≤ ⊕ .  p  p  p FC ˜ ˜ (F A) F˜ (t) ⊕ G(t) dt (F A) F˜ (t) dt (F A) G(t) dt [a,b]

[a,b]

17

[a,b]

(18)



ACCEPTED MANUSCRIPT

Proof. From definition of ≤FC , it follows that (18) holds if and only if Z Z Z    p+1  p+1 α  p+1 α ˜ ˜ (F A) (F A) F˜ (t) dt (F A) F˜ (t) ⊕ G(t) dt G(t) dt     [a,b] [a,b] [a,b]     Z Z Z ⊕  p  p  p   ≤FC   ˜ ˜ ˜ ˜ (F A) F (t) ⊕ G(t) dt (F A) F (t) dt (F A) G(t) dt [a,b]

[a,b]

[a,b]

(19)

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for all α ∈ [0, 1]. However, from definition of ⊕ and definition of the division between fuzzy intervals, from Corollary 3.3, and from Theorem 3.2, it follows that (19) is equivalent to Z Z Z p+1 p+1 p+1 (IA) (IA) (Fα (t)) dt (IA) (Gα (t)) dt (Fα (t) + Gα (t)) dt [a,b] [a,b] [a,b] Z Z Z ≤I + p p p (IA) (Fα (t) + Gα (t)) dt (IA) (Fα (t)) dt (IA) (Gα (t)) dt [a,b]

[a,b]

(20)

[a,b]

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for all α ∈ [0, 1]. Therefore, (18) holds if and only if (20) holds for all α ∈ [0, 1]. On the other hand, from assumptions, it follows that the interval-valued functions Fα , Gα : [0, 1] → KC given, respectively, by Fα (t) = [f α (t), f α (t)] and Gα (t) = [g α (t), g α (t)] for all t ∈ [a, b] and for all α ∈ [0, 1], they are IA−integrable and p − IA−integrable over [a, b] for all α ∈ [0, 1], with f α (t) > 0 and g α (t) > 0 for all t ∈ [a, b] and for all α ∈ [0, 1]. Then from Theorem 2.6 and Remark 2.3, it follows

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that (20) holds for all α ∈ [0, 1], and consequently, it implies that (18) also holds. 4. Some other inequalities of type fuzzy integral

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There are other inequalities of type fuzzy integral in the literature. For example, by using the R Lebesgue measure µ and the Sugeno integral [47], which in this presentation is denoted by (S) , inequality

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Flores-Franuliˇc and Rom´ an-Flores [20] provided the following fuzzy version of the Chebyshev’s

Theorem 4.1. (Fuzzy Chebyshev’s inequality [20]) Let f, g : [0, 1] → R be two real-valued functions

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and let µ be the Lebesgue measure on R. If f, g are both continuous and strictly increasing functions,

AC

then the inequality

(S)

Z

0

1

 Z f gdµ ≥ (S)

1

0

 Z f dµ (S)

0

1

 gdµ

(21)

holds.

Rom´an-Flores et al. [42] also presented the following fuzzy version of the Jensen’s inequality, in

which the assumption of convexity of a real function is replaced by the monotonicity. 18

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Theorem 4.2. (Fuzzy Jensen inequality [42]) Let (X, Σ, Z µ) be a fuzzy measure space and let f : X → [0, +∞] be a µ−measurable function such that (S)

f dµ = p. If Φ : [0, +∞] → [0, +∞]

is a strictly increasing function such that Φ(x) ≤ x for every x ∈ [0, p], then   Z Z Φ (S) f dµ ≤ (S) (Φ ◦ f )dµ.

(22)

the Miskowski’s inequality

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Also using the Sugeno integral, Agahi and Yaghoob [5] provided the following fuzzy version of

Theorem 4.3. (Fuzzy Miskowski’s inequality [5]) Let µ be an arbitrary fuzzy Z a measure on [0, a] and (f + g)s dµ ≥ 1. If f, g let f, g : [0, a] → [0, +∞) be two µ−measurable functions such that (S) 0

are both non-decreasing functions, then the inequality (S)

Z

a s

(f + g) dµ

0

 1s

≥

holds for all 1 ≤ s < ∞.



(S)

Z

1

 1s  Z f dµ + (S) s

a

 1s g dµ

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0

0

s

(23)

The inequalities (21) and (23) have been generalized [2, 3, 13, 30, 36, 50] and, in general, such generalizations are given based on the concept of comonotonicity for real-valued functions. By using

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the comonotonicity on the real functions, Wu et al. [50] presented the following fuzzy version of the H¨older’s inequality, where (X, F, µ) denotes a fuzzy measurable space, A ∈ F, and F(X) denotes

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the set of all real functions µ−measurable with respect to F. Theorem 4.4. (Fuzzy H¨ older’s inequality [50]) Given f, g : X → [0, 1], let µ : F(X) → [0, 1] be an arbitrary fuzzy measure and let ? : [0, 1]2 → [0, 1] be continuous and nondecreasing in both

PT

arguments and bounded from below by the maximum. If f, g are comonotone, then the inequality (S)

Z

CE

A

f ? gdµ ≥



(S)

Z

A

 p1   q1 Z f p dµ ? (S) g q dµ

(24)

A

holds for all p, q ≤ 1.

Remark 4.1. ([50]) The requirement

1 p

+ 1q = 1 of the classical H¨ older inequality can be abandoned

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and the addition “+” could be replaced by a pseudo-additive operation “?”. However, f, g must to be comonotone. Inequality (22) has motived other extensions [25, 39] and recently, Abbaszadeh et al. [1] provided

a fuzzy version of the Jensen’s inequality using the Sugeno integral. However, different from (22), 19

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Abbaszadeh et al. used the concepts of convexity and concavity given by Rocaffeler [41] to present such fuzzy inequality. There are other fuzzy versions of these inequalities, which are given by means of the other kinds of fuzzy integrals, such as the versions [31, 39, 49, 51] given by means of the Choquet integral, the versions [18, 23] given by means of a seminormed fuzzy integral, and the versions [6, 38] given by

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means of the universal integral [26]. However, in all of these fuzzy versions of integral inequalities the integrands are real-valued functions. One approach for dealing with fuzzy inequalities different from those cited above is the approach used in this article, which is also found in [35], where Osuna-G´omez et al. use the Kaleva integral, the concept of s−convexity for fuzzy-interval-valued functions [16], and the fuzzy inclusion ⊆F given level-wise by the inclusion of numerical sets in

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order to present the following fuzzy version of the Hadamard’s inequality.

Theorem 4.5. (Hadamard’s inequality for fuzzy-interval-valued functions [35]) Given I ⊆ [0, ∞)

and a, b ∈ I, with a < b, let F˜ : [a, b] → FC (R) be a s−convex integrably bounded function (s ∈ (0, 1)). Then

  Z b  a+b 1 1 ˜ F (a) + F˜ (b) ⊆F (F A) F˜ (t)dt ⊆F 2s−1 F˜ . s+1 2 a (b − a)

(25)

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In this same context, by using the concept of 1−concavity for fuzzy-interval-valued functions introduced in [35] and the Kaleva integral, Costa [17] presented the following fuzzy version of

Theorem

4.6.

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Jensen’s inequality. (Jensen’s

inequality

for

fuzzy-interval-valued

functions

[17])

Let

g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given the concave fuzzy-interval-valued func-

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tion F˜ : [a, b] −→ FC (R), whose α−levels Fα : [a, b] −→ KC are given by Fα (t) = [f1α (t), f2α (t)] for

all t ∈ [a, b], where f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are

[0,1]

F˜ (g(t))dt.

(26)

[0,1]

AC

CE

Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then ! Z Z ˜ F g(t)dt ⊆F (F A)

Thus, the inequalities (14), (15), and (18) are part of the development of the fuzzy inequalities

in this conceptual framework. Next some numerical examples are presented to illustrate the theory developed in this study.

20

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5. Numerical examples ˜ : [0, 1] → R be the fuzzy-interval-valued functions whose Example 5.1. Given p = 2, let F˜ , G   i h  2 2 2 , t − α t − t+t α−levels Fα , Gα : [0, 1] → KC are given, respectively, by Fα (t) = t2 + α t+t 2 −t 2 

 3t and Gα (t) = t + α 3t for all t ∈ [0, 1] and for all α ∈ [0, 1]. Thus, F˜ and 2 − t , 2t − α 2t − 2 

 (F A)

Z

[a,b]



˜ F˜ (t) ⊕ G(t)

2

! 12 α

 =

dt

(IA)

Z

(Fα (t) + Gα (t)) dt

[a,b]

[a,b]



Z 2  12    ⊕(F A) F˜ (t) dt  

[v α , v α ] =

"r

[a,b]



˜ G(t)

2

"r

2 2 43 31 α + α+ , 15 60 30

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Z

 1 α  2    =(IA)   

 dt 

1 2 1 1 α + α+ + 120 20 5

r

M

   (F A)  

ED



! 21

2

= [uα , uα ] =

and

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˜ are 2 − F A−integrable over [0, 1], with f (t) = t2 ≥ 0 and g (t) = t ≥ 0 for all t ∈ [0, 1]. Then G 0 0

Z

2

[a,b]

1 (α + 2)2 , 12

1 2

 (Fα (t)) dt 

r

r

2 2 5 α − α+3 15 4

#

(27)



 + (IA)

Z

1 2

2

[a,b]

1 2 1 1 α − α+ + 120 12 3

 (Gα (t)) dt 

r

# 1 2 (α − 4) . 12 (28)

Since [uα , uα ] ≤I [v α , v α ] for all α ∈ [0, 1] (see Figure 1), from definition of ≤F , if follows that

˜ F˜ (t) ⊕ G(t)

2

dt

! 12

≤F

(F A)

Z

[a,b]

AC

CE

[a,b]



PT

(F A)

Z

21



F˜ (t)

2

dt

! 21

⊕

(F A)

Z

[a,b]



˜ G(t)

2

! 21

dt

.

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0,0075

0,96

0,88

0,8

0,72

0,64

0,56

0,48

0,4

0,32

0,24

0,16

0

0,08

0

0,0025

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0,005

Figure 1: Blue represents v α − uα and red represents v α − uα

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The α−levels [uα , uα ] and [v α , v α ] are illustrated in Figure 2 for all α ∈ [0, 1]. 1,75

0,96

0,88

0,8

0,72

0,64

0,56

0,48

0,32

0,24

0,16

ED 0,08

PT

0

1,25

0,4

M

1,5

Figure 2: Red represents [uα , uα ] and blue represents [v α , v α ]

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Example 5.2. Given p =

1 2,

˜ : [1, 3] → R be the fuzzy-interval-valued functions whose let F˜ , G

α−levels Fα , Gα : [1, 3] → KC are given, respectively, by Fα (t) = [t + α(5 − t), 5t − α(5t − 5)] and 

 3t ˜ are Gα (t) = t + α 3t for all t ∈ [1, 3] and for all α ∈ [0, 1]. Thus, F˜ and G 2 − t , 2t − α 2t − 2 − F A− integrable over [1, 3] with f 0 (t) = t > 0 and g 0 (t) = t > 0 for all t ∈ [1, 3]. Then

AC

1 2

 1 3 3 R    3 f (t) 2 dt = 2 (4α+1) 2 − 2 (2α+3) 2 if 0 ≤ α < 1 3α−3 3α−3 1 α uα = 1 √ R    3 f (t) 2 dt = 2 5 if α = 1 1 1 22

(29)

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1

1

r    21 √ 1 2 g α (t) dt = (1 + α) 2 3 − wα = 2 3 1 r   Z 3 √ 1 1 2 wα = (g α (t)) 2 dt = (2 − α) 2 3 − 2 3 1 3   Z 3  32  2  √ 1 2 1+ α g α (t) dt = 9 3−1 zα = 5 2 1   23  Z 3  √ 3 2 1 2− α 9 3−1 zα = (g α (t)) 2 dt = 5 2 1 √ 3 Z 3  21 √ (9α + 4) 2 3 2 − (7α + 12) 2 xα = f α (t) + g α (t) dt = 2 3α − 12 3α − 12 1 3 3 Z 3 √ (42 − 23α) 2 √ (14 − α) 2
1 xα = f α (t) + g α (t) 2 dt = 2 − 2 33α − 42 33α − 42 1 √ √ Z 3  32 5 5 1 1 2 2 2 yα = f α (t) + g α (t) dt = (9α + 4) − (7α + 12) 2 10 α − 4 10 α − 4 1 5 5 Z 3 √ (14 − α) 2 √ (42 − 23α) 2
3 − 2 . yα = f α (t) + g α (t) 2 dt = 2 110α − 140 110α − 140 1 3



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Given

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M

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Z





y y [k α , k α ] = = α, α = [xα , xα ] xα xα

[σ α , σ α ]

AC

and

CE

[y α , y α ]

=

=

(30)

(31)

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   3 √ R  (15−10α) 2 5  3 (f α (t)) 12 dt = 10 if 0 ≤ α < 1 15α−15 − 2 15α−15 1 uα = 1 √
R   3 f (t) 2 dt = 2 5 if α = 1 1 1  3  5 5 R   3 f (t) 2 dt = 2 (4α+1) 2 − 2 (2α+3) 2 if 0 ≤ α < 1 5α−5 5α−5 1 α vα = 3 √ R    3 f (t) 2 dt = 10 5 if α = 1 1 1  R 5 √ 3 (15−10α) 2  3 f (t)
2 dt = 50 5 − 2 if 0 ≤ α < 1 α 25α−25 25α−25 1 vα = 3  R 3 f (t)
2 dt = 10√5 if α = 1

(IA) (IA)

Z

Z[a,b]

3

(Fα (t) + Gα (t)) 2 dt 1

(Fα (t) + Gα (t)) 2 dt

[a,b]

      [v α , v α ] z zα v zα [z , z α ] v vα z vα + α, = α + α, + + α = α, [uα , uα ] [wα , wα ] uα uα wα wα uα wα uα wα Z Z 3 3 (IA) (Fα (t)) 2 dt (IA) (Gα (t)) 2 dt [a,b] [a,b] Z Z + , 1 1 (IA) (Fα (t)) 2 dt (IA) (Gα (t)) 2 dt [a,b]

[a,b]

23

(32)

(33) (34)

(35) (36) (37) (38) (39) (40)

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then from (29) - (40), it follows that √

5

√

5

2 1 2 (9α + 4) 2 − 10 α−4 (7α + 12) , 3 √ (14−α) 32 √ 2 2 33α−42 − 2 (42−23α) 33α−42

σα =

and σα =

      

        

5 5 (2α+3) 2 (4α+1) 2 5α−5 −2 5α−5 3 √ (15−10α) 2 5 10 15α−15 −2 15α−15

2

√ 3 9√ 3−1 5 2 3− 2 3

3 2

2 (9α+4) 3α−12 −

5 (15−10α) 2 25α−25 3 3 (4α+1) 2 (2α+3) 2 2 3α−3 −2 3α−3 √

√

√

5 2

2 (42−23α) 110α−140

2 3α−12

3

,

(7α + 12) 2

3

!

+

2 5

√ 9√ 3−1 2 2− 2 α 2 3− 3

1 2 (√ 2 α+1) 1

+ 5 if α = 1

5 −2 50 25α−25

√ 3 9√ 3−1 5 2 3− 2 3

kα = √

5 2

(14−α) 2 110α−140 −

+ √ 12 5

2 − 12 α

2 α+1

+ 5 if α = 1


32

, if 0 ≤ α < 1

,

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kα =

√ 2 1 10 α−4

√ 9√ 3−1 2 3− 23

if 0 ≤ α < 1

.

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Since [k α , k α ] ≤I [σ α , σ α ] for all α ∈ [0, 1] (see Figure 3), from definition of ≤F , if follows that Z Z Z   32   23   32 ˜ ˜ F˜ (t) ⊕ G(t) G(t) (F A) dt F˜ (t) dt (F A) dt (F A) [a,b] [a,b] [a,b] ≤FC ⊕ (41) Z Z Z  12  12  21 .    ˜ ˜ F˜ (t) ⊕ G(t) F˜ (t) dt (F A) G(t) (F A) dt (F A) dt [a,b]

[a,b]

[a,b]

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The α−levels of the terms in (41) are represented in Figure 4 2

1,2

0,8

PT

0,4

ED

1,6

0

0,96

0,88

0,8

0,72

0,64

0,56

0,48

0,4

0,32

0,24

0,16

0,08

0

Figure 3: Blue represents σ α − kα and red represents σ α − kα

AC

CE

-0,4

24

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25

0,8

0,72

0,64

0,56

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0,48

0,4

0,32

0,24

0,16

0,08

0

5

0,96

10

0,88

15

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20

Figure 4: Blue represents [σ α , σ α ] and red represents [kα , kα ]

Conclusion

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The Minkowski inequality presented in this study allows one to deal with classes of problems semantically different from those studied with the fuzzy versions of the Minkowski’s inequality pre-

ED

sented previously in the literature. Moreover, with the introduction of the concept of p-integrability for fuzzy-interval-valued functions, an interesting avenue has been opened to the study of Lp-type fuzzy intervals spaces. On the other hand, the concept of width-integral of convex bodies allows

PT

for an important application of Beckenbach’s inequality to convex geometry on Rn (see [52]). In this context we wish to extend these ideas to the study of some problems connected with convex

CE

geometry in the space FCn , a topic which will be subject of forthcoming study.

AC

Acknowledgments

The authors are grateful to the reviewers and the editor for the constructive criticisms regarding this article. The authors are also grateful to Professor Weldon A. Lodwick for his suggestions that helped to improve this presentation. Moreover, the authors are greatly acknowledge the financial

25

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support of the Coordination for the Improvement of Higher Education Personnel (CAPES) and the financial support of the CONICYT-CHILE via project FONDECYT 1151159, which helped in the develop this study. References

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