Jensen's inequality type integral for fuzzy-interval-valued functions

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Jensen’s inequality type integral for fuzzy-interval-valued functions T. M. Costa Department of Mathematical and Statistical Sciences, University of Colorado Denver, Denver, CO 80204, USA Received 21 May 2016; received in revised form 28 January 2017; accepted 5 February 2017

Abstract This study presents fuzzy versions of Jensen inequalities type integral for convex and concave fuzzy-interval-valued functions. To this end, the concepts of fuzzy inclusion order relation, convexity, and concavity for fuzzy-interval-valued functions are used. Some examples showing the applicability of the theory developed in this study are presented. Since the fuzzy results are obtained through level sets of fuzzy-interval elements, the versions of these results in the interval context are presented here for the first time. © 2017 Elsevier B.V. All rights reserved. Keywords: Fuzzy interval space; Fuzzy-interval integrability; Fuzzy Jensen’s inequalities type integral

1. Introduction It is known that Jensen’s inequalities [11,20] play an important role in optimization (see, e.g., [15]), in probability theory (see, e.g., [17]) among others filed of mathematics. The important concepts involved in the Jensen inequality are order relation, convexity, concavity and, in some cases, integrability. All these concepts are developed in this study since these are the associated concepts needed. Jensen’s inequality has had an increasing number of versions over the years in several context such as measurable spaces and, in particular, probability spaces [4]. This last version offers an interesting way to work with the Jensen inequality using random data. That is, probability space offers tools to deal with Jensen’s inequality composed of data with probabilistic uncertainty. It is also known that interval and fuzzy analysis are fields in the mathematics, which provide tools to deal with data with uncertainty. In general, interval analysis is used to deal with the models whose data are composed of inaccuracies, which can arise from some type of measurement. On the other hand, the fuzzy analysis can be used to deal with the models which were obtained without total information about the problem. One can find, in the literature, some fuzzy versions of Jensen’s inequality type integral (see, e.g., [13,16,19,21–23]). In general, the difference between these is the type of integrals used. For example, [19] uses the Sugeno integral, in [16,21] a concept of pseudo-integral is used, and in [13,22,23] the Choquet integral is used. The characteristic in E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2017.02.001 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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common between these articles is that, in the Jensen’s inequality, the convexity (or concavity) hypothesis is required for real-valued functions and not for fuzzy-valued functions. This study proposes a new fuzzy version of Jensen’s inequality type integral, which is the main result of this presentation. To be more precise, in this study the fuzzy integral [12] is used and, differently from the articles mentioned above, the convexity (or concavity) hypothesis pertains to fuzzy-interval-valued functions. Thus, a new class of Jensen’s inequality is generated and this enables us to attack a class of problems that is different from the class studied in the previous articles. The fuzzy results presented in this research are developed through the level sets of fuzzy-interval elements, thus, the interval results related to these fuzzy results are developed first. The development of fuzzy results through the level sets of fuzzy-interval elements allows us to obtain an equivalence between Jensen’s inequality for fuzzy-intervalvalued functions and two classical real Jensen’s inequalities. Thus, it is possible to deal with Jensen’s inequality for fuzzy-interval-valued functions using only tools from real analysis. This study is organized as follows. In Section 2 some concepts of interval analysis such as limit, continuity, and integrability for interval-valued functions and some results involving these concepts are reviewed. Section 3 reviewed the concepts of convexity and concavity for interval-valued functions, both used in Jensen’s inequality for interval-valued functions. In Section 4 some concepts of fuzzy analysis such as, continuity and integrability for fuzzy-interval-valued functions and some results related to these concepts are presented. Moreover, the fuzzy order relation, which is used in the concepts of convexity and concavity for fuzzy-interval-valued functions is found in this section. In Section 5 our main result for Jensen’s inequality for fuzzy-interval-valued functions together with some examples are presented. Finally, Section 6 presents an overview about this study and some additional comments. 2. Preliminaries This research uses KC to denote the space composed of bounded closed real intervals equipped with the algebraic operations “+” and “·” given, respectively, by  [λa1 , λa2 ] if 0 ≤ λ [a1 , a2 ] + [b1 , b2 ] = [a1 + b1 , a2 + b2 ] and λ · [a1 , a2 ] = [λa2 , λa1 ] if λ < 0, that is, the space KC = {[a1 , a2 ] : a1 , a2 ∈ R, and a1 ≤ a2 } equipped with “+” and “·”. Given A and B nonempty subsets of R the Haudorff 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 between the a∈A

nonempty subsets of R, A and B, is defined by dH (A, B) = max{dH ∗ (B, A), dH ∗ (A, B)}. It is known that the Pompeiu–Hausdorff distance dH is a metric and the Pompeiu–Hausdorff metric dH , where dH : KC × KC −→ R is given by dH ([a1 , a2 ], [b1 , b2 ]) = max{|a1 − b1 |, |a2 − b2 |}. Moreover. It is known that (KC , dH ) is a complete metric space [1,9]. A map F : U ⊆ R −→ KC with F (t) = [f1 (t), f2 (t)], where f1 , f2 : U −→ R are real functions with f1 (t) ≤ f2 (t) for all t ∈ U , it is called an interval-valued function. Given the interval-valued functions F, G : U ⊆ R −→ KC and λ ∈ R, the notation used for the algebraic operations between interval-valued functions is given by (F + G)(t) := F (t) + G(t) and (λ · F )(t) := λ · F (t). Definition 2.1. [1] Let F : U ⊆ R −→ KC be an interval-valued function. L ∈ KC is 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. [1] Let F : U ⊆ R −→ KC be an interval-valued function with F (t) = [f1 (t), f2 (t)], where f1 , f2 : U −→ R. Then L = [l1 , l2 ] ∈ KC is a limit of F at t0 ∈ U if and only if li is the limit of fi at t0 , i ∈ {1, 2}. Moreover, if L is the limit of F at t0 , then   lim F (t) = lim f1 (t), lim f2 (t) . t→t0

t→t0

t→t0

Definition 2.2. [1] An interval-valued function F : U ⊆ R −→ KC is called dH -continuous at t0 ∈ U if lim F (t) = F (t0 ).

t→t0

If F is dH -continuous at all t ∈ U , then we say that F is dH -continuous. Theorem 2.2. [1] Let F : U ⊆ R −→ KC be an interval-valued function with F (t) = [f1 (t), f2 (t)], where f1 , f2 : U −→ R. Then F is dH -continuous at t0 ∈ U if and only if fi is continuous at t0 , i = 1, 2. Moreover, if F is dH -continuous at t0 , then     lim F (t) = lim f1 (t), lim f1 (t) = f1 (t0 ), f2 (t0 ) . t→t0

t→t0

t→t0

Remark 2.1. If “⊆” is the usual inclusion of sets, then [a1 , a2 ] ⊆ [b1 , b2 ] if and only if b1 ≤ a1 and a2 ≤ b2 .

(1)

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 all Lebesgue-measurable subsets of R and B denotes the σ -algebra composed of all Borel-measurable subsets of R. F is said to be integrably bounded on [a, b] if and only if there exists a Lebesgue-integrable function h : [a, b] −→ [0, +∞) such that |x| ≤ h(t) for all x and t such that x ∈ F (t). [3] Given an interval-valued function F : [a, b] −→ KC , the Aumann integral ((I A)-integral, for short) of F over [a, b] is defined by ⎧ ⎫ ⎪ ⎪  ⎨ ⎬ (I A) F (t)dt = f (t)dt : f ∈ S(F ) , (2) ⎪ ⎪ ⎩ ⎭ [a,b]

[a,b]

where S(F ) := {f ∈ L1 ([a, b]) : f (t) ∈ F (t) for almost every t ∈ [a, b]} and L1 ([a, b]) is the space of all functions f : [a, b] −→ R that are Lebesgue-integrable over [a, b]. We say that the (IA)-integral of F over [a, b] exists (or that F is (I A)-integrable over [a, b]) if S(F ) = ∅. Theorem 2.3. (see, e.g., [9,12]) If an interval-valued function F : [a, b] −→ KC is measurable and integrably bounded on [a, b], then F is (I A)-integral over [a, b]. Theorem 2.4. (see, e.g., [9,12]) Let F : [a, b] −→ KC be an interval-valued function given by F (t) = [f1 (t), f2 (t)] with f1 , f2 : [a, b] −→ R. Then F is I A-integrable over [a, b] if and only if f1 and f2 belongs to L1 ([a, b]). Moreover, if F is (I A)-integrable over [a, b], then ⎡ ⎤    ⎢ ⎥ (3) (I A) F (t)dt = ⎣ f1 (t)dt, f2 (t)dt ⎦ . [a,b]

[a,b]

[a,b]

Remark 2.2. From Theorem 2.4 and from Theorem 2.2, it is easy to see that, if F is dH -continuous, then F is (I A)-integrable over [a, b].

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3. Jensen’s inequality for interval-valued functions This section presents Jensen’s inequality for the class of convex interval-valued functions and for the class of concave interval-valued functions. The results presented in this section are necessary tools for the fuzzy version of Jensen’s inequality. Given a real-valued function f : U −→ R, where U ⊆ Rn is a convex set, f is said to be a convex real-valued function if and only if f ((1 − λ)t1 + λt2 ) ≤ (1 − λ)f (t1 ) + λf (t2 ) ∀t1 , t2 ∈ U, ∀λ ∈ [0, 1]

(4)

and, f is said to be a concave real-valued function if and only if (1 − λ)f (t1 ) + λf (t2 ) ≤ f ((1 − λ)t1 + λt2 ) ∀t1 , t2 ∈ U, ∀λ ∈ [0, 1].

(5)

Definition 3.1. (see, e.g., [6]) Given an interval-valued function F : U −→ KC , where U ⊆ R is a convex set, we say that F is a convex interval-valued function if and only if (1 − λ) · F (t1 ) + λ · F (t2 ) ⊆ F ((1 − λ)t1 + λt2 )

∀t1 , t2 ∈ U, ∀λ ∈ [0, 1].

(6)

Definition 3.2. (see, e.g., [6]) Given an interval-valued function F : U −→ KC , where U ⊆ R is a convex set, we say that F is a concave interval-valued function if and only if F ((1 − λ)t1 + λt2 ) ⊆ (1 − λ) · F (t1 ) + λ · F (t2 )

∀t1 , t2 ∈ U, ∀λ ∈ [0, 1].

(7)

Theorem 3.1. Let F : U −→ KC be an interval-valued function given by F (t) = [f1 (t), f2 (t)] with f1 , f2 : U −→ R and U ⊆ R convex. Then F is a convex interval-valued function if and only if f1 is a convex real-valued function and f2 is a concave real-valued function. Proof. Consider t1 , t2 ∈ [a, b] and λ ∈ [0, 1]. Then, F is a convex interval-valued function if and only if λ · F (t1 ) + (1 − λ) · F (t2 ) ⊆ F (λt1 + (1 − λ)t2 ), that is, 

   λf1 (t1 ) + (1 − λ)f1 (t2 ), λf2 (t1 ) + (1 − λ)f2 (t2 ) ⊆ f1 (λt1 + (1 − λ)t2 ), f2 (λt1 + (1 − λ)t2 ) .

This is equivalent to f1 (λt1 + (1 − λ)t2 ) ≤ λf1 (t1 ) + (1 − λ)f1 (t2 ) and λf2 (t1 ) + (1 − λ)f2 (t2 ) ≤ f2 (λt1 + (1 − λ)t2 ). This means that f1 is a convex function real valued and f2 is a concave real-valued function. Therefore, F is a convex interval-valued function if and only if f1 is a convex real-valued function and f2 is a concave real-valued function. 2 Theorem 3.2. Let F : U −→ KC be an interval-valued function given by F (t) = [f1 (t), f2 (t)], where f1 , f2 : U −→ R and U ⊆ R convex. Then F is a concave interval-valued function if and only if f1 is a concave real-valued function and f2 is a convex real-valued function. Proof. The proof is similar to the proof of Theorem 3.1.

2

Theorem 3.3. Let F : [a, b] −→ KC be an interval-valued function, with F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R. If F is convex or concave, then F is (I A)-integrable over [a, b]. Proof. Suppose that F is convex or concave. From Theorem 3.1 and from Theorem 3.2, it follows that f1 and f2 are continuous. Then, from Theorem 2.2 and from Remark 2.2, it follows that F is I A-integrable over [a, b]. 2

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Remark 3.1. Let F : [a, b] −→ KC be an interval-valued function, with F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R. Given a real function g : U −→ R such that g(U ) ⊆ [a, b], then (F ◦ g)means the interval-valued  function (F ◦ g) : U   −→ KC that associates each t ∈ U to the interval (F ◦ g) (t) = (f1 ◦ g)(t), (f2 ◦ g)(t) = f1 (g(t)), f2 (g(t)) ∈ KC . Theorem 3.4. (Jensen’s interval inequality) Let g : [0, 1] −→ (a, b) be a Lebesgue-integrable real-valued function. Given a concave interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R are real functions such that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1], then ⎛ ⎞   ⎜ ⎟ F⎝ g(t)dt ⎠ ⊆ (I A) F (g(t))dt. (8) [0,1]

[0,1]

Proof. From the assumptions that (f1 ◦ g) and (f2 ◦ g) are real-valued functions Lebesgue-integrable over [0, 1], it implies that   S(F ◦ g) = (f ◦ g) ∈ L1 ([0, 1]) : (f ◦ g)(t) ∈ (F ◦ g) (t) for almost every t ∈ [0, 1] = ∅. That is, (F ◦ g) : [0, 1] −→ KC is (I A)-integrable over [0, 1]. From Theorem 2.4, it follows that ⎡ ⎤    ⎢ ⎥ (I A) F (g(t))dt = ⎣ f1 (g(t))dt, f2 (g(t))dt ⎦ . [0,1]

[0,1]

(9)

[0,1]

Thus, the existence of the right side of (8) is proved. Also from the assumption that F is concave and from Theorem 3.2, it follows that f1 is concave and f2 is convex. In particular, f1 is concave on (a, b) and f2 is convex on (a, b). Moreover, g, (f1 ◦ g), and (f2 ◦ g) are real-valued functions Lebesgue-integrable over [0, 1]. Then from the classical Jensen’s inequality [20], it follows that ⎞ ⎞ ⎛ ⎛     ⎟ ⎟ ⎜ ⎜ f1 (g(t))dt ≤ f1 ⎝ g(t)dt ⎠ and f2 ⎝ g(t)dt ⎠ ≤ f2 (g(t))dt. (10) [0,1]

[0,1]

[0,1]

[0,1]

⎛

⎜ Therefore, from (10), from definition of ⊆ and from (9), it follows that F ⎝



⎞ ⎟ g(t)dt ⎠ ⊆ (I A)

[0,1]

 F (g(t))dt .

2

[0,1]

Corollary 3.1. Let g : [0, 1] −→ (a, b) be a Lebesgue-integrable real-valued function. Given an (I A)-integrable interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R are real functions such that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1], then ⎛ ⎞   ⎜ ⎟ F⎝ g(t)dt ⎠ ⊆ (I A) F (g(t))dt if and only if [0,1]

 [0,1]

⎛

⎜ f1 (g(t))dt ≤ f1 ⎝

[0,1]



[0,1]

⎞

⎛

⎟ g(t)dt ⎠

⎜ f2 ⎝

and



[0,1]

⎞ ⎟ g(t)dt ⎠ ≤

 f2 (g(t))dt.

[0,1]

Proof. This result follows from Theorem 2.4, from definition of ⊆ and from the proof of Theorem 3.4.

2

Theorem 3.5. (Reverse Jensen’s interval inequality) Let g : [0, 1] −→ (a, b) be a Lebesgue-integrable realvalued function. Given a convex interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R are real functions such that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1], it follows that

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 (I A)

⎛ ⎜ F (g(t))dt ⊆ F ⎝

[0,1]



⎞ ⎟ g(t)dt ⎠ .

(11)

[0,1]

Proof. From the assumption that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1] it results that   S(F ◦ g) = (f ◦ g) ∈ L1 ([0, 1]) : (f ◦ g)(t) ∈ (F ◦ g) (t) for almost every t ∈ [0, 1] = ∅. That is, F ◦ g : [0, 1] −→ KC is I A-integrable over [0, 1] and, from Theorem 2.4, it follows that ⎡ ⎤    ⎢ ⎥ (I A) F (g(t))dt = ⎣ f1 (g(t))dt, f2 (g(t))dt ⎦ . [0,1]

[0,1]

(12)

[0,1]

Thus, the existence of the left side of (11) is proved. On the other hand, from the assumption that F is convex and from Theorem 3.2, it follows that f1 is convex on [a, b] and f2 is concave. In particular, f1 is convex on (a, b) and f2 is concave on (a, b). Thus, since g, (f1 ◦ g), and (f2 ◦ g) are Lebesgue-integrable over [0, 1], then, from classical Jensen’s inequality [20], it follows that ⎞ ⎞ ⎛ ⎛     ⎟ ⎟ ⎜ ⎜ (13) f1 ⎝ g(t)dt ⎠ ≤ f1 (g(t))dt and f2 (g(t))dt ≤ f2 ⎝ g(t)dt ⎠ . [0,1]

[0,1]

[0,1]

[0,1]

⎛



⎜ F (g(t))dt ⊆ F ⎝

Therefore, from (13), from definition of ⊆ and from (12), it follows that (I A) [0,1]



⎞ ⎟ g(t)dt ⎠.

2

[0,1]

Corollary 3.2. Let g : [0, 1] −→ (a, b) be a Lebesgue-integrable real-valued function. Given an (I A)-integrable interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [a, b] −→ R are real functions such that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1], then ⎛ ⎞   ⎜ ⎟ (I A) F (g(t))dt ⊆ F ⎝ g(t)dt ⎠ if and only if ⎛ ⎜ f1 ⎝

[0,1]



[0,1]

[0,1]

⎞ ⎟ g(t)dt ⎠ ≤



 f1 (g(t))dt

[0,1]

and [0,1]

⎛ ⎜ f2 (g(t))dt ≤ f2 ⎝



⎞ ⎟ g(t)dt ⎠ .

[0,1]

Proof. This result follows from Theorem 2.4, from the definition of ⊆ and from the proof of Theorem 3.5.

2

Corollary 3.3. Let g : [0, 1] −→ (a, b) be a Lebesgue-integrable real-valued function. Given an interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [c, d] −→ R are real functions such that (f1 ◦ g) and (f2 ◦ g) are Lebesgue-integrable over [0, 1]. If F is convex and concave on [a, b], it follows that ⎛ ⎞   ⎜ ⎟ (14) (I A) F (g(t))dt = F ⎝ g(t)dt ⎠ . [0,1]

[0,1]

Proof. This result directly follows from Theorem 3.4 and from Theorem 3.5.

2

Corollary 3.4. Let g : [0, 1] −→ (a, b) be a continuous real-valued function. Given an interval-valued function F : [a, b] −→ KC by F (t) = [f1 (t), f2 (t)], where f1 , f2 : [c, d] −→ R are real functions, it follows that

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(i) if F is concave, then is ensured the inequality (8); (ii) if F is convex, then is ensured the inequality (11); (iii) if F is concave and convex, then is ensured the equality (14). Proof. If F is concave or convex, from Theorem 3.1 and from and Theorem 3.2, it follows that f1 and f2 are continuous. Since g is continuous, it follows that f1 ◦ g and f2 ◦ g are Lesbesgue-integrable over [0, 1]. Then if F is concave we have the assumptions of Theorem 3.4 and thus is ensured the inequality (8). If we have that F is convex we have the assumptions of Theorem 3.5 and thus is ensured the inequality (11). On the other hand, if we have that F is concave and convex, then we have the assumptions of Corollary 3.3, then is ensured the equality (14). 2 Remark 3.2. Given the function s : R × KC −→ R defined by s(x, A) = sup{x · a} a∈A

and denoting by C(S 0 ) the Banach space of continuous functions f : S 0 −→ R with the supremum norm f ∞ = sup{|f (x)|; x ∈ S 0 }, where S 0 = {x ∈ R; |x| = 1} = {−1, 1}, it is known (see [7,9]) that the function j : KC −→ C(S 0 ), defined by j (A) = s(·, A) for each A ∈ KC , has the following properties: 1. j is injective; 2. j (αA + βB) = αj (A) + βj (B) for all α ≥ 0, β ≥ 0 and A, B ∈ KC ; 3. dH (A, B) = j (A) − j (B) ∞ for all A, B ∈ KC . Another known result (see [2,9]) is that, for any fixed x ∈ R, s(x, A) ≤ s(x, B) whenever A ⊆ B.

(15)

Moreover, it is also known (see Proposition 8.6.2 in [2] and Theorem 4.3.2 in [9]) that an interval-valued function G : [0, 1] −→ KC is (IA)-integrable over [0, 1] if and only if s(·, G(·)) : [0, 1] −→ C(S 0 ) is Bochner integrable ((B)-integral, for short). In this case, it follows that ⎛ ⎞     ⎜ ⎟ s ⎝ · , (I A) G(t)dt ⎠ = (B) s · , G(t) dt, (16) [0,1]

[0,1]

and from (16), it follows that ⎛ ⎞     ⎜ ⎟ s ⎝x, (I A) G(t)dt ⎠ = s x, G(t) dt, [0,1]

(17)

[0,1]

for all x ∈ S 0 , where the integral on the right side is the Lebesgue-integral of real-valued functions. That is, we obtain the pointwise equality in (17). Then given the assumptions of Theorem 3.4 (Jensen inequality for interval-valued function), it follows that ⎛ ⎞   ⎜ ⎟ F⎝ g(t)dt ⎠ ⊆ (I A) F (g(t))dt. (18) [0,1]

[0,1]

Then from (18) (15), (16), and from (17), we can reduces the Jensen inequality for interval-valued function to classical Jensen inequality for real-valued function given by ⎛ ⎛ ⎞⎞     ⎜ ⎜ ⎟⎟ s ⎝x, F ⎝ g(t)dt ⎠⎠ ≤ s x, F (g(t)) dt (19) [0,1]

[0,1]

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⎛ ⎜ for each x ∈ S 0 . In particular, for x = −1 we have that −f1 ⎝ ⎛ ⎜ have that f2 ⎝



[0,1]

⎞ ⎟ g(t)dt ⎠ dt ≤



[0,1]

⎞ ⎟ g(t)dt ⎠ dt ≤

 −f1 (g(t))dt and for x = 1 we [0,1]

 f2 (g(t))dt, where each one of these cases is a particular case of the Jensen [0,1]

inequality for real-valued function given in [20] since by assumptions (−f1 ) and f2 are convex real-valued functions. Therefore, the Jensen inequality for interval-valued functions presented here can be reduced to Jensen inequality for real-valued functions. By using similar arguments we also can reduce the Reverse Jensen inequality for interval-valued function presented here to Jensen inequality for real-valued functions and if we use the specific propriety of the function s that says “given A, B ∈ KC , then A = B if and only if s(x, A) = s(x, B) for all x ∈ R” (see [9]), then the equality obtained in Corollary 3.3 can be reduced to a real-valued version. 4. Fuzzy interval space: preliminaries This section recalls some concepts and results from fuzzy literature, which are necessary to development of the Jensen’s inequality for fuzzy-interval-valued functions. Definition 4.1. (see e.g., [5,9]) A fuzzy subset of R is a function u˜ : R −→ [0, 1]. A fuzzy subset u˜ of R is called a fuzzy interval if it satisfies the following properties: (i) u˜ is normal, i.e., there exists x¯ ∈ R such that u( ˜ x) ¯ = 1; (ii) u˜ is fuzzy convex, i.e., min{u(x ˜ 1 ), u(x ˜ 2 )} ≤ u(λx ˜ 1 + (1 − λ)x2 ) for all x1 , x2 ∈ R and for all λ ∈ [0, 1]; ˜ − u(y) ˜ < (iii) u˜ is upper semicontinuous on R, i.e., given y ∈ R, for every  > 0 there exists δ > 0 such that u(x) for all x ∈ R with |x − y| < δ; (iv) u˜ is compactly supported, i.e., cl{x ∈ R : 0 < u(x)} ˜ is compact, where cl(A) denotes the closure of the set A. Here, F(R) denotes the space of real fuzzy intervals. Given u, ˜ v˜ ∈ F(R) and λ ∈ R, addition u˜ ⊕ v˜ and scalar multiplication λ  u˜ are defined, respectively, by  x  u˜ λ if λ = 0 (u˜ ⊕ v)(x) ˜ = sup {u(y), ˜ v(z)} ˜ and (λ  u)(x) ˜ = , χ˜ {0} (x) if λ = 0 y+z=x where χ˜ {0} is the characteristic function of {0}. Definition 4.2. (see e.g., [5,9]) Given u˜ ∈ F(R), the level sets of u˜ are given by [u] ˜ α = {x ∈ R : α ≤ u(x)} ˜ for all 0 α ∈ (0, 1] and by [u] ˜ = cl{x ∈ R : 0 < u(x)}. ˜ These sets are called α-level sets of u˜ for all α ∈ [0, 1]. In particular, ˜ 1 is called the core of u. ˜ [u] ˜ 0 is called the support of u˜ and [u] Theorem 4.1. ([14]) If u˜ ∈ F(R) and [u] ˜ α are its α-levels, then: (i) (ii) (iii) (iv)

[u] ˜ α is a closed interval [u] ˜ α = [uα1 , uα2 ] for all α ∈ [0, 1]. ˜ α2 ⊆ [u] ˜ α1 . If 0 ≤ α1 ≤ α2 ≤ 1, then [u] For any sequence (αn )n∈N which converges from below to α ∈ (0, 1], wehave ∩∞ ˜ αn = [u] ˜ α. n=1 [u] ∞ αn 0 = [u] ˜ . ˜ For any sequence (αn )n∈N which converges from above to 0, we have cl ∪n=1 [u]

Proposition 4.1. (see e.g., [9]) Let u, ˜ v˜ ∈ F(R) be given such that [u] ˜ α = [uα1 , uα2 ] and [v] ˜ α = [v1α , v2α ] for all α ∈ [0, 1]. Then, ˜ α for all α ∈ [0, 1]; (i) u˜ = v˜ if and only if [u] ˜ α = [v]

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(ii) [u˜ ⊕ v] ˜ α = [uα1 + v1α , uα2 + v2α ] and [λ  u] ˜ α = λ · [uα1 , uα2 ], for all α ∈ [0, 1], and λ ∈ R. Next two results provide necessary and sufficient conditions to characterize an interval fuzzy via real-valued functions. Theorem 4.2. ([5,10]) Let u be a fuzzy interval and let [u] ˜ α = [uα1 , uα2 ]. Then the functions u1 , u2 : [0, 1] −→ R, defining the endpoints of the α-level sets, satisfy the following conditions: (i) u1 (α) = uα1 ∈ R is bounded, non-decreasing, left-continuous function on (0, 1] and it is right-continuous at 0. (ii) u2 (α) = uα2 ∈ R is bounded, non-increasing, left-continuous function on (0, 1] and it is right-continuous at 0. (iii) u1 (1) ≤ u2 (1). Theorem 4.3. ([5,10]) Consider the functions u1 , u2 : [0, 1] −→ R satisfying the following conditions: (i) u1 (α) = uα1 ∈ R is bounded, non-decreasing, left-continuous function on (0, 1] and it is right-continuous at 0. (ii) u2 (α) = uα2 ∈ R is bounded, non-increasing, left-continuous function on (0, 1] and it is right-continuous at 0. (iii) u1 (1) ≤ u2 (1). ˜ α = [uα1 , uα2 ] for every α ∈ [0, 1]. Then there is a fuzzy interval u˜ ∈ F(R) such that [u] Proposition 4.2. The relation ⊆F given in F(R) × F(R) by u˜ ⊆F v˜ if and only if [u] ˜ α ⊆ [v] ˜ α for all α ∈ [0, 1],

(20)

is a partial order relation. Proof. This result directly follows from Remark 2.1.

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Theorem 4.4. (see. e.g, [9]) The space F(R) equipped with the supremum metric, i.e.,  α  ˜ , [v] d∞ (u, ˜ v) ˜ = sup dH [u] ˜ α , α∈[0,1]

is a complete metric space. A fuzzy-interval-valued map F˜ : U ⊆ R −→ F(R) is called a fuzzy-interval-valued function. Definition 4.3. (see e.g., [5,9]) A fuzzy-interval-valued function F˜ : U ⊆ R −→ F(R) is said to be continuous at t0 ∈ U if for any  > 0, there exists δ(, t0 ) = δ > 0 such that  d∞ F˜ (t), F˜ (t0 ) < , for all t ∈ U with |t − t0 | < δ. Remark 4.1. Given a fuzzy-interval-valued function F˜ : U ⊆ R −→ F(R), from (i) of Theorem 4.1, it follows that for each α ∈ [0, 1] an interval-valued function Fα : U −→ KC can be defined by Fα (t) = [F˜ (t)]α for all t ∈ U . On the other hand, for each interval valued function Fα : U ⊆ R −→ KC two real-valued functions f1α , f2α : U −→ R can be defined such that Fα (t) = [f1α (t), f2α (t)] for each t ∈ U . Given these considerations, from definition of d∞ and from Definition 4.3, it follows that F˜ is continuous at t0 ∈ U if and only if Fα is continuous at t0 ∈ U for all α ∈ [0, 1] and, from Theorem 2.2, this is equivalent to f1α and f2α being continuous at t0 ∈ U for all α ∈ [0, 1]. An fuzzy-interval-valued function F˜ : [a, b] −→ F(R) is said to be integrably bounded if there exists a Lebesgueintegrable function h : [a, b] −→ [0, +∞) such that |x| ≤ h(t) for all x and t such that x ∈ F0 (t). [12] A fuzzy-interval-valued function F˜ : [a, b] −→ F(R) is said to be strongly measurable if and only if the interval-valued function Fα : [a, b] −→ KC is measurable for all α ∈ [0, 1]. This concept is equivalent to the concept

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of measurability given in [18] and such equivalence can be obtained through Theorem III-2 and Theorem III-30 given in [7]. Definition 4.4. ([12]) Let F˜ : [a, b] −→ F(R) be a fuzzy-interval-valued function. The integral of F˜ over [a, b], b  ˜ denoted by (FA) F (t)dt or (FA) F˜ (t)dt, is defined level-wise by [a,b]

⎡ ⎢ (FA) ⎣



[a,b]

a

⎤α ⎥ F˜ (t)dt ⎦ =

 Fα (t)dt =

[a,b]

⎧ ⎪ ⎨ ⎪ ⎩

[a,b]

⎫ ⎪ ⎬

f (t)dt : f ∈ S (Fα ) , ⎪ ⎭

for all α ∈ [0, 1], where Fα (t) is the α-level of F˜ (t). F˜ is integrable over [a, b] if and only if (FA)



F˜ (t)dt ∈ F(R).

[a,b]

[12] It is known that the concept of fuzzy integral given in Definition 4.4 is equivalent to the concept of fuzzy Aumann integral defined in [18], for this reason that notation for the integral of F˜ over [a, b] given in Definition 4.4 is used in this presentation. Theorem 4.5. ([18]) Given a fuzzy-interval-valued function F˜ : [a, b] −→ F(R), if F˜ is strongly measurable and integrably bounded, then F˜ is integrable over [a, b]. A consequence of Definition 4.4, Remark 4.1, and Remark 2.2 is that, given a fuzzy-interval-valued function F˜ : [a, b] −→ F(R), whose α-levels are given by Fα : U −→ KC , if Fα is dH -continuous for all α ∈ [0, 1], then F˜ is integrable over [a, b]. Theorem 4.6. (see, e.g., [9,12]) Let F˜ : [a, b]−→ F(R) be a fuzzy-interval-valued function , whose α-levels Fα : U −→ KC are given by Fα (t) = f1α (t), f2α (t) . Then F˜ is integrable over [a, b] if and only if f1α and f2α belongs to L1 ([a, b]) for all α ∈ [0, 1]. If F˜ is integrable over [a, b], then ⎤ ⎡ ⎤α ⎡    ⎢ ⎥ ⎢ ⎥ f1α (t)dt, f2α (t)dt ⎦ F˜ (t)dt ⎦ = ⎣ ⎣(FA) [a,b]

[a,b]

[a,b]

for all α ∈ [0, 1]. 5. Jensen’s inequality for fuzzy-interval-valued functions This section presents the Jensen’s inequality for fuzzy-interval-valued functions and this is the main result of this study. Since the α-levels of a fuzzy interval are elements in KC , then, the results presented in Section 3 are used in this section. Definition 5.1. (see, e.g., [8]) Given a fuzzy-interval-valued function F˜ : U −→ F(R), where U ⊆ R is a convex set, we say that F˜ is convex on U if (1 − λ)  F˜ (t1 ) ⊕ λF˜ (t2 ) ⊆F F˜ ((1 − λ)t1 + λt2 ) for all t ∈ U and λ ∈ [0, 1]. Definition 5.2. (see, e.g., [8]) Given a fuzzy-interval-valued function F˜ : U −→ F(R), where U ⊆ R is a convex set, we say that F˜ is concave on U if F˜ ((1 − λ)t1 + λt2 ) ⊆F (1 − λ)  F˜ (t1 ) ⊕ λF˜ (t2 ) for all t ∈ U and λ ∈ [0, 1].

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Theorem 5.1. Let U ⊆ R be a convex set and let F˜ : U −→ F(R) be a fuzzy-interval-valued function whose α-levels are given by Fα : U −→ KC . Then F˜ is convex (concave respectively) on U if and only if Fα is convex (concave respectively) on U for all α ∈ [0, 1]. Proof. This result directly follows from Definition 5.1, from Definition 5.2 and from definition of ⊆F .

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Remark 5.1. Let F˜ : [a, b] −→ F(R) be a fuzzy-interval-valued function, whose α-levels are the interval-valued   functions Fα : [a, b] −→ KC such that Fα (t) = f1α (t), f2α (t) with f1α , f2α : [a, b] −→ R for all α ∈ [0, 1]. From Theorem 4.2, it follows that (I) f1 (α, t) = f1α (t) ∈ R is bounded, non-decreasing, left-continuous function on (0, 1] and it is right-continuous at 0 with respect to α for all t ∈ [a, b]. (II) f2 (α, t) = f2α (t) ∈ R is bounded, non-increasing, left-continuous function on (0, 1] and it is right-continuous at 0 with respect to α for all t ∈ [a, b]. (III) f1 (1, t) ≤ f2 (1, t) for all t ∈ [a, b]. In particular, given a real-valued function g : U −→ R such that g(U ) ⊆ [a, b], we can define the fuzzy-intervalvalued function (F˜ ◦ g) : U −→ F(R), which associates to each t ∈ U the value F˜ ◦ g (t) ∈ F(R) such that "α !    F˜ ◦ g (t) = (Fα ◦ g)(t) = f1α (g(t)), f2α (g(t)) ∈ KC for all t ∈ U and for all α ∈ [0, 1]. F˜ ◦ g is well defined because f1 (α, t) and f2 (α, t) satisfy I–III for all t ∈ [a, b]. In particular, it follows that f1 (α, g(t)) and f2 (α, g(t))  ˜ satisfy I–III for all t ∈ U , and from Theorem 4.3, it follows that F ◦ g (t) ∈ F(R). Next, we present the two main results of this study. Theorem 5.2. (Jensen’s fuzzy inequality) Let g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given a concave function F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) =   α fuzzy-interval-valued f1 (g(t)), f2α (g(t)) , where f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then ⎞ ⎛   ⎟ ⎜ ˜ g(t)dt ⎠ ⊆F (FA) F⎝ F˜ (g(t))dt. (21) [0,1]

[0,1]

Proof. Since (f1α ◦g) and (f2α ◦g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then the fuzzy-interval-valued function F˜ ◦ g : [0, 1] −→ F(R) is integrable over [0, 1]. Thus, the existence of the right side of (21) is proved. On the other hand, F˜ : [a, b] −→ F(R) is concave, from Theorem 5.1, this is equivalent to Fα : [a, b] −→ KC being concave for all α ∈ [0, 1]. Moreover, since g, (f1α ◦ g), and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], from Remark 5.1 and from Theorem 3.4, it follows that ⎞ ⎛   ⎟ ⎜ Fα ⎝ g(t)dt ⎠ ⊆ (I A) Fα (g(t))dt for all α ∈ [0, 1]. [0,1]

[0,1]

Then, from Theorem 4.6 and from definition of ⊆ F , it follows that ⎞ ⎛   ⎟ ⎜ g(t)dt ⎠ ⊆F (FA) F˜ ⎝ F˜ (g(t))dt. 2 [0,1]

[0,1]

Corollary 5.1. Let g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given a fuzzy-interval-valued  function F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) = f1α (g(t)), f2α (g(t)) , where

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f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then ⎞ ⎞ ⎛ ⎛     ⎟ ⎟ ⎜ ⎜ ˜ ˜ g(t)dt ⎠ ⊆F (FA) g(t)dt ⎠ ⊆ (I A) Fα (g(t))dt F⎝ F (g(t))dt if and only if Fα ⎝ [0,1]

[0,1]

[0,1]

[0,1]

for all α ∈ [0, 1]. Proof. This result follows from Theorem 4.6, from proof of Theorem 5.2 and from definition of ⊆F .

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Example 5.1. Given g : [0, 1] −→ R by g(t) = et , let F˜ : [1, e] −→ F(R) be the fuzzy-interval-valued function whose α-levels are the interval-valued functions Fα : [1, e] −→ KC , Fα (t) = [f1α (t), f2α (t)], where f1α , f2α : [1, e] −→ R are defined by f1α (t) = α − t 2 and f2α (t) = 2 − α + t 2 for all α ∈ [0, 1]. Since f1α is concave for all α ∈ [0, 1] and f2α is convex for all α ∈ [0, 1], then from Theorem 3.2, it follows that Fα is concave for all α ∈ [0, 1], and consequently, F˜ is concave. Moreover, since g, (f1α ◦⎛g), and (f2α⎞◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then   ⎟ ⎜ g(t)dt ⎠ ⊆ Fα (g(t))dt for all α ∈ [0, 1], and consequently, from from Theorem 3.4, it follows that Fα ⎝ ⎛ ⎜ Corollary 5.1, it follows that F˜ ⎝  approximation of ⎛ ⎜ Fα ⎝

[0,1]



[0,1]

⎡ ⎢ =⎣

⎟ g(t)dt ⎠ ⊆F

[0,1]

[0,1]



⎛

⎜ F˜ (g(t))dt . This means that, F˜ ⎝

[0,1]



⎞ ⎟ g(t)dt ⎠ is an inner

[0,1]

F˜ (g(t))dt that can be characterized by ⎞

⎤ ⎡ ⎤        2  2 ⎥ ⎢ ⎥ dt, 2 − α + e2t dt ⎦ = ⎣ α − et dt, 2 − α + et dt ⎦

α − e2t

[0,1]





Fα (et )dt = [0,1]

⎞

 ! "  e2 3 e2 1 ⎟ g(t)dt ⎠ = Fα (e − 1) = α − (e − 1)2 , 2 − α + (e − 1)2 ⊆ α − + , −α + + 2 2 2 2

 

=

[0,1]



[0,1]

[0,1]

[0,1]

Fα (g(t))dt

[0,1]

for all α ∈ [0, 1] as illustrated in Fig. 1. Theorem 5.3. (Reverse Jensen’s fuzzy inequality) Let g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given a fuzzy-interval-valued function F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) =  convex α α f1 (g(t)), f2 (g(t)) , where f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then ⎛ ⎞   ⎜ ⎟ F˜ (g(t))dt ⊆F F˜ ⎝ (FA) g(t)dt ⎠ . (22) [0,1]

[0,1]

Proof. Since (f1α ◦ g), and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then the fuzzy-intervalvalued function F˜ ◦ g : [0, 1] −→ F(R) is integrable over [0, 1]. Thus, the existence of the left side of (22) is proved. On the other hand, F˜ : [a, b] −→ F(R) is convex, from Theorem 5.1, this is equivalent to Fα : [a, b] −→ KC being convex for all α ∈ [0, 1]. Moreover, since g, (f1α ◦g), and (f2α ◦g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1],

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Fig. 1. Illustration of Example 5.1.

 from Theorem 3.5, it follows that (I A)

⎛ ⎜ Fα (g(t))dt ⊆ Fα ⎝

[0,1]

and from definition of ⊆ F , it follows that ⎞ ⎛   ⎟ ⎜ (FA) g(t)dt ⎠ . F˜ (g(t))dt ⊆F F˜ ⎝ [0,1]



⎞ ⎟ g(t)dt ⎠ for all α ∈ [0, 1]. Then from Theorem 4.6

[0,1]

2

[0,1]

Corollary 5.2. Let g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given a fuzzy-interval-valued  function F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) = f1α (g(t)), f2α (g(t)) , where f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then  (FA)

⎛ ⎜ F˜ (g(t))dt ⊆F F˜ ⎝

[0,1]



⎞ ⎟ g(t)dt ⎠ if and only if (I A)

[0,1]



⎛ ⎜ Fα (g(t))dt ⊆ Fα ⎝

[0,1]



⎞ ⎟ g(t)dt ⎠

[0,1]

for all α ∈ [0, 1]. Proof. This result follows from Theorem 4.6, from proof of Theorem 5.3 and from definition of ⊆F .

2

√ Example 5.2. Given g : [0, 1] −→ R by g(t) = x, let F˜ : [0, 1] −→ F(R) be the fuzzy-interval-valued function whose α-levels are the interval-valued functions Fα : [0, 1] −→ KC Fα (t) = [f1α (t), f2α (t)], where f1α , f2α : √ [0, 1] −→ R are defined by f1α (t) = α + et and f2α (t) = 4 − α + t for all α ∈ [0, 1]. Since f1α is convex for all α ∈ [0, 1] and f2α is concave for all α ∈ [0, 1], from Theorem 3.1, it follows that Fα is convex for all α ∈ [0, 1], and consequently, F˜ is convex. Moreover, we have that g, (f1α ◦ g), and (f2α ◦ g)⎛are Lebesgue-integrable over [0, 1] ⎞   ⎜ ⎟ for all α ∈ [0, 1], then, from Theorem 3.5, it follows that Fα (g(t))dt ⊆ Fα ⎝ g(t)dt ⎠ for all α ∈ [0, 1], and [0,1]

consequently, from Corollary 5.1, it follows that  [0,1]

⎛ ⎜ F˜ (g(t))dt ⊆F F˜ ⎝



[0,1]

⎞ ⎟ g(t)dt ⎠ .

[0,1]

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Fig. 2. Illustration of Example 5.2.

⎛ ⎜ This means that F˜ ⎝

⎞



⎟ g(t)dt ⎠ is an exterior approximation of

[0,1]

F˜ (g(t))dt that can be characterized by

[0,1]



 Fα (g(t))dt =

[0,1]



⎡

⎤ % #    $ √ √ √ ⎢ ⎥ Fα t dt = ⎣ t dt ⎦ α + e t dt, 4−α+

[0,1]

[0,1]

[0,1]

⎞ ⎛ ' (   & # %  2 24 2 2 ⎟ ⎜ = α + 2, −α + g(t)dt ⎠ ⊆ α + e 3 , −α + 4 + = Fα = Fα ⎝ 5 3 3 [0,1]

for all α ∈ [0, 1] as illustrated in Fig. 2. Corollary 5.3. Let g : [0, 1] −→ (a, b) be a Lesbesgue-integral function. Given a concave and convex fuzzy-interval valued function F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) = f1α (g(t)), f2α (g(t)) , where f1α , f2α : [a, b] −→ R are real-valued functions such that (f1α ◦ g) and (f2α ◦ g) are Lebesgue-integrable over [0, 1] for all α ∈ [0, 1], then ⎞ ⎛   ⎟ ⎜ ˜ g(t)dt ⎠ = (FA) F⎝ F˜ (g(t))dt. (23) [0,1]

[0,1]

Proof. This result directly follows from Theorem 5.2 and from Theorem 5.3.

2



0 if t ∈ Q , where Q is the set of all rational numbers, let F˜ : 1 if t ∈ /Q [0, 1] −→ F(R) be the fuzzy-interval-valued function whose α-levels are the interval-valued functions Fα : [0, 1] −→ KC , Fα (t) = [f1α (t), f2α (t)], where f1α , f2α : [0, 1] −→ R are defined by f1α (t) = α − t and f2α (t) = −α + 2 + t for all α ∈ [0, 1]. Since f1α is concave and convex for all α ∈ [0, 1] and f2α is convex and concave for all α ∈ [0, 1], from Theorem 3.1 and from 3.2, it follows that Fα is concave and convex for all α ∈ [0, 1], and consequently, F˜ is over [0, 1] for concave and convex. Moreover, since we have that g, (f1α ◦ g) and (f2α ◦ g) are⎛Lebesgue-integrable ⎞   ⎟ ⎜ g(t)dt ⎠ = Fα (g(t))dt for all α ∈ [0, 1], then, from Theorem 3.4 and from Theorem 3.5, it follows that Fα ⎝ Example 5.3. Given g : (0, 1) −→ R by g(t) =

all α ∈ [0, 1], and consequently, from Proposition 4.1, it follows that

[0,1]

[0,1]

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⎛



⎜ F˜ (g(t))dt = F˜ ⎝

[0,1]

15

⎞



⎟ g(t)dt ⎠ .

[0,1]

⎛



⎜ F˜ (g(t))dt can be found by performing the calculus of F˜ ⎝

This means that the value of [0,1]



in general is easier than to perform the calculus of

F˜ (g(t))dt . In this case we have that

[0,1]



⎞ ⎟ g(t)dt ⎠, which

 [0,1] F˜ (g(t))dt is given [0,1]

level-wise by ⎡



⎢ Fα (g(t))dt = ⎣

[0,1]



 (α − g(t))dt,

[0,1]

⎤ ⎥ (−α + 2 + g(t))dt ⎦ = Fα (1) = [α − 1, −α + 3]

[0,1]

for all α ∈ [0, 1]. Corollary 5.4. Let g : [0, 1] −→ (a, b) be a continuous real-valued function. Given a fuzzy-interval-valued function   F˜ : [a, b] −→ F(R) whose α-levels are Fα : [a, b] −→ KC given by Fα (t) = f1α (g(t)), f2α (g(t)) , where f1α , f2α : [a, b] −→ R are real-valued functions, it follows that (i) if F˜ is concave, then the inequality (21) is ensured; (ii) if F˜ is convex, then the inequality (22) is ensured; (iii) if F˜ is concave and convex, then the equality (23) is ensured. Proof. If F is concave (convex respectively), then Theorem 5.1 implies that Fα is concave (convex respectively) for all α ∈ [0, 1]. Then from Theorem 3.1 and from Theorem 3.2, it follows that f1α and f2α are continuous on [a, b] for all α ∈ [0, 1]. Since g is continuous, it follows that f1α ◦ g and f2α ◦ g are Lesbesgue-integrable over [0, 1] for all α ∈ [0, 1]. Then if F˜ is concave we have the assumptions of Theorem 5.2 and thus is ensured the inequality (21). If we have that F is convex we have the assumptions of Theorem 5.3 and thus is ensured the inequality (22). On the other hand, if we have that F is concave and convex, then we have the assumptions of Corollary 5.3, then is ensured the equality (23). 2 Remark 5.2. Considering the assumptions given in Theorem 5.2, then we have the Jensen’s fuzzy inequality ⎞ ⎛   ⎟ ⎜ ˜ g(t)dt ⎠ ⊆F (FA) F˜ (g(t))dt. On the other hand, from definition of ⊆F , it follows that F⎝ [0,1]

[0,1]

⎛ ⎜ Fα ⎝



⎞ ⎟ g(t)dt ⎠ ⊆F (I A)

[0,1]

 Fα (g(t))dt for all α ∈ [0, 1].

(24)

[0,1]

Then for each α ∈ [0, 1], Remark 3.2 implies that (24) can be reduced to the Jensen inequality for real-valued function ⎛ ⎞⎞ ⎛     ⎜ ⎟⎟ ⎜ s ⎝x, Fα ⎝ g(t)dt ⎠⎠ ≤ s x, Fα (g(t) dt [0,1]

[0,1]

for each x ∈ S 0 , where the integral on the right side is the Lebesgue-integral of real-valued functions. Therefore, the Jensen’s inequity for fuzzy-interval-valued functions can be reduced to a family of Jesen’s inequalities for real-valued functions.

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By using similar arguments, the Reverse Jensen inequality for fuzzy-interval-valued function presented here can be reduced to a family of Jensen inequalities for real-valued functions and if we use the specific property of the function s that says “given A, B ∈ KC , then A = B if and only if s(x, A) = s(x, B) for all x ∈ R” (see [9]), then the equality obtained in Corollary 5.3 can also be reduced to a real-valued version. 6. Conclusion This study presents a version of Jensen’s inequality for interval-valued functions using the interval inclusion order relation, the concepts of convexity and concavity for interval-valued functions, given in [6], and the Aumann integral [3]. By using the support function for sets, the Jensen’s inequality for interval-valued functions presented here can be reduced to the Jensen inequality for real-valued functions. Since the level sets of a fuzzy interval are intervals of real numbers, the interval inclusion order relation is the order relation used level-wise to obtain the fuzzy inclusion order relation and, with this fuzzy order relation, the concepts of convexity and concavity for fuzzy-interval-valued functions studied in [6,8] as well as the concept integral for fuzzy-interval-valued functions given in [12] were used in this presentation. Using these concepts of convexity and concavity for fuzzy-interval-valued functions and the fuzzy integral, the Jensen’s inequality for fuzzy-interval-valued functions was derived. This is the main result of this study. This presentation also shows that the Jensen’s inequality for fuzzy-interval-valued functions can be to reduced to a family of classical Jensen’s inequality for real-valued functions, thus it was possible to deal with Jensen’s inequality for fuzzy-interval-valued functions using only tools from real analysis. The interval and fuzzy versions of Jensen’s inequality presented in this study, are tools to work in uncertainty environment. Moreover, since these inequalities are given using different assumptions than those used in the previous research articles, our results are original. Acknowledgements The author would like to thank the editors-in-chief, the area editor, and the anonymous referees for their critical comments and helpful suggestions. The author greatly acknowledge the financial support of the Brazilian National Council for Science and Technology Development (CNPq) grant number 249300/2013-3. References [1] J.P. Aubin, A. Cellina, Differential Inclusions: Set-Valued Maps and Viability Theory, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1984. [2] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. [3] R.J. Aumann, Integrals of set-valued functions, J. Math. Anal. Appl. 12 (1) (1965) 1–12. [4] P. Auscher, T. Coulhon, A. 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[20] W. Rudin, Real and Complex Analysis, McGraw-Hill, Inc., New York, NY, USA, 1987. [21] M. Štrboja, T. Grbi´c, I. Štajner-Papuga, G. Gruji´c, S. Medi´c, Jensen and Chebyshev inequalities for pseudo-integrals of set-valued functions, Fuzzy Sets Syst. 222 (2013) 18–32. [22] R.S. Wang, Some inequalities and convergence theorems for Choquet integral, J. Appl. Math. Comput. 35 (1) (2011) 305–321. [23] X. Zhao, Q. Zhang, Hölder type inequality and Jensen type inequality for Choquet integral, in: Knowledge Engineering and Management: Proceedings of the Sixth International Conference on Intelligent Systems and Knowledge Engineering, Shanghai, China, Dec 2011 (ISKE2011), Springer, Berlin/Heidelberg, 2011.