Hermite-Hadamard Type Fuzzy Inequalities based on s-convex Function in the Second Sense

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1 Mthemtic Letter 7; 3(6): doi: 648/jml7364 ISSN: X (Print); ISSN: (Online) Hermite-Hdmrd Type Fuzzy Inequlitie ed on -Convex Function in the Second Sene Lnping Li School of Mthemtic nd Sttitic, Hunn Univerity of Finnce nd Economic, Chngh, Chin Emil ddre: To cite thi rticle: Lnping Li Hermite-Hdmrd Type Fuzzy Inequlitie ed on -Convex Function in the Second Sene Mthemtic Letter Vol 3, No 6, 7, pp 77-8 doi: 648/jml7364 Received: Octoer 7, 7; ccepted: Novemer 4, 7; Pulihed: Decemer 5, 7 trct: Integrl inequlitie hve importnt ppliction in propility nd engineering field Sugeno integrl i n importnt fuzzy integrl in fuzzy theory, which h mny ppliction in vriou field The oject of thi pper i to develop ome new integrl inequlitie for Sugeno integrl Bed on clicl Hermite-Hdmrd type inequlity, thi pper intend to extend it for the Sugeno integrl Some new Hermite-Hdmrd type inequlitie re derived for Sugeno integrl ed on - convex function in the econd ene n exmple i ued to illutrte the effectivene of the new inequlitie Keyword: Fuzzy Integrl, Sugeno Integrl, Herimite-Hdmrd Inequlity, -Convex Function Introduction In ome prcticl ppliction prolem, the dt informtion ometime cnnot e preciely expreed due to humn error or the limittion of deciion mker knowledge or other reon For intnce, the temperture in room cnnot e meured exctly ecue of the fluctution [] Zdeh firt introduce the concept of fuzzy et Since then fuzzy et h received gret ttention nd got lot of ppliction in vriou field [-5] Mny extenion of Zdeh fuzzy et, uch vgue et, intuitionitic fuzzy, heitnt fuzzy et, etc, hve een propoed nd een utilized mngement deciion nd engineering cience prolem [6- ] Fuzzy integrl, firt introduced y Sugeno in 974, i n importnt nlyticl tool to meure uncertin informtion [3-8] Hermite-Hdmrd inequlitie re importnt integrl inequlity for convex function Mny reult re derived for Hermite-Hdmrd inequlitie ed on vriou type convex function For exmple, Wng et l [7] etlihed ome new Hermite-Hdmrd type inequlitie involving Riemnn-Liouville frctionl integrl vi -convex function in the econd ene Sriky [8] derived etlihed Hermite-Hdmrd type inequlitie for the cl of function whoe econd derivtive re -convex in the econd ene Ltif [9] etlihed everl new inequlitie of the Hermite Hdmrd type for function whoe derivtive re -convex in the econd ene in the olute vlue Hoeini et l [] derived everl verion of Hermite Hdmrd type inequlity for peudo-frctionl integrl The tudy of inequlitie for Sugeno integrl, which w initited y Romn-Flore et l [], i mot populr From ove dicuion, we ee tht mot of extended Hermite- Hdmrd inequlitie re etlihed for definite integrl, ut the reerch ed on fuzzy integrl i till rre [-6] -convex function in the econd ene i kind of importnt convex function, nd ordinry convex function i it pecil ce [7] Under definite integrl, mny uthor re intereted in uilding inequlity for thi function In thi pper, we will extend the Hermite-Hdmrd type inequlity for -convex function in the econd ene for Sugneo integrl The orgniztion of thi rticle i follow: Second will recll the definition nd propertie of Sugeno integrl nd convex function Section 3 will etlih ome new Hermite-Hdmrd type inequlitie for -convex function in the econd ene ed on Sugeno integrl Finlly, concluion re provided in Section 4 Preliminrie In thi ection, ome concept nd propertie of Sugeno integrl nd -convex function in the econd ene re introduced In the follow we lwy denote yr the et of rel numer Suppoed tht X i nonempty et nd

2 Mthemtic Letter 7; 3(6): R + = [, ), Σ i σ lger of uet of X Definition [3] We cll mpping µ : Σ R+ nondditive meure, if it i non-negtive et function, nd tifie the following propertie: (i) µ ( Φ ) = (ii) B, Σ nd B µ ( ) µ ( B) (iii) For ll n, n Σ nd µ ( n ) = lim µ ( n ) n= n, (iv) For lln, n Σ, nd µ ( ) < +, µ ( n ) = lim µ ( n ) n= n The triple ( X, Σ, µ ) i clled fuzzy meure pce For ny α, we denote Fα = { x x X, f ( x) α} = { f α} Then it i eily proved tht if α β then F F Definition [, 8] Let ( X, Σ, µ ) e fuzzy meure pce, f : X R + e non-negtive meurle function, Σ The Sugeno integrl of function f on et i defined In prticulr, if = X, then µ = α µ F α () α ( S) fd [ ( )] µ = α µ F α () X α ( S) fd [ ( )] Here nd re the opertion up nd inf on R + = [, ), repectively Propoition [8] Let ( X, Σ, µ ) e fuzzy meure pce, B, Σ f nd g re non-negtive meurle function Then (i) ( S) fd µ µ ( ) (ii) ( S) kdµ k µ ( ), k R +, (iii) If f g on the et, then ( S) fd µ ( S) gd µ (iv) If B, Σ, B, then ( S) fdµ ( S) fd µ (v) If µ ( { f α}) α, then B ( S) fd µ α β α (vi) If µ ( { f α}) α, then ( S) fd µ α (vii) ( S) fd µ < α There exit γ < α, t µ ( { f γ }) < α (IX) ( S) fd µ > α There exit γ > α, t µ ( { f γ }) > α (X) µ ( ) < +, then ( S) fd µ α µ ( { f α}) α Remrk Let the ditriution ocited to f on i F( α ) = µ ( { f α}), then ccording to the propertie (V) nd (VI) of Propoition, we hve tht F( α) = α ( S) fd µ = α (3) Then from numericl point of view, the Sugeno integrl () cn e olved the olution of the eqution F( α) = α Definition3 [7] Let (,] e rel numer function f : I R+ R i id to e -convex in the econd ene if f ( λx + ( λ) y) λ f ( x ) + ( λ) f ( y) (4) hold for ll ( x, y) I nd λ [,] Remrk If =, then one cn otin the definition of ordinry convex function Denote y K the et of ll - convex function in the econd ene Lemm [9] Let x, y, then the inequlity θ θ θ ( x + y) x + y (5) hold for θ (,] Lemm [9] Let x [,], then the inequlity hold for (,] ( x) x (6) 3 Hermite-Hdmrd Type Inequlitie for Sugeno IntegrlBed on -Convex in the Second Sene Hermite-Hdmrd Type inequlity provide etimte ofthe men vlue of nonnegtive nd convex function f :[, ] R with the following inequlity ([3-33]) f ( ) f ( ) f ( + ) f ( xdx ) + (7) Unfortuntely, Exmple will how tht Hermite- Hdmrd typeinequlity for Sugeno integrl ed on - convex function in the econd enei not vlid

3 79 Lnping Li: Hermite-Hdmrd Type Fuzzy Inequlitie ed on -Convex Function in the Second Sene Exmple Conider the univere et X = [,] nd let µ e the Leegue meure on X If we tke the function x f ( x ) =, then y reference [7], we cn eily know tht 4 f ( x) K Clculte the Sugeno integrl ( S) fdµ y Remrk, we get On the other hnd, ( S) fdµ = 3 7 f () + f () = 5 Thi prove tht the Hermite-Hdmrd type inequlity i not tifiedfor Sugeno integrl ed on -convex function in the econd ene In thi ection, we will derived ome new Hermite- Hdmrd type inequlitie for the Sugeno integrl ed on -convex function in the econd ene Theorem Let (,] nd f :[,] [, ) i - convex function in the econd ene, uch tht f () > f () nd let µ e the Leegue meure on R, then ( S) fdµ min{ β,} (8) where β i the poitive rel olution of the eqution Proof / β - f () f () f () f ( x) K for x [,], we hve f ( x) = f (( x) + x ) ccording to Lemm, we hve ( x) f () + x f () (9) () f ( x) f () + x ( f () f ()) = g( x) () Theny (iii) of Propoition nd Definition, we hve, () ( S) f ( x) dµ ( S) g( x) dµ µ = β µ g β (3) β ( S) g( x) d [ ([,] { })] In order to clculte the integrl ( S) g( x) dµ, we conider the ditriution function F ocited to g( x) on [, ] which i given y Tht i F( β ) = µ ([,] { g β}) F( β ) = µ ([,] { x f () + x ( f () f ()) β}) When f () > f (), then - β f () F( β ) = µ [,] x x f () f () - β f () = f () f () / / β - f () f () f () By (i) of Propoition nd (4), we hve / ( S) f ( x) dµ ( S) g( x) dµ = min{ β,} (4) Then we complete the proof Theorem Let (,] nd f :[,] [, ) i - convex function in the econd ene, uch tht f () < f () nd let µ e the Leegue meure on R, then ( S) fdµ min{ β,} (5) where β i the poitive rel olution of the eqution β - f () f () f () / (6) Proof Similrly to the proof of Theorem, we conider the function g( x) = f () + x ( f () f ()) (7) In thi ce, the ditriution function F ocited to g( x) on [, ] which i given y Tht i F( β ) = µ ([,] { g β}) F( β ) = µ ([, { g β}) = µ ([,] { x f () + x ( f () f ()) β}) When f () < f (), then

4 Mthemtic Letter 7; 3(6): β f () F( β ) = µ [,] x x f () f () - β f () = f () f () / β - f () f () f () / By (i) of Propoition nd (8), we otined / ( S) f ( x) dµ ( S) g( x) dµ = min{ β,} (8) Thi complete the proof Remrk 3 In the ce f () = f (), then function g( x ) of Theorem nd i g( x) = f () ccording to (ii) of Propoition, we hve µ µ = f (9) ( S) f ( x) d ( S) g( x) d min{ (),} Now, we will prove the generl ce of Theorem nd Theorem Theorem 3 Let (,] nd f :[, ] [, ) i - convex function in the econd ene, uch tht f ( ) > f ( ) nd let µ e the Leegue meure on R, then ( S) f dµ min{ β, } () where β i the poitive rel olution of the eqution Proof / β - f ( ) ( ) f ( ) f ( ) f ( x) K for x [, ], we hve x m x m f ( x) = f + m m x m x m f ( ) + f ( ) m m () Where f ( x) g( x) (3) x m g( x) = f ( ) + ( f ( ) f ( )) m Theny (iii) of Propoition nd Definition, we hve ( S) f ( x) dµ ( S) g( x) dµ (4) (5) µ = β µ g β (6) β ( S) g( x) d [ ([, ] { })] In order to clculte the integrl ( S) g( x) dµ, we conider the ditriution function F ocited to g( x) on [, ] which i given y F( β ) = µ ([, ] { g β}) Tht i x m F( β ) = µ [, ] x f ( ) + ( f ( ) f ( )) β m When f ( ) > f ( ), then - β f ( ) F( β ) = µ [, ] x x m + ( m) f ( ) f ( ) - β f ( ) = ( m) f ( ) f ( ) / / β - f ( ) ( m) f ( ) f ( ) By (i) of Propoition nd (7), we otined ( S) f ( x) dµ ( S) g( x) dµ = min{ β, } / (7) Then we complete the proof Theorem 4 Let (,] nd f :[, ] [, ) i - convex function in the econd ene, uch tht f ( ) < mf ( ) nd let µ e the Leegue meure on R, then ccording to Lemm, we hve Then x x () ( S) fdµ min{ β, } (8) where β i the poitive rel olution of the eqution

5 8 Lnping Li: Hermite-Hdmrd Type Fuzzy Inequlitie ed on -Convex Function in the Second Sene / β - f ( ) ( m) f ( ) f ( ) (9) Proof Similrly to the proof of Theorem, we conider the function x m g( x) = f ( ) + ( f ( ) f ( )) m (3) In thi ce, the ditriution function F ocited to g( x) on [, ] which i given y Tht i F( β ) = µ ([, ] { g β}) F( β ) = µ ([, ] { g β}) x m = µ [, ] x f ( ) + ( f ( ) f ( )) β m When f ( ) < f ( ), then - β f ( ) F( β ) = µ [, ] x x m + ( m) f ( ) f ( ) - β f ( ) = ( m) f ( ) f ( ) / / ( S) fdµ min{ β, } = 68 Strightforwrd clcultion how tht ( S) fdµ = 68 Thi lo implie the inequlity cn get well etimte of Sugeno integrl ( S) fdµ 4 Concluion Sugeno integrl i n importnt fuzzy integrl in fuzzy theory, which h mny ppliction in mngement deciion nd engineering field Integrl inequlitie re n importnt tool for etimting the vlue of integrl Hermite-Hdmrd type inequlitie provide etimte of the men vlue of nonnegtive nd ordinry convex function, nd thu received gret ttention in definite integrl The min contriution of thi pper i to develop Hermite- Hdmrd type inequlitie for Sugeno integrl Thi pper etlihed n upper pproximtion for the Sugeno integrl of -convex function in the econd ene In the future tudy, we will tudy the other propertie nd inequlitie out -convex function in the econd ene for ome other fuzzy integrl, uch Choquet integrl, eminormed fuzzy integrl, etc / β - f ( ) ( m) f ( ) f ( ) By (i) of Propoition nd (3), we otined ( S) f ( x) d µ ( S) g( x) d µ = min{ β, } (3) cknowledgement The uthor i grteful to the reviewer for very creful reding of the mnucript nd the uggetion tht led to the improvement of the pper Thi tudy i prtilly upported y 933rd item of Eduction Reform Project in Hunn Province in 6 Thi complete the proof To how the vlid nd effectivene of the ove etlihed Hermite-Hdmrd type fuzzy inequlitie, we ee the following exmple Exmple Conider X = [,] nd let µ e the Leegue meure on X If we tke the function f ( x) = x, then y Remrk 3, we know tht f ( x) K for (, ] Here we let =, then f () > f () By Theorem, through olving the eqution β f (), f () f () 5 we get β = = 68 Then Reference [] Wu H C Fuzzy Byein etimtion on lifetime dt [J] Computtionl Sttitic, 4, 9(4): 63 [] Duoi D, Prde H, Etev F, et l Fuzzy et modelling in ce-ed reoning [J] Interntionl Journl of Intelligent Sytem, 5, 3(4): [3] Setz S, Semling M, Mülhupt R Fuzzy et pproch for fitting continuou repone urfce in dheion formultion[j] Journl of Chemometric, 5, (5): [4] Dlmn H, Güzel N, Sivri M fuzzy et-ed pproch to multi-ojective multi-item olid trnporttion prolem under uncertinty [J] Interntionl Journl of Fuzzy Sytem, 6, 8(4): [5] Wng W, Liu X Fuzzy forecting ed on utomtic clutering nd xiomtic fuzzy et clifiction [J] Informtion Science, 5, 94(94): 78-94

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