Research Article Refinements of Aczél-Type Inequality and Their Applications

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1 Hidwi Pulishig Corportio Jourl of Applied Mthetics Volue 04, Article ID 58354, 7 pges Reserch Article Refieets of Aczél-Type Iequlity d Their Applictios Jigfeg Ti d We-Li Wg College of Sciece d Techology, North Chi Electric Power Uiversity, Bodig, Heei 0705, Chi Deprtet of Ifortio Egieerig, Chi Uiversity of Geoscieces Gret Wll College, Bodig 07000, Chi Correspodece should e ddressed to Jigfeg Ti; tifhx cepu@63co Received April 04; Revised 8 Jue 04; Accepted 9 Jue 04; Pulished 6 Jue 04 Acdeic Editor: Shhe Wu Copyright 04 J Ti d W-L Wg This is ope ccess rticle distriuted uder the Cretive Coos Attriutio Licese, which perits urestricted use, distriutio, d reproductio i y ediu, provided the origil work is properly cited We preset soe ew shrpeed versios of Aczél-type iequlity Moreover, s pplictios, soe refieets of itegrl type of Aczél-type iequlity re give Itroductio Let e positive iteger, d let i, i i=,,,)e rel uers such tht i= i >0or i= i >0The, the fous Aczél iequlity ] c e stted s follows: i ) i ) i= i= i= i i ) ) Aczél s iequlity plys very iportt role i the theory of fuctiol equtios i o-euclide geoetry Due to the iportce of Aczél s iequlity ), it hs received cosiderle ttetio y y uthors d hs otivted lrge uer of reserch ppers givig it vrious geerliztios, iproveets, d pplictios see ] d the refereces therei) I 959, Popoviciu 0] first otied expoetil extesio of the Aczél iequlitys follows Theore B Let pq>, /p) /q) =, dlet i, i i=,,,)e positive uers such tht p i= p i > 0 d q i= q i >0The p /p p i ) q /q q i ) i= i= i= i i ) Lter, i 98, Vsić dpečrić 6] estlishedthe followig reversed versio of iequlity ) Theore C Let q<0, p>0, /p) /q) =, dlet i, i i=,,,)e positive uers such tht p i= p i > 0 d q i= q i >0The p /p p i ) q /q q i ) i= i= i= i i 3) I other pper, Vsić d Pečrić 5] geerlized iequlity )ithefollowigfor Theore D Let r > 0, β > 0, β β r > 0, r =,,,,,,,,dlet /β )The β β /β r ) r 4) I 0, Ti 3] preseted the reversed versio of iequlity 4) s follows Theore E Let r > 0, β =0, β < 0 =,3,,), /β ), β β r > 0, r =,,,, =,,,The β β /β r ) r 5) Moreover, i 3] Ti estlished itegrl type of iequlity 5)

2 Jourl of Applied Mthetics Theore F Let β > 0, β < 0 =,3,,), /β )=,lett >0,,,),dletf x) =,,,)e positive Rie itegrle fuctios o, ] such tht t β t β fβ x)dx > 0The f β x) dx)/β t f x) dx 6) Rerk I fct, the itegrl for of iequlity 4) islso vlid; tht is, oe hs the followig Le 4 see 8]) Let 0x<, α>0the x) /α x x α, ) Le 5 Let 0<β β β, /β ),,let0 < x < =,,,),dletξ) = / if eve )/ if odd The Theore G Let β > 0 =,,,), /β )=, let t > 0 =,,,),dletf x) =,,,)e positive Rie itegrle fuctios o, ] such tht t β fβ x)dx > 0The t β f β x) dx)/β t f x) dx 7) x β )/β x ξ) ξ ) x β, xβ Proof Fro the ssuptios we hve tht ) ] 3) The i purpose of this work is to give ew refieets of iequlities 4) d5) As pplictios, ew refieets of iequlities 6)d7)re lso give Refieets of Aczél-Type Iequlity I order to preset our i results, we eed soe les s follows Le see 6]) Let i,x i i=,,,)e rel uers such tht i 0d x i >If i= i,the i= x i ) i i= i x i 8) If either i i =,,,)or i 0 i =,,,)d if ll x i re positive or egtive with x i >, the the reverse iequlity of 8) holds Le 3 see 5]) Let i > 0 i =,,,, =,,,) ) If λ 0d if λ,the λ i i= i= ) If λ 0,,,),the λ i i= i= i )λ 9) i )λ 0) c) If λ >0, λ 0 =,3,,),d λ, the λ i i= i= i )λ ) >0, β β β β 0,,,) β β 4) Cse I) let e eve) I view of /β /β )/β /β /β 3 /β 4 )/β 4 /β 4 /β /β )/β /β =/β /β /β y usig iequlity 9), we get / x β = / /β ) ] x β )xβ ] /β x β )xβ ]/β x β )xβ ]/β /β =x β )xβ ]/β x β )xβ ]/β x β )xβ ]/β /β x β 3 3 )xβ 4 4 ]/β 4 x β 4 4 )xβ 3 3 ]/β 4 x β 3 3 )xβ 3 3 ]/β 3/β 4 x β )xβ ] /β x β )x β ]/β

3 Jourl of Applied Mthetics 3 = x β )xβ ]/β /β / / x β )/β x β )/β / x β )/β /β ] x β )/β x β )/β x β x β )/β /β ] )/β x 5) /β /β /β =/β /β /β,y usig iequlity 9), we hve )/ = = x β )/ /β ) ] x β x β )x β ] /β )/ /β ) ] x β )xβ ]/β x β )xβ ]/β O the other hd, pplyig Le 4 d the rithetic-geoetric es iequlity we oti x β )xβ ]/β /β / x β / /β ) ] x β, xβ ) ] / x β, xβ ) ] / = / x β, xβ Applyig Le4 gi, we get / x β, xβ / x β, xβ ) ] ) ] / ) ] / 6) 7) = x β )x β ] /β )/ )/ x β x β )/β x β )/β x β /β ] )/β x β ) /β x β )/β x β )/β x β /β ] )/β x β ) /β )/β x 8) O the other hd, pplyig Le 4 d the ritheticgeoetric es iequlity we oti Coiig 5), 6), d 7) yields ieditely iequlity 3) Cse II) let e odd) I view of /β /β )/β /β /β 3 /β 4 )/β 4 /β 4 /β /β ) )/ x β /β ) ] )/ x β, xβ ) ]

4 4 Jourl of Applied Mthetics )/ x β, = )/ x β ) ] x β, x β Applyig Le4 gi, we hve )/ )/ x β, xβ x β, xβ )/ ) ] ) ] )/ ) ] 9) )/ 0) Hece, coiig 8), 9), d 0) yields ieditely iequlity 3) Siilr to the proof of Le 5 ut usig Le i plce of Le 4, we ieditely oti the followig result Le 6 Let β >0, 0>β β 3 β, /β ),,let0<x <, x > =,3,,),dlet ξ) = / if eve )/ if odd The x β )/β x ξ) x β ) ) β Usig the se ethods s i Le 6, wegetthe followig Le Le 7 Let 0>β β β,,letx > =,,,),dletξ) = / if eve )/ if odd The x β )/β x ξ) x β ) ) β Now, we preset soe ew refieets of iequlities 4) d 5) Theore 8 Let r >0, r =,,,, =,,,,,, let0 < β β β, /β ), β β r >0,dletξ) = / if eve )/ if odd The β β /β r ) r ξ ) ξ) x β, β r) β ) β r) β ) Proof Fro the ssuptios we fid tht 0< β β r )/β β )/β ) ] ] 3) <,,,) 4) Thus, y usig Le 5 with sustitutio x β β r )/β )/β,,,)i 3), we oti β r )/β β β ξ ) ξ) x β, β r ) = ξ) ξ ) x β, β β r) β ) β r) β ) ) /β β r) β ) β r) β ) ) ] ] ) ], ] 5)

5 Jourl of Applied Mthetics 5 which iplies β β /β r ) β /β r ) ξ ) ξ) x β, β r) β ) β r) β ) ) ] ] 6) Theore 0 Let t > 0 =,,,), β > 0, 0 > β β 3 β, /β ) =,letf x) =,,, ) e positive itegrle fuctios defied o, ] with t β The t β fβ x)dx >0,dletξ) = / if eve )/ if odd f β /β x) dx) t t t,,t ξ) f x) dx β fβ t β x) fβ x) t β )dx] ] 9) O the other hd, we get fro Le 3 tht β /β r ) r 7) Coiig 6) d7) yields ieditely the desired iequlity 3) Theore 9 Let r > 0, 0 > β β β, β β r >0, r =,,,, =,,,,let,, d let ξ) = / if eve )/ if odd The β β /β r ),, r 8) Proof For y positive iteger, we choose equidistt prtitio of, ] s Sice t β tht < < < k < < ) <, x i = i, i=0,,,, 30) Δx k =, k=,,, 3) fβ x)dx > 0 =,,,),itfollows ξ) β β r) β ) β r) β ) ) ] ] Iequlity 8) is lso vlid for β >0, 0>β β 3 β, /β ) Proof The proof of Theore 9 is siilr to the oe of Theore 8, d we oit it t β f β k= li k ) ) >0,,,) Therefore, there exists positive iteger N such tht 3) 3 Applictios I this sectio, we show two pplictios of the iequlities ewly otied i Sectio Firstly, we preset ew refieet of iequlity 6) y usig Theore 9 t β f β k= for ll >Nd,,, k ) ) >0, 33)

6 6 Jourl of Applied Mthetics Moreover, for y >N,itfollowsfroTheore 9 tht we get t β t β f β k= k= k= Notig tht t β t β f β k= k ) ) /β ] f k ) )) ) /β/β /β t t,,t t β k= f β t β t t,,t ξ) k= t β f β t β f β k ) ) ξ) β k ) ) ) ] ] 34) =, 35) β k ) ) /β ] f β f β k ) )) ) k ) ) k ) ) ) ] ] 36) I view of the ssuptio tht f x) =,,,) re positive Rie itegrle fuctios o, ],wefidtht f x) d f λ x) relsoitegrleo, ] Lettig o oth sides of iequlity 36), we get the desired iequlity 9) Next, we give ew refieet of iequlity 7)yusig Theore 8 Theore Let t > 0 =,,,), 0<β β β, /β ) =,,dletf x) =,,, ) e positive itegrle fuctios defied o, ] with t β fβ / if eve x)dx > 0,dletξ) = )/ if odd The t β f β x) dx)/β t t,,t ξ ) ξ) β fβ t β x) t fβ x) t β f x) dx )dx] ] 37) Proof The proof of Theore is siilr to the oe of Theore 0, d we oit it Ackowledgets The uthors would like to thk the reviewers d the editors for their vlule suggestios d coets This work ws supported y the Fudetl Reserch Fuds for the Cetrl Uiversities Grt o 3ZD9) Coflict of Iterests The uthors declre tht there is o coflict of iterests regrdig the pulictio of this pper Refereces ] Y Aczél, Soe geerl ethods i the theory of fuctiol equtios i oe vrile New pplictios of fuctiol equtios, Uspekhi Mteticheskikh Nuk, vol,o3,pp 3 68,956Russi) ] E F BeckechdR Bell, Iequlities, Spriger, Berli, Gery, 983 3]GFrid,JPečrić, d A Ur Reh, O refieets of Aczél, Popoviciu, Bell s iequlities d relted results, Jourl of Iequlities d Applictios, vol00,articleid , 00 4] Z Hu d A Xu, Reeets of Aczél d Bell s iequlities, Coputers & Mthetics with Applictios,vol59,o9, pp , 00 5] F Mirzpour, A Morssei, d M S Moslehi, More o opertor Bell iequlity, Questioes Mthetice, vol 37, o, pp 9 7, 04 6]DSMitriović, J E Pečrić, d A M Fik, Clssicl d New Iequlities i Alysis, Kluwer Acdeic Pulishers, Dordrecht, The Netherlds, 993 7] B Mod, J Pečrić, J Šude, d S Vrošec, Opertor versios of soe clssicl iequlities, Lier Alger d Its Applictios,vol64,o,pp7 6,997

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