Extensions of Chebyshev inequality for fuzzy integral and applications

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1 Extensions of Chebyshev inequlity for fuzzy integrl nd pplictions H. Román-Flores, Y. Chlco-Cno, nd. Flores-Frnulič Instituto de lt Investigción, Universidd de Trpcá bstrct. New extensions of Chebyshev type inequlities for the Sugeno integrl on bstrct spces re studied. More precisely, necessry nd sufficient conditions under which the inequlity ( ( Φ(f gdµ Φ(fdµ Φ(gdµ or its reverse hold for n rbitrry fuzzy mesure-bsed type Sugeno integrl µ nd binry opertion : [, 2 [, nd nonnegtive function Φ : [, [,, re given. lso, nd s n ppliction, we prove Milne s type inequlity for fuzzy integrl. Keywords: Fuzzy integrl, comonotone functions, Chebyshev s inequlity, Milne s inequlity. The theory of fuzzy mesures nd fuzzy integrls ws introduced by Sugeno [24] s tool for modeling nondeterministic problems. Sugeno s integrl is nlogous to Lebesgue integrl which hs been studied by mny uthors, including Pp [18], Rlescu nd dms [19] nd, Wng nd Klir [25], mong others. Román- Flores et l [9, 2 23], strted the studies of inequlities for Sugeno integrl, nd then followed by the uthors [1 4, 6, 11, 13 17]. In generl, ny integrl inequlity cn be very strong tool for pplictions. In prticulr, when we think n integrl opertor s predictive tool then n integrl inequlity cn be very importnt in mesuring nd dimensioning such processes. The im of this pper is to study new frmeworks of Chebyshev type inequlities for Sugeno integrl nd some pplictions. 1 Preliminries In this section, we re going to review some well known results from the theory of fuzzy mesures, Sugeno s integrls. For detils, we refer to [19], [24], [25]. s usul we denote by R the set of rel numbers. Let X be non-empty set nd Σ be σ-lgebr of subsets of X. Let N denote the set of ll positive This work ws prtilly supported by Fondecyt-Chile by projects nd

2 Extensions of Chebyshev inequlity for fuzzy integrl nd pplictions 845 integers nd R + denote [,+ ]. Throughout this pper, we fix the mesurble spce (X,Σ, nd ll considered subsets re supposed to belong to Σ. Definition 1. (Rlescu nd dms [19] set function µ : Σ R + is clled fuzzy mesure if the following properties re stisfied: (FM1 µ( = ; (FM2 B implies µ( µ(b; (FM3 1 2 implies µ( n=1 n = lim n µ( n ; nd (FM4 1 2, nd µ( 1 < + imply µ( n=1 n = lim n µ( n. When µ is fuzzy mesure, the triple (X,Σ,µ then is clled fuzzy mesure spce. Let (X,Σ,µ be fuzzy mesure spce, by F + (X we denote the set of ll nonnegtive mesurble functions f : X [, with respect to Σ. In wht follows, ll considered functions belong to F + (X. If f F + (X, we will denote the set {x X f(x α} by F α for α. Clerly, F α is nonincresing with respect to α, i.e., α β implies F α F β. Definition 2. (Pp [18], Sugeno [24], Wng nd Klir [25] Let (X,Σ,µ be fuzzy mesure spce nd Σ. If f F + (X then the Sugeno integrl of f on, with respect to the fuzzy mesure µ, is defined s = fdµ (α µ( F α. α When = X, then fdµ = fdµ = α. X α (α µ(f Some bsic properties of Sugeno integrl re summrized in [18, 25], we cite some of them in the next theorem. Theorem 1. (Pp [18], Wng nd Klir [25] Let (X,Σ,µ be fuzzy mesure spce, then (i µ( F α α = fdµ α; (ii µ( F α α = fdµ α; (iii fdµ < α there exists γ < α such tht µ( F γ < α; (iv fdµ > α there exists γ > α such tht µ( F γ > α; (v If µ( <, then µ( F α α fdµ α; (vi If f g, then fdµ gdµ; (vii fdµ µ(. Definition 3. Functions f,g: X R re sid to be comonotone if for ll x,y X, (f(xf(y(g(xg(y. nd f nd g re sid to be countermonotone if for ll x,y X, (f(xf(y(g(xg(y.

3 846 H. Román-Flores The comonotonicity of functions f nd g is equivlent to the nonexistence of points x,y X such tht f(x < f(y nd g(x > g(y. lso, it is cler tht if f nd g re comonotone then (see [11,17] for ll nonnegtive rel numbers s,t either F s G t or F t G s. Observe tht the concept of comonotonicity ws first introduced in [8]. Notice tht comonotone functions cn be defined on ny bstrct spce. It is well known tht Sugeno integrl is comonotone mxitive (cf. [7], i.e., if f nd g re comonotone then f gdµ = fdµ gdµ. In recent pper [11], Mesir nd Ouyng proved the following Chebyshev type inequlity for Sugeno integrl: Theorem 2. Let f,g F + (X nd µ be n rbitrry fuzzy mesure such tht fdµ nd gdµ re finite. Let : [, 2 [, be continuous nd nondecresing in both rguments nd bounded from bove by minimum. If f,g re comonotone, then the inequlity f gdµ fdµ gdµ (1 holds. lso, it is known tht f gdµ fdµ gdµ where f,g re comonotone functions whenever mx (see [15]. Notice tht Ineq.(1 together with the monotonicity of Sugeno integrl (Theorem 1(vi imply tht f gdµ = fdµ gdµ. I.e., Sugeno integrl lso hs the comonotone minitive property (cf.[11]. In this contribution, we will prove new frmeworks of Chebyshev type inequlities for the Sugeno integrl. For this, we precise the following definition: Definition 4. Let : [, 2 [, be binry opertion nd consider Φ : [, [,. Then we sy tht Φ is subdistributive over if Φ(x y Φ(x Φ(y for ll x,y [,. nlogously, we sy tht Φ is superdistributive over if for ll x,y [,. Φ(x y Φ(x Φ(y Now, our results cn be stted s follows. (2

4 Extensions of Chebyshev inequlity for fuzzy integrl nd pplictions Min results In this section, we provide new frmeworks of Chebyshev type inequlities for the Sugeno integrl. Now, we stte the min result of this pper. Theorem 3. Let f,g F + (X nd µ be n rbitrry fuzzy mesure. nd let Φ : [, [, be continuous nd nondecresing function nd : [, 2 [, be continuous nd nondecresing in both rguments nd bounded from bove by minimum. If f, g re comonotone nd Φ is superdistributive over, then the inequlity holds. Φ(f gdµ Φ(fdµ Φ(gdµ Proof. Firstly, becuse Φ is nondecresing function nd f, g re comonotone, then Φ(f nd Φ(g re lso comonotone functions. Thus, due to is bounded from bove by minimum, Φ is superdistributive over, nd inequlity (1, we obtin Φ(f gdµ Φ(f Φ(gdµ This completes the proof. Φ(fdµ Φ(gdµ. Remrk 1. If one tkes Φ(x = x in Theorem 3, then Ineq. (1 will be cquired. Remrk 2. Let Φ(fdµ = p nd Φ(gdµ = q, nd let c mx(p,q. Then therequirementof [, c] 2 minisenoughtoensurethevlidityoftheorem3.if = min, then Ineq. (3 remins true when Φ(fdµ nd/or Φ(gdµ re/is finite. Therefore Ineq. (3 together with the monotonicity of Sugeno integrl (Theorem 1(vi imply tht Φ(f gdµ = Φ(fdµ Φ(fdµ. This property is lso equivlent to comonotone minitivity. Remrk 3. If µ is minitive in Theorem 3, then Ineq. (3 holds even when f,g re not comonotone. We cn use the sme exmples in [11] to show the necessities of min nd the comonotonicity of f,g, nd so we omit them here. Theorem 4. Let f,g F + (X nd µ be n rbitrry fuzzy mesure. nd let Φ : [, [, be continuous nd nondecresing function nd : [, 2 [, be continuous nd nondecresing in both rguments nd bounded from (3

5 848 H. Román-Flores below by mximum. If f,g re comonotone nd Φ is subdistributive over, then the inequlity Φ(f gdµ Φ(fdµ Φ(gdµ (4 holds. Proof. Firstly, becuse Φ is nondecresing function nd f, g re comonotone, then Φ(f nd Φ(g re lso comonotone functions. Thus, due to is bounded from below by mximum, Φ is subdistributive over, nd inequlity (2, we obtin Φ(f gdµ Φ(f Φ(gdµ This completes the proof. Φ(fdµ Φ(gdµ. Remrk 4. If one tkes Φ(x = x in Theorem 4, then Ineq. (2 will be cquire. Remrk 5. Let Φ(f gdµ = r nd let c r. Then the requirement of [, c] 2 mx is enough to ensure the vlidity of Theorem 4. If = mx, then Ineq. (4 remins true when Φ(f gdµ is infinite. Therefore Ineq. (4 together with the monotonicity of Sugeno integrl (Theorem 1(vi imply tht Φ(f gdµ = Φ(fdµ Φ(gdµ. This property is equivlent to the comonotone mxitivity of Sugeno integrl. Remrk 6. If µ is mxitive in Theorem 4, then Ineq. (4 holds even when f,g re not comonotone. Exmple 1. Let = + nd the two comonotone functions f,g: [,1] R + be defined s f(x = 1x nd g(x = 1x 2 nd µ( = m( where m is the Lebesgue mesure on R. Let Φ(x = x 1 2. strightforwrd clculus shows tht (i f 1 2 (xdm = α [,1] = α [,1] [α m({(1x 1 2 α}] [α ( 1α 2 ] = =.6183, (ii g 1 2 (xdm = α [,1] = α [,1] [α m({ ( 1x 21 2 α}] [ α ] (1α 2 = 1 2 =.7711, 2

6 Therefore: Extensions of Chebyshev inequlity for fuzzy integrl nd pplictions 849 (iii.7878= (f +g 1 2 (xdm = α [, 2] = α [, 2] [α m({ ( x 2 x α}] [ ( α ] (94α2 2 = = (f +g 1 2 (xdm = ( ( g 1 2 (xdm + f 1 2 (xdm. 3 Milne s type inequlities vi extended Chebyshev s inequlity The clssicl Milne s integrl inequlity [5, 12] estblish tht b f(xg(x f(x+g(x dx b (f(x+g(xdx b f(xdx b g(xdx, (5 where f,g re two positive nd integrble functions on [,b]. However, Milne s inequlity is not vlid for Sugeno integrl s we show in the following exmple. Exmple 2. Consider µ the Lebesgue mesure on [,1] nd let f,g : [,1] R defined by f(x = x 2 nd g(x = 1 2 x 2. Then strightforwrd clculus shows tht i fdµ = α [,1] [α µ({f α}] = α [,1] [α (22α] = ii gdµ = α [,1] [α µ({g α}] = [ ] α [,1] α ( 12α = iiif(x+g(x = 1 (f +gdµ = 1 1 fg iv ( dµ = ] α [,1][ α ( 14α = Thus, f+g ( fg 1 dµ +gdµ =.2367> f +g (f = fdµ nd, consequently, Milne s inequlity (5 is not vlid for the Sugeno integrl. Now we will give converse Milne s type inequlity Sugeno integrl by using the previous results. Theorem 5. (Converse Milne s inequlity Let b > nd let f,g : [,b] R be two positive nd mesurble functions with respect to µ, where µ is the Lebesgue mesure. If f,g re comonotone on [,b], then the inequlity ( b f(xg(x b b b b f(x+g(x dµ f(xdµ+ g(xdµ f(xdµ g(xdµ (6 gdµ

7 85 H. Román-Flores holds. Proof. If we define x y = xy x+y, then x y min{x,y} for ll x,y R +. Thus, tking = [,b], Φ = Id, nd using Theorem 3, we obtin i.e., Φ(f gdµ [,b] b [,b] f(xg(x f(x+g(x dµ nd the proof is completed. Φ(fdµ ( [,b] Φ(gdµ ( b f(xdµ b g(xdµ b f(xdµ+ b g(xdµ,, 4 Conclusion We hve introduced new frmeworks of Chebyshev type inequlities for Sugeno integrls on bstrct spces. More precisely, necessry nd sufficient conditions under which the inequlity Φ(f gdµ Φ(fdµ Φ(gdµ or its reverse hold for n rbitrry fuzzy mesure-bsed type Sugeno integrl µ ndbinryopertion : [, 2 [, ndnonnegtivefunction Φ : [, [,, re given. lso, some ides to explore Milne s type inequlities hve been presented. (7 References 1. H. ghi, R. Mesir, Y. Ouyng, Generl Minkowski type inequlities for Sugeno integrls, Fuzzy Sets nd Systems 161 ( H. ghi, R. Mesir, Y. Ouyng, New generl extensions of Chebyshev type inequlities for Sugeno integrls, Interntionl Journl of pproximte Resoning 51 ( H. ghi, H. Román-Flores,. Flores-Frnulič, Generl Brnes-Godunov-Levin type inequlities for Sugeno integrl, Informtion Sciences 181 ( H. ghi, R. Mesir, Y. Ouyng, type inequlities for Sugeno integrl nd T-(S- evlutors, Informtion Sciences 19 ( H. lzer,. Kovčec, The inequlity of Milne nd its converse, J. of Inequl. & ppl. 7 ( H. ghi,. Mohmmdpour, S. Mnsour Vezpour, generliztion of the Chebyshev type inequlities for Sugeno integrls, Soft Computing 16(4 ( P. Benvenuti, R. Mesir, D. Vivon, Monotone set functions-bsed integrls In: E. Pp, editor, Hndbook of Mesure Theory, Vol II, Elsevier, (

8 Extensions of Chebyshev inequlity for fuzzy integrl nd pplictions C. Dellcherie, Quelques commentires sur les prolongements de cpcités, in: Seminire de Probbilites (1969/7, Strsbourg, Lecture Notes in Mthemtics, Vol. 191 (Springer, Berlin, Flores-Frnulič, H. Román-Flores, Chebyshev type inequlity for fuzzy integrls, pplied Mthemtics nd Computtion 19 ( E.P. Klement, R. Mesir, E. Pp, Tringulr norms, Trends in Logic. Studi Logic Librry, Vol. 8, Kluwer cdemic Publishers, Dodrecht, R. Mesir, Y. Ouyng, Generl Chebyshev type inequlities for Sugeno integrls, Fuzzy Sets nd Systems 16 ( E.. Milne, Note on Rosselnd s integrl for the stellr bsorption coefficient, Monthly Notices Roy. stronom. Soc. 85 ( Y. Ouyng, J. Fng, Sugeno integrl of monotone functions bsed on Lebesgue mesure, Computers nd Mthemtics with pplictions 56 ( Y. Ouyng, J. Fng, L. Wng, Fuzzy Chebyshev type inequlity, Interntionl Journl of pproximte Resoning 48 ( Y. Ouyng, R. Mesir, On the Chebyshev type inequlity for seminormed fuzzy integrl, pplied Mthemtics Letters 22 ( Y. Ouyng, R. Mesir, Sugeno integrl nd the comonotone commuting property, Interntionl Journl of Uncertinty, Fuzziness nd Knowledge-Bsed Systems 17 ( Y. Ouyng, R. Mesir, J. Li, On the comonotonic- -property for Sugeno integrl, pplied Mthemtics nd Computtion 211 ( E. Pp, Null-dditive Set Functions, Kluwer, Dordrecht, D. Rlescu, G. dms, The fuzzy integrl, Journl of Mthemticl nlysis nd pplictions 75 ( H. Román-Flores, Y. Chlco-Cno, H-continuity of fuzzy mesures nd set defuzzifinction, Fuzzy Sets nd Systems 157 ( H. Román-Flores, Y. Chlco-Cno, Sugeno integrl nd geometric inequlities, Interntionl Journl of Uncertinty, Fuzziness nd Knowledge-Bsed Systems 15 ( H. Román-Flores,. Flores-Frnulič, Y. Chlco-Cno, The fuzzy integrl for monotone functions, pplied Mthemtics nd Computtion 185 ( H. Román-Flores,. Flores-Frnulič, Y. Chlco-Cno, Jensen type inequlity for fuzzy integrls, Informtion Sciences 177 ( M. Sugeno, Theory of fuzzy integrls nd its pplictions, Ph.D. thesis. Tokyo Institute of Technology, Z. Wng, G. Klir, Fuzzy Mesure Theory, Plenum Press, New York, 1992.

FURTHER DEVELOPMENT OF CHEBYSHEV TYPE INEQUALITIES FOR SUGENO INTEGRALS AND T-(S-)EVALUATORS

K Y BERNETIK VOLUM E 46 21, NUMBER 1, P GES 83 95 FURTHER DEVELOPMENT OF CHEBYSHEV TYPE INEQULITIES FOR SUGENO INTEGRLS ND T-S-EVLUTORS Hamzeh gahi, Radko Mesiar and Yao Ouyang In this paper further development