Comp. Appl. Math. DOI 10.1007/s40314-016-0396-7

Some integral inequalities for interval-valued functions H. Román-Flores1 · Y. Chalco-Cano1 · W. A. Lodwick2

Received: 18 October 2013 / Revised: 22 March 2015 / Accepted: 26 October 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016

Abstract In this paper, we explore some integral inequalities for interval-valued functions. More precisely, using the Kulisch–Miranker order on the space of real and compact intervals, we establish Minkowski’s inequality and then we derive Beckenbach’s inequality via an interval Radon’s inequality. Also, some examples and applications are presented for illustrating our results. Keywords Interval-valued functions · Minkowski’s inequality · Radon’s inequality · Beckenbach’ inequality Mathematics Subject Classification 26D15 · 26E25 · 28B20

1 Introduction The importance of the study of set-valued analysis from a theoretical point of view as well as from their application is well known (see Aubin and Cellina 1984; Aubin and Franskowska 1990). Also, many advances in set-valued analysis have been motivated by control theory and

Communicated by Marko Rojas-Medar. This work was supported in part by Conicyt-Chile through Projects Fondecyt 1120674 and 1120665. Also, W. A. Lodwick was supported in part by FAPESP 2011/13985.

B

H. Román-Flores [email protected] Y. Chalco-Cano [email protected] W. A. Lodwick [email protected]

1

Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile

2

Department of Mathematical and Statistical Sciences, University of Colorado, Denver, CO 80217, USA

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dynamical games and, in addition, optimal control theory and mathematical programming were a motivating force behind set-valued analysis since the sixties (see Aubin and Franskowska 2000). Interval Analysis is a particular case and it was introduced as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena. The first monograph dealing with interval analysis was given by Moore (1966). Moore is recognized as the first to use intervals in computational mathematics, now called numerical analysis. He also extended and implemented the arithmetic of intervals to computers. One of his major achievements was to show that Taylor series methods for solving differential equations not only are more tractable, but also more accurate (see Moore 1985). On the other hand, several generalizations of classical integral inequalities were obtained in the recent years by Agahi et al. (2010, 2011a, b, 2012), Flores-Franuliˇc and Román-Flores (2007), Flores-Franuliˇc et al. (2009), Mesiar and Ouyang (2009), Román-Flores and ChalcoCano (2007), Román-Flores et al. (2007a, b, 2008, 2013), in the context of non-additive measures and integrals (also see the following related references: Pap 1995; Ralescu and Adams 1980; Román-Flores and Chalco-Cano 2006; Sugeno 1974; Wang and Klir 2009). In general, any integral inequality can be a very powerful tool for applications and, in particular, when we think an integral operator as a predictive tool then an integral inequality can be very important in measuring, computing errors and delineating such processes. Interval-valued functions (or fuzzy-interval valued functions) may provide an alternative choice for considering the uncertainty into the prediction processes and, in connection with this, the Aumann integral for interval-valued function is the natural-associated expectation (see for example Puri and Ralescu 1986). Also, several integral inequalities involving functions and their integrals and derivatives, such as Wirtinger’s inequality, Ostrowski’s inequality, and Opial’s inequality, among others, have been extensively studied during the past century (see for example Anastassiou 2011; Mitrinovi´c et al. 1991). All these studies have been fundamental tools in the development of many areas in mathematical analysis. Recently, some differential-integral inequalities have been extended to the set-valued context. For example Anastassiou (2011), using the Hukuhara derivative, extended an Ostrowski type inequality to the context of fuzzy-valued functions. Chalco-Cano et al. (2012) using the concept of generalized Hukuhara differenciability (see Chalco-Cano et al. 2011, 2013) establish some Ostrowski type inequalities for interval-valued functions. This presentation generalizes Minkowski’s inequality for interval-valued functions and, as an application, establishes the Beckenbach’s inequality for interval-valued functions via an interval Radon’s inequality.

2 Preliminaries 2.1 Interval operations Let R be the one-dimensional Euclidean space. Following Diamond and Kloeden (1994), let KC denote the family of all non-empty compact convex subsets of R, that is, KC = {[a, b] | a, b ∈ R and a ≤ b}.

(1)

The Pompeiou–Hausdorff metric on KC (Pompeiu was the first mathematician introducing the concept of set distance, see Birsan and Tiba 2006), frequently called Hausdorff metric, is defined by

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Some integral inequalities for interval-valued functions

H (A, B) = max {d(A, B), d(B, A)} ,

(2)

where d(A, B) = maxa∈A d(a, B) and d(a, B) = minb∈B d(a, b) = minb∈B |a − b|. Remark 1 An equivalent form for the Hausdorff metric defined in (2) is         M a, a , b, b = max a − b , a − b which is also known as the Moore metric on the space of intervals (see Moore and Kearfott 2009, Eq. (6.3), pp. 52). It is well known that (KC , H ) is a complete metric space (see Aubin and Cellina 1984; Diamond and Kloeden 1994). If A ∈ KC then we define the norm of A as A = H (A, 0) = H ([0, 0]). The Minkowski sum and scalar multiplication are defined on KC by means A + B = {a + b | a ∈ A, b ∈ B}

and

λA = {λa | a ∈ A}.

(3)

Also, if A = [a, a] and B = [b, b] are two compact intervals then we define the difference   A − B = a − b, a − b , (4) the product

     A · B = min ab, ab, ab, ab , max ab, ab, ab, ab ,

and the division





 a a a a a a a a A , , , , , , = min , max , B b b b b b b b b

(5)

(6)

whenever 0 ∈ / B. An order relation “≤ ” is defined on KC as follows (see Kulisch and Miranker 1981): [a, a] ≤ [b, b] ⇔ a ≤ b

and

a ≤ b.

(7)

Remark 2 We note that if [a, b], [c, d], and [x, y] are intervals with positive enpoints then [x, y] [a, b] ≥ [c, d] [c, d] [a, b] [a, b] [c, d] ≤ [x, y] ⇔ ≥ . [c, d] [x, y] [a, b] ≥ [x, y] ⇔

(8) (9)

The space KC is not a linear space since it does not possess an additive inverse and therefore subtraction is not well defined (see Aubin and Cellina 1984; Markov 1979). However, one very important property of interval arithmetic is that: A, B, C, D ∈ KC , A ⊆ B, C ⊆ D ⇒ A ∗ C ⊆ B ∗ D

(10)

where ∗ can be sum, subtraction, product or division. Property (10) is called the “inclusion isotony ”of interval operations and it is recognized as the fundamental principle of interval analysis (see Moore and Kearfott 2009). One consequence of this is that any function f (x) described by an expression in the variable x which can be evaluated by a programmable real calculation can be embedded in interval calculations using the natural correspondence between operation so that if x ∈ X ∈ KC then f (x) ∈ f (X ), where f (X ) is interpreted as the calculation of f (x) with x replaced by X

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and the operations replaced by interval operations. The evaluation f (X ) is called the natural interval extension of the expression f (x). Also it is necessary to observe that two different expressions for a same real function can result in very different interval-valued functions. For instance, if we consider f (x) = x(1 − x) and f (x) = x − x 2 and taking X = [3, 5] then, on the one hand we obtain f (X ) = [3, 5] ([1, 1] − [3, 5]) = [−20, −6], and on the other we have f (X ) = [3, 5] − [3, 5]2 = [−22, −4]. Denote the range of a function f (x) over an interval X as R( f ) | X = { f (x) | x ∈ X } .

(11)

Then from (10) it follows for the natural interval extension that R( f ) | X ⊆ f (X ).

As an example consider the function f (x) =

(12)

+ x and X = [−1, 1]. Then 1 R( f ) | X = − , 2 , 4 x2



whereas using the natural interval extension of f (x) we obtain f (X ) = [−1, 1]2 + [−1, 1] = [−2, 2] , which is a more larger interval but nevertheless contains R( f ) | X . For the particular case when f (x) is monotone and continuous over an interval X = [a, b], we can define f ([a, b]) = [min { f (a), f (b)} , max { f (a), f (b)}]

(13)

and, in this case, f (X ) = R( f ) | X . For example: (a) if f (x) = x r , r > 0, and 0 ≤ a ≤ b then

  f ([a, b]) = [a, b]r = a r , br .

(b) If g(x) = e x then the “exponential” of an interval [a, b] is defined as 

g ([a, b]) = e[a,b] = ea , eb .

(14)

(15)

For more details on interval operations and interval analysis see Markov (1979), Moore (1966) and Rokne (2001).

2.2 Integral of interval-valued functions If T = [a, b] is a closed interval and F : T → KC is an interval-valued function, then we will denote F(t) = [ f (t), f (t)], where f (t) ≤ f (t), ∀t ∈ T . The functions f and f are called the lower and the upper (endpoint) functions of F, respectively. For interval-valued functions it is clear that F : T → KC is continuous at t0 ∈ T if limt→t0 F(t) = F(t0 ),

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(16)

Some integral inequalities for interval-valued functions

where the limit is taken in the metric space (KC , H ). Consequently, F is continuous at t0 ∈ T if and only if its endpoint functions f and f are continuous functions at t0 ∈ T . We denote by C ([a, b], KC ) the family of all continuous interval-valued functions. Definition 1 Let M the class of all Lebesgue measurable sets of T , then (a) the function f : T → R is measurable if and only if f −1 (C) ∈ M for all closed subset C of R. (b) the interval-valued function F : T → KC is measurable if and only if F ω (C) = {t ∈ T /F(t) ∩ C = ∅} ∈ M, ∀ C ⊆ R, C closed. (c) Also, if F : T → KC is an interval-valued function and f : T → R, then we say that f is a selector (or selection) of F if and only if f (t) ∈ F(t) for all t ∈ T . In this case if, additionally f is a measurable function, then we say that f is a measurable selector of F. Finally, an integrable selector of F is a measurable selector of F for which there is T f (t). Definition 2 (Aubin and Cellina 1984) Let F : T → KC be an interval-valued function. The integral (Aumann integral) of F over T = [a, b] is defined as

 b   b F(t)dt = f (t)dt | f ∈ S(F) , (17) a

a

where S(F) is the set of all integrable selectors of F, i.e., S(F) = { f : T → R | f integrable and f (t) ∈ F(t) for all t ∈ T } . If S(F) = ∅, then the integral exists and F is said to be integrable (Aumann integrable). Note that if F is integrable then it has a measurable selector which is integrable and, consequently, S(F) = ∅. b b Also, in above definition, the integral symbol a F(t)dt and/or a f (t)dt denotes the integral with respect to the Lebesgue measure. Definition 3 We say that a mapping F : T → KC is integrally bounded if there exists a positive integrable function g : T → R such that F(t) ≤ g(t), for all t ∈ T . Theorem 1 (Aubin and Cellina 1984) Let F : T → KC be a measurable and integrally b bounded interval-valued function. Then it is integrable and a F(t)dt ∈ KC . Corollary 1 (Aubin and Cellina 1984; Diamond and Kloeden 1994) A continuous intervalvalued function F : T → KC is integrable. The Aumann integral satisfies the following properties. Proposition 1 (Aubin and Cellina 1984; Diamond and Kloeden 1994) Let F, G : T → KC be two measurable and integrally bounded interval-valued functions. Then t t t (i) t12 (F(t) + G(t)) dt = t12 F(t)dt + t12 G(t)dt, a ≤ t1 ≤ t2 ≤ b  t2 τ  t2 (ii) t1 F(t)dt = t1 F(t)dt + τ F(t)dt, a ≤ t1 ≤ τ ≤ t2 ≤ b.

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Theorem 2 (Bede and Gal 2005) Let F : T → KC be a measurable and integrally bounded interval-valued function such that F(t) = [ f (t), f (t)]. Then f and f are integrable functions and  t2  t2  t2 F(t)dt = f (t)dt, f (t)dt . (18) t1

t1

t1

Remark 3 Above Theorem 2 is a direct consequence of two relevant results:  (a) (Aumann 1965, Theorem 1, pp. 2) T F(t)dt is convex.  (b) (Aumann 1965, Theorem 4, pp. 2) If F is closed-valued then T F(t)dt is compact.   In fact, because f , f ∈ S(F) then, by convexity of T F(t)dt, we obtain [ T f (t)dt,   T f (t)dt] ⊆ T F(t)dt. On the other hand, if f ∈ S(F) then f (t) ≤ f (t) ≤ f (t), for all t ∈ T , which implies that    f (t)dt ∈ f (t)dt, f (t)dt and, consequently, 2 holds.

T

 T

T

T

  F(t)dt ⊆ [ T f (t)dt, T f (t)dt]. Therefore equality (18) in Theorem

To finalize this section, we give an example of prediction under uncertainty using interval tools (see Puri and Ralescu 1986). Example 1 Toss a fair coin. Denote the outcomes Tail by T and Head by H. Suppose a player loses approximately 10 EUR if the outcome is T, and wins an amount much larger than 100 EUR but not much larger than 1000 EUR if the outcome is H. The question here is: what is the expected value for the next outcome? To represent the uncertainty contained in the above linguistic descriptions, we can define the interval random variable X : {T, H } → KC , where (a) X (T ) = approximately − 10, and (b) X (H ) = much larger than 100 but not much larger than 1000. Furthermore, if E = {T, H } then we can consider the measure space (E, P (E), μ) taking as is usual: μ(T ) = μ(H ) = 21 , μ(X ) = 1 and μ(∅) = 0. Suppose we interpret the linguistic variables as X (T ) = [−12, −8] and X (H ) = [250, 1010]. Now, we can write X (z) = [ f (z), f (z)] where f , f : {T, H } → R are defined by f (T ) = −12, f (H ) = 250, f (T ) = −8, and f (H ) = 1010. So, using properties of the Aumann integral we have    1 1 f dμ = f dμ + f dμ = (−12) + (250) = 119 2 2 E {T } {H } and



 f dμ = E

{T }

 f dμ +

{H }

f dμ =

1 1 (−8) + (1010) = 501. 2 2

Thus, the expected value for the next outcome is the interval E(X ) = [119, 501].

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Some integral inequalities for interval-valued functions

3 Minkowski’s inequality The well-known inequality due to Minkowski can be stated as follows (see Hardy et al. 1934, pp. 31): Theorem 3 Let f (x), g(x) ≥ 0 and p ≥ 1, then 1   1  1  p p p p p p ≤ + f (x) dx g(x) dx ( f (x) + g(x)) dx

(19)

with equality if and only if f and g are proportional, and if 0 < p < 1, then 1   1  1  p p p ≥ + f (x) p dx g(x) p dx ( f (x) + g(x)) p dx

(20)

with equality if and only if f and g are proportional. We recall that two non-negative functions f and g are proportional if and only if there is a non-negative real constant k such that f = kg (or g = k f ). Now, using above theorem and properties of interval integration, we can prove the following interval version of Minkowski’s inequality: Theorem 4 (Interval Minkowski’s inequality) If F, G : [a, b] → KC are two integrable interval-valued functions, with F = [ f , f ], G = [g, g], f (x), g(x) ≥ 0 and p ≥ 1, then 1

 [a,b]

p

(F(x) + G(x)) p dx

 ≤

1

[a,b]

F(x) p dx



p

+

1

[a,b]

with equality if F and G are proportional, and if 0 < p < 1, then 1   1   p p ≥ F(x) p dx + (F(x) + G(x)) p dx [a,b]

[a,b]

p

G(x) p dx

(21)

1

[a,b]

p

G(x) p dx

(22)

with equality if F and G are proportional. Proof Due to (3), (7), (13), (14), Theorem 2, and Theorem 3 (19), we have 

1

[a,b]

(F(x) + G(x)) p dx



=

[a,b]

 = =

[a,b]

= 

[a,b]

1 p  p p f (x) + g(x) , f (x) + g(x) dx



 p f (x) + g(x) dx,



 p  1p  , f (x) + g(x) dx

[a,b]



≤

 p  1p f (x) + g(x), f (x) + g(x) dx



 

 [a,b]

p f (x) + g(x) dx

[a,b]

1 f (x) dx p

[a,b]

p

p

 +



p

p f (x) + g(x) dx

 1  p g(x) dx ,

1  p

1

p

[a,b]

 1

f (x) dx p

[a,b]

p

1 

 +

p

[a,b]

g(x) dx

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p

H. Román-Flores et al.

 =

[a,b]

f (x) p dx

 =

[a,b]

f (x) p dx,

[a,b]

 =

[a,b]

1

[a,b]

[a,b]

[a,b]

f (x) , f (x) p

p

f (x) p dx 1



p

dx

1 F(x)] p dx

p

p

+

+

[a,b]

 +

[a,b]

[a,b]

[a,b]

G(x) p dx

g(x) p dx

 1  p ,

g(x) p dx,

g(x) , g(x) p

p

[a,b]

1  g(x) p dx

p

1



[a,b]



 +

p



 p  1p  + f (x), f (x) dx

 =

1  f (x) p dx



 =

 1  p ,

[a,b]

g(x) p dx

p

1



p

dx

 p  1p g(x), g(x) dx 1

p

Analogously, using (3), (7), (13), (14), Theorems 2 and 3 (20), we can prove the second part of our theorem for 0 < p < 1. Finally, a straightforward calculation shows that equality is reached if F and G are proportional. This completes the proof.  

4 Beckenbach’s inequality The well-known Beckenbach’s inequality can be stated as follows (see Beckenbach and Bellman 1992, pp. 27): Theorem 5 (Beckenbach and Bellman 1992) If 0 < p < 1, and f (x), g(x) > 0, then    f (x) p+1 dx g(x) p+1 dx ( f (x) + g(x)) p+1 dx    ≤ + . (23) p p ( f (x) + g(x)) dx f (x) dx g(x) p dx The aim of this section is to show a Beckenbach type inequality for interval-valued functions, and for this we will use an interval version of the Radon’s inequality. We recall that the classical Radon’s inequality (published by Radon 1913), establishes that Theorem 6 (Radon 1913) For every real numbers p > 0, xk ≥ 0, ak > 0, for 1 ≤ k ≤ n, the inequality  n  p+1 p+1 n  xk k=1 x k ≥ (24)  p n p ak k=1 ak k=1 holds. Inequality (24) has been widely studied by many authors because of its utility in practical and theoretical applications (see for example Mortici 2011; Zhao 2012). The next result is an extension of Radon’s inequality to the interval context. Theorem 7 (Interval Radon’s inequality) Let [x k , x k ], [a k , a k ] ∈ KC , with x k ≥ 0, a k > 0, for all 1 ≤ k ≤ n. If p > 0, then the inequality  p+1  n n  [x k , x k ] p+1 k=1 [x k , x k ] ≥ n (25) p [a k , a k ] p k=1 [a k , a k ] k=1

holds.

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Some integral inequalities for interval-valued functions

Proof Working on the left-side of (25), we have p+1 p+1 n n   [x k , x k ] [x k , x k ] p+1 = p p [a k , a k ] p [a k , a k ] k=1 k=1  p+1 p+1  n  xk xk = p , p a ak k k=1  n p+1 n p+1  x x k k = . p , p a a k k k=1 k=1

On the other hand, working on the right-side of (25), we obtain  n  n  p+1  p+1  [ k=1 x k , nk=1 x k ] k=1 [x k , x k ]  p =  n p  n  [ k=1 a k , nk=1 a k ] k=1 [a k , a k ]

  p+1 n  p+1  n , k=1 x k k=1 x k = n  p  n p , k=1 a k k=1 a k   n  p+1 n  p+1  k=1 x k k=1 x k = .  n  p ,  n p k=1 a k k=1 a k Now, by Radon’s inequality (24) we have n

p+1 n  xk

≥

p

k=1

ak

and

n

p+1 n  xk

≥ k=1 n

p

k=1

 p+1 k=1 x k  n p k=1 a k

ak

xk

(26)

 p+1

k=1 a k

p

(27)  

which implies that inequality (25) holds.

Theorem 8 (Interval Beckenbach’s inequality) If F, G : [a, b] → KC are two integrable interval-valued functions, with F = [ f , f ], G = [g, g], f (x), g(x) > 0 and 0 < p < 1, then    p+1 p+1 dx p+1 dx dx [a,b] (F(x) + G(x)) [a,b] F(x) [a,b] G(x)    ≤ + . (28) p p p [a,b] (F(x) + G(x)) dx [a,b] F(x) dx [a,b] G(x) dx Proof Taking  I1 =

 [a,b]

F(x) p+1 dx

 I2 =

[a,b]

 G(x)

p+1

1 p+1

1 p+1

dx

 ,

J1 =

1 [a,b]

F(x) p dx

 ,

J2 =

p

(29) 1

p

[a,b]

p

G(x) dx

(30)

and using the intervalar Radon inequality (25) we have p+1

I1

p

J1

p+1

+

I2

p

J2

≥

(I1 + I2 ) p+1 , (J1 + J2 ) p

(31)

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H. Román-Flores et al.

that is to say,  [a,b]



F(x) p+1 dx

 [a,b] + 

G(x) p+1 dx

p F(x) p dx [a,b] G(x) dx   1   1  p+1  p+1 p+1 p+1 p+1 dx + [a,b] G(x) dx [a,b] F(x) . ≥   1   1 p  p p p p + [a,b] G(x) dx [a,b] F(x) dx [a,b]

(32)

Now, because 0 < p < 1 then 1 < p + 1 < 2 and , due to (21) and (22), we obtain 



[a,b]



1 p+1

(F(x) + G(x)) p+1 dx

≤



[a,b]

F(x) p+1 dx

1 p+1

 +

 [a,b]

G(x) p+1 dx

1 p+1

,

(33) 

1

[a,b]

p

(F(x) + G(x)) p dx

 ≥

1 [a,b]

F(x) p dx

p

 +

[a,b]

1 G(x) p dx

p

.

(34)

Finally, due (8), (9), (33) and (34) we obtain that  1   1  p+1 p+1 dx p+1 + p+1 dx p+1  F(x) G(x) p+1 [a,b] [a,b] dx [a,b] (F(x) + G(x))  ≥ ,   p p  1  1 + G(x)) dx  (F(x) [a,b] p dx p + p dx p F(x) G(x) [a,b] [a,b]

 

(35)  

and the proof is completed.

Example 2 Let p = 1/2 and let F, G : [0, 1] → KC two interval-valued functions defined by F(x) = [x, 2x] and G(x) = [x 2 , x], with x ∈ [0, 1]. Using (14), interval operations and properties of the Aumann integral, a straightforward calculation shows that:   2 32 2 12 p+1 p 2 2 ,2 ,2 , (36) F(x) dx = F(x) dx = 5 5 3 3 [0,1] [0,1]

 [0,1]

G(x) p+1 dx =

 1 2 1 2 G(x) p dx = , , , . 4 5 2 3 [0,1]

(37)

On the other hand, 

 [0,1]

(F + G) p+1 dx

1 p+1

 =

√ 3 3 39 √ 3 ln( 2 + ) + ln2 + 2 128 2 128 64

2  2  3 2√ 3 , 3 3 (38)

(F + G) dx p

[0,1]

123



1



p

=

  3√ 1 3 2 12 1 √ 2 − ln2 − ln( 2 + ) , . 4 8 8 2 9

(39)

Some integral inequalities for interval-valued functions

Also, 



[0,1]

F(x)

p+1

dx

and 



[0,1]

G(x)

1 p+1

p+1

1 p+1

dx

  2   2   1 p 2 3 2 3 4 8 p = ,2 F(x) dx = , , 5 5 9 9 [0,1]   2   2   1 p 1 3 2 3 1 4 p , . = , G(x) dx = , 4 5 4 9 [0,1]

(40)

(41)

Thus, from (38), (40) and (41), because 

 2  2  2  √ 3 3 3 2 3 1 3 39 √ 3 + 2+ 2 ≤ + ln ln2 + 128 2 128 64 5 4

and 

2√ 3 3

 2  2 2 3 2 3 ≤2 + , 5 5

2

3

we obtain that  1   p+1 (F + G) p+1 dx ≤ [0,1]



[0,1]

F p+1 dx

1 p+1

 +



[0,1]

G p+1 dx

1 p+1

and, consequently, Minkowski’s inequality (21) is verified. Analogously, from (39)to (41), because    √ 3√ 1 1 3 2 4 1 2 − ln2 − ln 2+ ≥ + 4 8 8 2 9 4 and 12 8 4 ≥ + , 9 9 9 we obtain that 

1 (F + G) dx p

[0,1]

p

 ≥

1 p

[0,1]

F dx

p

 +

1 p

[0,1]

p

G dx

and, consequently, Minkowski’s inequality (22) is verified. Additionally,    √  p+1 dx p+1 dx 3 3 8 3 4 [0,1] F [0,1] G   , + = + √ , p p 8 5 5 2 5 [0,1] F dx [0,1] G dx and



[0,1] (F



(42)

+ G) p+1 dx

+ G) p dx  √ ⎡ √ 3 3 3 ln 2 + 128 2 + 128 ln 2 + =⎣ √ 2 3 3 [0,1] (F

39 64

√

2

6 5

√

⎤ 3

⎦ . √ , √ 3 1 1 3 2 − ln2 − ln 2 + 4 8 8 2

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(43)

H. Román-Flores et al.

Thus, from (42) and (43), because 3 3 √ + ≥ 5 2 8 and

we obtain

3 128 ln

√ 3 8 4 + ≥ 5 5  [0,1]



F p+1 dx

[0,1]

F p dx

3 4

√

√

 [0,1]

+ 

2+

3 2



√ 3 + 128 ln 2 + √ 2 3 3 6 5

2−

3

1 1 8 ln2 − 8 ln

G p+1 dx

[0,1]

√

G p dx

√

 [0,1] ≥ 

2+

39 64

3 2

√ 2



(F + G) p+1 dx

[0,1] (F

+ G) p dx

and, consequently, Beckenbach’s inequality (28) is verified.

5 Conclusion In this paper, using the Kulisch–Miranker order on the space KC of non-empty compact and convex subsets of R, we have proved the Minkowski’s inequality (see Theorem ***4) for non-negative interval-valued functions, i.e., for interval-valued functions taken values in + KC = {[a, b] ∈ KC | 0 ≤ a ≤ b}. This fact shows that the functional · p defined by  F p =

[a,b]

1 F(x) p dx

p

  is a seminorm (for p ≥ 1) on the convex and positive cone I [a, b], KC+ of non-negative and integrable interval-functions, opening an interesting route toward the class of L p -type interval spaces. On the other hand, Radon and Beckenbach inequalities have important applications in convex geometry on Rn through the concept of width-integral of convex bodies (see Zhao 2012) and, in this context, in the near future we wish to extend these ideas for studying some problems connected with convexity on the space KCn .

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