Some integral inequalities for interval-valued functions

Com. Al. Math. (08) 7:06 8 htts://doi.org/0.007/s404-06-096-7 Some integral inequalities for interval-valued functions H. Román-Flores Y. Chalco-Cano W. A. Lodwic Received: 8 October 0 / Revised: March 05 / Acceted: 6 October 06 / Published online: 6 November 06 SBMAC - Sociedade Brasileira de Matemática Alicada e Comutacional 06 Abstract In this aer, we exlore some integral inequalities for interval-valued functions. More recisely, using the Kulisch Miraner order on the sace of real comact intervals, we establish Minowsi s inequality then we derive Becenbach s inequality via an interval Radon s inequality. Also, some examles alications are resented for illustrating our results. Keywords Interval-valued functions Minowsi s inequality Radon s inequality Becenbach inequality Mathematics Subject Classification 6D5 6E5 8B0 Introduction The imortance of the study of set-valued analysis from a theoretical oint of view as well as from their alication is well nown (see Aubin Cellina 984; Aubin Fransowsa 990). Also, many advances in set-valued analysis have been motivated by control theory Communicated by Maro Rojas-Medar. This wor was suorted in art by Conicyt-Chile through Projects Fondecyt 0674 0665. Also, W. A. Lodwic was suorted in art by FAPESP 0/985. B H. Román-Flores hroman@uta.cl Y. Chalco-Cano ychalco@uta.cl W. A. Lodwic Weldon.Lodwic@ucdenver.edu Instituto de Alta Investigación, Universidad de Taraacá, Casilla 7D, Arica, Chile Deartment of Mathematical Statistical Sciences, University of Colorado, Denver, CO 807, USA

Some integral inequalities for interval-valued functions 07 dynamical games, in addition, otimal control theory mathematical rogramming were a motivating force behind set-valued analysis since the sixties (see Aubin Fransowsa 000). Interval Analysis is a articular case it was introduced as an attemt to hle interval uncertainty that aears in many mathematical or comuter models of some deterministic real-world henomena. The first monograh dealing with interval analysis was given by Moore (966). Moore is recognized as the first to use intervals in comutational mathematics, now called numerical analysis. He also extended imlemented the arithmetic of intervals to comuters. 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 985). On the other h, several generalizations of classical integral inequalities were obtained in the recent years by Agahi et al.(00, 0a, b, 0), Flores-Franulič Román-Flores (007), Flores-Franulič et al.(009), Mesiar Ouyang (009), Román-Flores Chalco- Cano (007), Román-Flores et al. (007a, b, 008, 0), in the context of non-additive measures integrals (also see the following related references: Pa 995; Ralescu Adams 980; Román-Flores Chalco-Cano 006; Sugeno 974; Wang Klir 009). In general, any integral inequality can be a very owerful tool for alications, in articular, when we thin an integral oerator as a redictive tool then an integral inequality can be very imortant in measuring, comuting errors delineating such rocesses. Interval-valued functions (or fuzzy-interval valued functions) may rovide an alternative choice for considering the uncertainty into the rediction rocesses, in connection with this, the Aumann integral for interval-valued function is the natural-associated exectation (see for examle Puri Ralescu 986). Also, several integral inequalities involving functions their integrals derivatives, such as Wirtinger s inequality, Ostrowsi s inequality, Oial s inequality, among others, have been extensively studied during the ast century (see for examle Anastassiou 0; Mitrinović et al. 99). All these studies have been fundamental tools in the develoment of many areas in mathematical analysis. Recently, some differential-integral inequalities have been extended to the set-valued context. For examle Anastassiou (0), using the Huuhara derivative, extended an Ostrowsi tye inequality to the context of fuzzy-valued functions. Chalco-Cano et al. (0) using the concet of generalized Huuhara differenciability (see Chalco-Cano et al. 0, 0) establish some Ostrowsi tye inequalities for interval-valued functions. This resentation generalizes Minowsi s inequality for interval-valued functions, as an alication, establishes the Becenbach s inequality for interval-valued functions via an interval Radon s inequality. Preliminaries. Interval oerations Let R be the one-dimensional Euclidean sace. Following Diamond Kloeden (994), let K C denote the family of all non-emty comact convex subsets of R,thatis, K C {[a, b a, b R a b}. () The Pomeiou Hausdorff metric on K C (Pomeiu was the first mathematician introducing the concet of set distance, see Birsan Tiba 006), frequently called Hausdorff metric, is defined by

08 H. Román-Flores et al. H(A, B) max {d(a, B), d(b, A)}, () where d(a, B) max a A d(a, B) d(a, B) min b B d(a, b) min b B a b. Remar An equivalent form for the Hausdorff metric defined in ()is M ([ a, a, [ b, b ) max { a b, a b } which is also nown as the Moore metric on the sace of intervals (see Moore Kearfott 009, Eq. (6.),. 5). It is well nown that (K C, H) is a comlete metric sace (see Aubin Cellina 984; Diamond Kloeden 994). If A K C then we define the norm of A as A H (A, 0) H ([0, 0). The Minowsi sum scalar multilication are defined on K C by means A B {a b a A, b B} λa {λa a A}. () Also, if A [a, a B [b, b are two comact intervals then we define the difference the roduct A B [ a b, a b, (4) A B [ min { ab, ab, ab, ab }, max { ab, ab, ab, ab }, (5) the division [ { A a B min b, a b, a b, a } { a, max b b, a b, a b, a }, (6) b whenever 0 / B. An order relation isdefinedonk C as follows (see Kulisch Miraner 98): [a, a [b, b a b a b. (7) Remar We note that if [a, b, [c, d, [x, y are intervals with ositive enoints then [a, b [x, y [a, b [x, y (8) [c, d [c, d [a, b [a, b [c, d [x, y [c, d [x, y. (9) The sace K C is not a linear sace since it does not ossess an additive inverse therefore subtraction is not well defined (see Aubin Cellina 984; Marov 979). However, one very imortant roerty of interval arithmetic is that: A, B, C, D K C, A B, C D A C B D (0) where can be sum, subtraction, roduct or division. Proerty (0) is called the inclusion isotony of interval oerations it is recognized as the fundamental rincile of interval analysis (see Moore Kearfott 009). One consequence of this is that any function f (x) described by an exression in the variable x which can be evaluated by a rogrammable real calculation can be embedded in interval calculations using the natural corresondence between oeration so that if x X K C then f (x) f (X), where f (X) is interreted as the calculation of f (x) with x relaced by X

Some integral inequalities for interval-valued functions 09 the oerations relaced by interval oerations. The evaluation f (X) is called the natural interval extension of the exression f (x). Also it is necessary to observe that two different exressions for a same real function can result in very different interval-valued functions. For instance, if we consider f (x) x( x) f (x) x x taing X [, 5 then, on the one h we obtain f (X) [, 5 ([, [, 5) [ 0, 6, on the other we have f (X) [, 5 [, 5 [, 4. Denote the range of a function f (x) over an interval X as R( f ) X { f (x) x X}. () Then from (0) it follows for the natural interval extension that R( f ) X f (X). () As an examle consider the function f (x) x x X [,. Then R( f ) X [ 4,, whereas using the natural interval extension of f (x) we obtain f (X) [, [, [,, which is a more larger interval but nevertheless contains R( f ) X. For the articular case when f (x) is monotone continuous over an interval X [a, b, we can define, in this case, f (X) R( f ) X. For examle: f ([a, b) [min { f (a), f (b)}, max { f (a), f (b)} () (a) if f (x) x r, r > 0, 0 a b then f ([a, b) [a, b r [ a r, b r. (4) (b) If g(x) e x then the exonential of an interval [a, b is defined as [ g ([a, b) e e a, e b. (5) For more details on interval oerations interval analysis see Marov (979), Moore (966)Rone (00).. Integral of interval-valued functions If T [a, b is a closed interval F : T K C 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 f are called the lower the uer (endoint) functions of F, resectively. For interval-valued functions it is clear that F : T K C is continuous at t 0 T if lim t t0 F(t) F(t 0 ), (6)

0 H. Román-Flores et al. where the limit is taen in the metric sace (K C, H). Consequently, F is continuous at t 0 T if only if its endoint functions f f are continuous functions at t 0 T. We denote by C ([a, b, K C ) the family of all continuous interval-valued functions. Definition Let M the class of all Lebesgue measurable sets of T,then (a) the function f : T R is measurable if only if f (C) M for all closed subset C of R. (b) the interval-valued function F : T K C is measurable if only if F ω (C) {t T/F(t) C } M, C R, C closed. (c) Also, if F : T K C is an interval-valued function f : T R, then we say that f is a selector (or selection) of F if 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 (Aubin Cellina 984) LetF : T K C 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), (7) a where S(F) is the set of all integrable selectors of F, i.e., a S(F) { f : T R f integrable f (t) F(t) for all t T }. If S(F), then the integral exists F is said to be integrable (Aumann integrable). Note that if F is integrable then it has a measurable selector which is integrable, consequently, S(F). Also, in above definition, the integral symbol b a F(t)dt /or b a f (t)dt denotes the integral with resect to the Lebesgue measure. Definition We say that a maing F : T K C is integrally bounded if there exists a ositive integrable function g : T R such that F(t) g(t), forallt T. Theorem (Aubin Cellina 984) Let F : T K C be a measurable integrally bounded interval-valued function. Then it is integrable b a F(t)dt K C. Corollary (Aubin Cellina 984; Diamond Kloeden 994) A continuous intervalvalued function F : T K C is integrable. The Aumann integral satisfies the following roerties. Proosition (Aubin Cellina 984; Diamond Kloeden 994) Let F, G : T K C be two measurable integrally bounded interval-valued functions. Then (i) t t (F(t) G(t)) dt t t F(t)dt t t G(t)dt, a t t b (ii) t t F(t)dt τ t F(t)dt t τ F(t)dt, a t τ t b.

Some integral inequalities for interval-valued functions Theorem (Bede Gal 005) Let F : T K C be a measurable integrally bounded interval-valued function such that F(t) [f (t), f (t).then f f are integrable functions t t [ t t t t F(t)dt f (t)dt, f (t)dt. (8) Remar Above Theorem is a direct consequence of two relevant results: (a) (Aumann 965, Theorem,. ) T F(t)dt is convex. (b) (Aumann 965, Theorem 4,. ) If F is closed-valued then T F(t)dt is comact. 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 h, if f S(F) then f (t) f (t) f (t), forallt T, which imlies that [ f (t)dt f (t)dt, f (t)dt T T T, consequently, T F(t)dt [ T f (t)dt, T f (t)dt. Therefore equality (8) in Theorem holds. To finalize this section, we give an examle of rediction under uncertainty using interval tools (see Puri Ralescu 986). Examle Toss a fair coin. Denote the outcomes Tail by T Head by H. Suose a layer loses aroximately 0 EUR if the outcome is T, wins an amount much larger than 00 EUR but not much larger than 000 EUR if the outcome is H. The question here is: what is the exected value for the next outcome? To reresent the uncertainty contained in the above linguistic descritions, we can define the interval rom variable X : {T, H} K C,where (a) X (T ) aroximately 0, (b) X (H) much larger than 00 but not much larger than 000. Furthermore, if E {T, H} then we can consider the measure sace (E, P(E), μ) taing as is usual: μ(t ) μ(h), μ(x) μ( ) 0. Suose we interret the linguistic variables as X (T ) [, 8 X (H) [50, 00. Now, we can write X (z) [f (z), f (z) where f, f : {T, H} R are defined by f (T ), f (H) 50, f (T ) 8, f (H) 00. So, using roerties of the Aumann integral we have f dμ f dμ f dμ ( ) (50) 9 E E f dμ {T } {T } {H} f dμ f dμ {H} ( 8) (00) 50. Thus, the exected value for the next outcome is the interval E(X) [9, 50.

H. Román-Flores et al. Minowsi s inequality The well-nown inequality due to Minowsi can be stated as follows (see Hardy et al. 94,. ): Theorem Let f (x), g(x) 0, then ( f (x) g(x)) f (x) g(x) (9) with equality if only if f g are roortional, if 0 < <, then ( f (x) g(x)) f (x) g(x) ) (0) with equality if only if f g are roortional. We recall that two non-negative functions f g are roortional if only if there is a non-negative real constant such that f g (or g f). Now, using above theorem roerties of interval integration, we can rove the following interval version of Minowsi s inequality: Theorem 4 (Interval Minowsi s inequality) If F, G :[a, b K C are two integrable interval-valued functions, with F [f, f, G[g, g, f(x), g(x) 0,then (F(x) G(x)) F(x) G(x) ) () with equality if F G are roortional, if 0 < <, then (F(x) G(x)) F(x) G(x) ) () with equality if F G are roortional. Proof Due to (), (7), (), (4), Theorem, Theorem (9), we have (F(x) G(x)) [ f (x) g(x), f (x) g(x) ([ [ [ [( ) ( ) f (x) g(x), f (x) g(x) ( ) ( ) f (x) g(x), f (x) g(x) ( ) ( ) f (x) g(x), f (x) g(x) f (x) g(x), f (x) g(x)

Some integral inequalities for interval-valued functions [ [ f (x), f (x) [, f (x) g(x), g(x) [ [ f (x), f (x) f (x) [ f (x), f (x) F(x) G(x) g(x), [ g(x), g(x) [ g(x), g(x) g(x) Analogously, using (), (7), (), (4), Theorems (0), we can rove the second art of our theorem for 0 < <. Finally, a straightforward calculation shows that equality is reached if F G are roortional. This comletes the roof. 4 Becenbach s inequality The well-nown Becenbach s inequality can be stated as follows (see Becenbach Bellman 99,. 7): Theorem 5 (Becenbach Bellman 99) If 0 < <, f (x), g(x) >0, then ( f (x) g(x)) f (x) g(x) ( f (x) g(x)) f (x) g(x). () The aim of this section is to show a Becenbach tye inequality for interval-valued functions, for this we will use an interval version of the Radon s inequality. We recall that the classical Radon s inequality (ublished by Radon 9), establishes that Theorem 6 (Radon 9) For every real numbers > 0, x 0, a > 0, for n, the inequality n x ( n ) x a ( n ) (4) a holds. Inequality (4) has been widely studied by many authors because of its utility in ractical theoretical alications (see for examle Mortici 0; Zhao 0). The next result is an extension of Radon s inequality to the interval context. Theorem 7 (Interval Radon s inequality) Let [x, x, [a, a K C, with x 0,a > 0, for all n. If > 0, then the inequality n [x, x ( n [x [a, a, x ) ( n [a, a ) (5) holds.

4 H. Román-Flores et al. Proof Woring on the left-side of (5), we have n [x, x n [x, x [a, a n [ n [a, a [ x a x a, x a n, x a. On the other h, woring on the right-side of (5), we obtain ( n [x, x ) ( [ n ( n [a, a ) x, n x ) ( [ n a, n a ) [ ( n ) ( x, n ) x Now, by Radon s inequality (4) wehave n x a n x a which imlies that inequality (5) holds. [( n a ), ( n a ) [ ( n ) ( x n ) x ( n ), ( a n ). a ( n x ) ( n a ) (6) ( n x ) ( n a ) (7) Theorem 8 (Interval Becenbach s inequality) If F, G :[a, b K C are two integrable interval-valued functions, with F [f, f, G[g, g, f(x), g(x) >0 0 < <, then Proof Taing (F(x) G(x)) (F(x) G(x)) I I F(x) G(x) F(x) G(x). (8) F(x), J F(x) ) (9) G(x) using the intervalar Radon inequality (5) wehave I J I J, J G(x) ) (0) (I I ) (J J ), ()

Some integral inequalities for interval-valued functions 5 that is to say, F(x) F(x) G(x) G(x) ( F(x) ( G(x) ) ( ( ). () F(x) G(x) Now, because 0 < < then< <, due to ()(), we obtain (F(x) G(x)) F(x) G(x), () (F(x) G(x)) F(x) G(x) ). (4) Finally, due (8), (9), ()(4) we obtain that ( ( ) F(x) ) ) G(x) ( ( ) F(x) G(x) (F(x) G(x)) (F(x) G(x)), (5) the roof is comleted. Examle Let / letf, G :[0, K C two interval-valued functions defined by F(x) [x, x G(x) [x, x, with x [0,. Using(4), interval oerations roerties of the Aumann integral, a straightforward calculation shows that: On the other h, (F G) F(x) G(x) [ 5, 5, [ 4,, 5 F(x) G(x) [, (6) [,. (7) [ ( 8 ln( ) ( ) 9 ln ), 8 64 (8) [ ( (F G) 4 8 ln 8 ln( ) ),. (9) 9

6 H. Román-Flores et al. Also, [ ( ) ( ) F(x),, 5 5 [ F(x) 4 9, 8 9 (40) G(x) [ ( ) ( ),, 4 5 [ G(x) 4, 4. (4) 9 Thus, from (8), (40)(4), because we obtain that ( 8 ln (F G) ( ) ( 9 ln ) 8 64 5 ( ( ) ( ) ), 5 5, consequently, Minowsi s inequality () is verified. Analogously, from (9)to (4), because ) ( ) 4 F G ( 4 8 ln ( )) 8 ln 4 9 4 we obtain that (F G) 9 8 9 4 9, F G, consequently, Minowsi s inequality () is verified. Additionally, F F G [ G 5, 8 5 [ 8, 4 5 (4) (F G) (F G) 8 ln ( ) 8 ln 64 9, 4 6 5 ( ). (4) 8 ln 8 ln

Some integral inequalities for interval-valued functions 7 Thus, from (4)(4), because we obtain 5 8 8 5 ( ) 8 ln 8 ln 64 9 4 6 5 5 ( ) 8 ln 8 ln 4 F F G G, consequently, Becenbach s inequality (8) is verified. (F G) (F G) 5 Conclusion In this aer, using the Kulisch Miraner order on the sace K C of non-emty comact convex subsets of R, we have roved the Minowsi s inequality (see Theorem ***4) for non-negative interval-valued functions, i.e., for interval-valued functions taen values in K C {[a, b K C 0 a b}. This fact shows that the functional defined by F F(x) is a seminorm (for ) on the convex ositive cone I ( [a, b, K C ) of non-negative integrable interval-functions, oening an interesting route toward the class of L -tye interval saces. On the other h, Radon Becenbach inequalities have imortant alications in convex geometry on R n through the concet of width-integral of convex bodies (see Zhao 0), in this context, in the near future we wish to extend these ideas for studying some roblems connected with convexity on the sace K n C. References Agahi H, Mesiar R, Ouyang Y (00) General Minowsi tye inequalities for Sugeno integrals. Fuzzy Sets Syst. 6:708 75 Agahi H, Ouyang Y, Mesiar R, Pa E, Štrboja M (0) Hölder Minowsi tye inequalities for seudointegral. Al. Math. Comut. 7:860 869 Agahi H, Román-Flores H, Flores-Franulič A (0) General Barnes Godunova Levin tye inequalities for Sugeno integral. Inf. Sci. 8:07 079 Agahi H, Mesiar R, Ouyang Y, Pa E, Strboja M (0) General Chebyshev tye inequalities for universal integral. Inf. Sci. 87:7 78 Anastassiou GA (0) Advanced Inequalities. World Scientific, New Jersey Aubin JP, Cellina A (984) Differential Inclusions. Sringer, New Yor Aubin JP, Fransowsa H (990) Set-Valued Analysis. Birhäuser, Boston Aubin JP, Fransowsa H (000) Introduction: set-valued analysis in control theory. Set Valued Anal. 8: 9 Aumann RJ (965) Integrals of set-valued functions. J. Math. Anal. Al. :

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