The Hermite-Hadamard's inequality for some convex functions via fractional integrals and related results
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1 AMSI 4 No 69 The Herie-Hdrd' ineliy or oe conve ncion vi rcionl inegrl nd reled rel E SET M Z SARIKAYA M E ÖZDEMIR AND H YILDIRIM Arc In hi pper we elih Herie-Hdrd ype ineliie or conve ncion in he econd ene nd conve ncion vi rcionl inegrl The nlyi ed in he proo i irly eleenry Mheic Sjec Cliicion : 6D7 6D 6D5 47A63 Addiionl Key Word nd Phre: conve ncion Herie-Hdrd ype ineliie Rienn-Lioville rcionl inegrl INTRODUCTION Le rel ncion e deined on oe nonepy inervl I o rel line R The ncion i id o e conve on I i ineliy hold or ll y y y I nd In [] Hdzik nd Mligrnd conidered ong oher he cl o ncion which re deined he ollowing: conve in he econd ene Thi cl o ncion i DEFINITION A ncion :[ R i id o e conve in he econd ene i y y or ll y[ [] nd or oe ied ] Thi cl o - conve ncion i lly denoed y K 478/ji-4-4 Unheniced Univeriy o SS Cyril nd Mehodi in Trnv Downlod De 8/3/8 5:5 PM
2 7 E Se M Z Sriky M E Özdeir nd H Yıldırı I cn e eily een h or conveiy redce o ordinry conveiy o ncion deined on [ In [] G Toder conidered he cl o ineredie eween he l conveiy nd rhped conveiy conve ncion: noher DEFINITION The ncion :[ ] R i id o e conve where [] i we hve i y y or ll y [ ] nd We y h i concve i conve Ovioly or Deiniion D recpre he concep o ndrd conve ncion on nd or he concep rhped ncion One o he o o ineliie or conve ncion i Herie- Hdrd ineliy Thi dole ineliy i ed ollow ee or eple [5] nd [4]: Le e conve ncion on oe nonepy inervl [ ] o rel line R where Then d Boh ineliie hold in he revered direcion i i concve We noe h Herie-Hdrd ineliy y e regrded reineen o he concep o conveiy nd i ollow eily ro enen' ineliy Herie-Hdrd ineliy or conve ncion h received renewed enion in recen yer nd rerkle vriey o reineen nd generlizion hve een ond ee or eple []-[6] In [8] Herie-Hdrd ineliy or ene i proved y Drgoir e l conve ncion in he econd Unheniced Downlod De 8/3/8 5:5 PM
3 AMSI 4 No 7 THEOREM Sppoe h :[ [ i n -conve ncion in he econd ene where nd le [ I L hen he ollowing ineliie hold: d The conn k i he e poile in he econd ineliy in In [] Kirci e l elihed new Herie-Hdrd ype ineliy which hold or -conve ncion in he econd ene I i given in he ne heore THEOREM Le : I R I [ e dierenile ncion on I ch h L [ ] where I I [ ] or oe ied nd hen: i conve on d 3 We give oe necery deiniion nd heicl preliinrie o rcionl clcl heory which re ed hrogho hi pper DEFINITION 3 Le L [ ] The Rienn-Lioville inegrl nd o order wih re deined y nd d d Unheniced Downlod De 8/3/8 5:5 PM
4 7 E Se M Z Sriky M E Özdeir nd H Yıldırı repecively where e d Here i In he ce o he rcionl inegrl redce o he clicl inegrl Properie concerning hi operor cn e ond [4][5] nd [6] For oe recen rel conneced wih rcionl inegrl ineliie ee [8]-[9] In [8] Srky e l proved vrin o he ideniy elihed y Drgoir e l in [6] or rcionl inegrl which i ed in he ollowing le LEMMA Le R : e dierenile pping on wih I L hen he ollowing eliy or rcionl inegrl hold: d 4 The i o hi pper i o elih Herie-Hdrd ineliy nd Herie-Hdrd ype ineliie or nd conve ncion in he econd ene cone ncion vi Rienn-Lioville rcionl inegrl Unheniced Downlod De 8/3/8 5:5 PM
5 AMSI 4 No 73 HERMITE-HADAMARD TYPE INEQUALITIES FOR SOME CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS For conve ncion Herie-Hdrd ineliy cn e repreened or rcionl inegrl or ollow: conve ncion in THEOREM 3 Le R : e poiive ncion wih nd L I i conve pping in he econd ene on [ ] hen he ollowing ineliie or rcionl inegrl wih hold: nd where i Eler Be ncion PROOF Since i conve pping in he econd ene on [ ] we hve or y [ ] wih y y Now le nd y wih Then we ge y h: or ll Mliplying oh ide o 3 y 3 ineliy wih repe o over [ ] we oin hen inegring he reling Unheniced Downlod De 8/3/8 5:5 PM
6 74 E Se M Z Sriky M E Özdeir nd H Yıldırı dv v v d d d ie nd he ir ineliy i proved For he proo o he econd ineliy in we ir noe h i i conve pping in he econd ene hen or i yield nd By dding hee ineliie we hve 4 Th liplying oh ide o 4 y nd inegring he reling ineliy wih repe o over ] [ we oin d d d Unheniced Downlod De 8/3/8 5:5 PM
7 AMSI 4 No 75 ie where he proo i copleed REMARK I we chooe in Theore 3 hen he ineliie ecoe he ineliie o Theore Uing Le we cn oin he ollowing rcionl inegrl ineliy or conve in he econd ene: THEOREM 4 Le R [ : e dierenile pping on wih ch h L I i conve in he econd ene on ] [ or oe ied nd hen he ollowing ineliy or rcionl inegrl hold: 5 PROOF Sppoe h Fro Le nd ing he properie o odl we hve 6 d Unheniced Downlod De 8/3/8 5:5 PM
8 76 E Se M Z Sriky M E Özdeir nd H Yıldırı Since i conve on ] [ we hve d d d 7 d d d d d d d d Unheniced Downlod De 8/3/8 5:5 PM
9 AMSI 4 No 77 Since d d d d d d nd d d We oin which coplee he proo or hi ce Sppoe now h Since i conve on we know h or every ] [ 8 o ing well know Hölder' ineliy ee or eple [7] or p nd 8 in 6 we hve cceively Unheniced Downlod De 8/3/8 5:5 PM
10 78 E Se M Z Sriky M E Özdeir nd H Yıldırı d d d d where we e he c h d d d which coplee he proo REMARK I we ke in Theore 4 hen he ineliy 5 ecoe he ineliy 3 o Theore Unheniced Downlod De 8/3/8 5:5 PM
11 AMSI 4 No 79 For conve ncion We r wih he ollowing heore: THEOREM 5 Le :[ ] R e poiive ncion wih nd L [ ] I i conve pping on [ ] hen he ollowing ineliie or rcionl inegrl wih nd ] hold: 9 PROOF Since i conve ncion we hve y y nd i we chooe we ge y Now le y nd y wih [] Then we ge Mliplying oh ide o nd y hen inegring he reling ineliie wih repec o over [ ] we oin Unheniced Downlod De 8/3/8 5:5 PM
12 8 E Se M Z Sriky M E Özdeir nd H Yıldırı d dv v v d d d which he ir ineliy i proved By he -conveiy o we lo hve or ll Mliplying oh ide o ove ineliy y nd inegring over ] [ we ge dv v v d which hi give he econd pr o 9 COROLLARY Under he condiion in Theore 5 wih hen he ollowing ineliy hold: d Unheniced Downlod De 8/3/8 5:5 PM
13 AMSI 4 No 8 REMARK 3 I we ke in Corollry kk hen he ineliie ecoe he ineliie THEOREM 6 Le R ] [ : e -conve ncion wih ] nd ] [ L R :[] y F re deined he ollowing: ] [ y y y F Then we hve d F or ll ] [ PROOF Since nd g re conve ncion we hve y y y y F nd o F I we chooe we hve F Th liplying oh ide o y hen inegring he reling ineliy wih repec o over ] [ we oin d d d F Unheniced Downlod De 8/3/8 5:5 PM
14 8 E Se M Z Sriky M E Özdeir nd H Yıldırı Th i we e he chnge o he vrile hen hve he conclion REFERENCES [] Alori M Dr M On he Hdrd' ineliy or log-conve ncion on he coordine ornl o Ineliie nd Applicion vol 9 Aricle ID pge 9 [] Azpeii AG Conve ncion nd he Hdrd ineliy Rev Coloin Mh [3] Bkl MK Özdeir ME Pečrić Hdrd pye ineliie or conve nd -conve ncion Ine Pre nd Appl Mh 94 8 Ar 96 [4] Bkl M K Pečrić Noe on oe Hdrd-ype ineliie ornl o Ineliie in Pre nd Applied Mheic vol 5 no 3 ricle 74 4 [5] Drgoir S S Perce C E M Seleced Topic on Herie-Hdrd Ineliie nd Applicion RGMIA Monogrph Vicori Univeriy [6] Drgoir S S Agrwl RP Two ineliie or dierenile pping nd pplicion o pecil en o rel ner nd o rpezoidl orl Appl Mh Le [7] Drgoir S S On oe new ineliie o Herie-Hdrd ype or conve ncion Tkng Mh 3 [8] Drgoir S S Fizprik S The Hdrd' ineliy or -conve ncion in he econd ene Deonrio Mh [9] Gill P M Perce C E M Pečrić Hdrd' ineliy or r -conve ncion ornl o Mheicl Anlyi nd Applicion vol 5 no pp [] Hdzik H Mligrnd L Soe rerk on conve ncion Aeione Mh [] Kirci US Bkl MK Özdeir ME Pečrić Hdrd-pye ineliie or - conve ncion Appl Mh nd Cop [] Toder G Soe generlizion o he conveiy Proceeding o The Colloi On Approiion And Opiizion Univ Clj-Npoc Clj-Npoc [3] Özdeir M E Avci M Se E On oe ineliie o Herie-Hdrd ype vi - conveiy Applied Mheic Leer vol 3 no 9 pp [4] Pečrić E Prochn F Tong YL Conve Fncion Pril Ordering nd Siicl Applicion Acdeic Pre Boon 99 [5] Se E Özdeir M E Drgoir S S On he Herie-Hdrd ineliy nd oher inegrl ineliie involving wo ncion ornl o Ineliie nd Applicion Aricle ID 48 9 pge [6] Se E Özdeir M E Drgoir S S On Hdrd-Type ineliie involving everl kind o conveiy ornl o Ineliie nd Applicion Aricle ID pge [7] Mirinović DS Pečrić E Fink AM Clicl nd New Ineliie in Anlyi Klwer Acdeic Pliher Dordrech 993 p 6 [8] Anio G Hoohndl MR Ghei A Mokhrzdeh F Monogoery ideniie or rcionl inegrl nd reled rcionl ineliie Ine Pre nd Appl Mh 4 9 Ar 97 [9] Belri S Dhni Z On oe new rcionl inegrl ineliie Ine Pre nd Appl Mh 3 9 Ar 86 [] Dhni Z New ineliie in rcionl inegrl Inernionl ornl o Nonliner Scinece [] Dhni Z On Minkowki nd Herie-Hdrd inegrl ineliie vi rcionl inegrion Ann Fnc Anl 5-58 [] Dhni Z Thri L T S Soe rcionl inegrl ineliie Nonl Sci Le A 55-6 [3] Dhni Z Thri L T S New generlizion o Gr ineliy in Rienn- Lioville rcionl inegrl Bll Mh Anl Appl Unheniced Downlod De 8/3/8 5:5 PM
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