On New Inequalities of Hermite-Hadamard-Fejér Type for Harmonically s-convex Functions via Fractional Integrals
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1 Krelm en ve Müh. Derg. 6(:879 6 Krelm en ve Mühendili Dergii Jornl home ge: h://fd.en.ed.r eerch Aricle n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl Keirli İnegrller Yol ile Hrmoni Konve oniyonlr için HermieHdmrdejér ili Yeni Eşiizliler Üzerine Mehme Kn Krdeniz eni Üniveriei Memi Bölümü rzon üriye Arc In hi er ome HermieHdmrdejér ye inegrl ineliie for hrmoniclly convex fncion in frcionl inegrl form re oined. Keyword: Hrmoniclly convex fncion HermieHdmrd ineliy HermieHdmrdejér ineliy iemnn Lioville frcionl inegrl Öz B çlışmd eirli inegrller yol ile hrmoni onve foniyonlr için zı HermieHdmrdejér ili yeni eşiizliler elde edilmişir. Anhr Kelimeler: Hrmoni onve foniyonlr HermieHdmrd eşiizliği HermieHdmrdejér eşiizliği iemnnlioville eirli inegrller. Inrodcion Le fi : " e convex fncion defined on he inervl I of rel nmer nd I wih <. he ineliy ( ( f f f fxdx ( (. i well nown in he lierre HermieHdmrd ineliy (Hdmrd 89. he mo wellnown ineliie reled o he inegrl men of convex fncion f re he HermieHdmrd ineliie or heir weighed verion he oclled HermieHdmrdejér ineliie. In (ejér 96 ejér elihed he following ejér ineliy which i he weighed generlizion of Hermie Hdmrd ineliy : heorem. Le f: " e convex fncion. hen he ineliy *Correonding Ahor: mn@.ed.r eceived / eliş rihi :.5.6 Acceed / Kl rihi : ( ( f f gxdx ( fxgxdx ( ( (. gxdx ( hold where g: " i nonnegive inegrle nd ymmeric o (/. or ome rel which generlize imrove nd exend he ineliie nd ee (Bomrdelli nd rošnec 9 İşcn İşcn 4 rıy eng e l.. Definiion. (Kil e l. 6. Le f! he iemnnlioville inegrl J f nd J f of order > wih re defined y J x f( x ( x f( d x C( Jf( x ( x f( d x C( x reecively where C ( i he mm fncion defined y C ( e d nd J f( x J fx ( fx (.
2 Kn / n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl Bece of he wide licion of HermieHdmrd ye ineliie nd frcionl inegrl mny reercher exend heir die o HermieHdmrd ye ineliie involving frcionl inegrl no limied o ineger inegrl. ecenly more nd more HermieHdmrd ineliie involving frcionl inegrl hve een oined for differen cle of fncion; ee (Dhmni İşcn. İşcn 4 rıy e l. ng e l. ng e l.. Definiion. (İşcn 5. Le I ( e rel inervl. A fncion fi : " i id o e hrmoniclly convex if xy fc m f( y ( fx ( x ( y for ll x y I [] nd for ome fixed (]. In (İşcn 4 İşcn gve definiion of hrmoniclly convex fncion nd elihed following Hermie Hdmrd ye ineliy for hrmoniclly convex fncion follow: Definiion. Le I " e rel inervl. A fncion fi : " i id o e hrmoniclly convex if xy fc m f( y ( fx ( (. x ( y for ll x y I nd []. If he ineliy in (. i revered hen f i id o e hrmoniclly concve. heorem. (İşcn 4. Le fi : " " e hrmoniclly convex fncion nd I wih <. If f L[] hen he following ineliie hold: fx ( f ( f ( f dx x (.4 In (Lif e l. 5 Lif e l. gve he following definiion: Definiion 4. A fncion g: " " i id o e hrmoniclly ymmeric wih reec o if gx ( g e o x hold for ll x []. In (Chen nd 4 Chn nd reened Hermie Hdmrdejér ineliy for hrmoniclly convex fncion follow: heorem. Le fi : " " e hrmoniclly convex fncion nd I wih <. If f L[] nd g: " " i nonnegive inegrle nd hrmoniclly ymmeric wih reec o l hen gx ( fxgx ( ( f dx dx x x f ( f ( gx ( dx. x (.5 In (Kn e l. 6 Kn e l. reened reecively HermieHdmrd ineliy in frcionl inegrl form for hrmoniclly convex fncion HermieHdmrd ejér ineliy in frcionl inegrl form for hrmoniclly convex fncion follow: heorem 4. Le fi : ( " e fncion ch h f L[ ] where I wih <. If f i hrmoniclly convex fncion on [ ] hen he following ineliie for frcionl inegrl hold: C^ h f f ( f ( J f% h / J f% h / ^ h^ h ^ h^ h (.6 wih > nd h(x/x x! : heorem 5. Le f: " e hrmoniclly convex fncion wih < nd f L[ ]. If g: " i nonnegive inegrle nd hrmoniclly ymmeric wih reec o hen he following ineliie for frcionl inegrl hold: f J g h / J 7 g h / h (.7 7J fg h / J fg h / h % ^ h^ f ( f ( 7J g% h / J g% h / ^ h^ h ^ h^ wih > nd h(x/x x! : Lemm. (Kn e l. 6. Le fi : ( " e differenile fncion on I o ch h fl! where I nd <. If g: " i inegrle nd hrmoniclly ymmeric wih reec o hen he following eliy for frcionl inegrl hold: f J g h / J 7 g h / h 7J fg h / J fg h / h c ^ g% hh^hdm^f% hhl^dhd C( c ` j ^g% hh^hdm^f% hhl^dd h (.8 88 Krelm en Müh. Derg. 6; 6(:879
3 Kn / n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl wih > nd h(x/x x! : In hi er we oin ome new ineliie conneced wih he lefhnd ide of HermieHdmrdejér ye inegrl ineliy for hrmoniclly convex fncion in frcionl inegrl.. el hrogho hi ecion we e g g( for conino fncion g: ". heorem 6. Le fi : ^ h " e differenile fncion on I o ch h fl! where I nd <. If fl i hrmoniclly convex on [] g: [] i conino nd hrmoniclly ymmeric wih reec o hen he following ineliy for frcionl inegrl hold: f J ( g h(/ J 7 % ( g h(/ % A 7J ( fg% h(/ J ( fg h( / % A (. ( g 6 ( ( ( ( ( C fl C C where C ( ( d d ( ( ( ( C( (. ( d ( ( ( d ( ( (. wih < nd h(x /x x! : Proof. rom Lemm we hve f J g h / J 7 % g h / ^ h^ h h A 7J fg h / J fg h / h c ( ^g% hh( dm h l ( d C( c ( ^g% hh( dm h( l d c ( dm h( l d g C( c ( dm h( l d ( f ( g d l C( ( f ( d l ^ h eing nd d d give f J ( g h(/ J 7 % ( g h( / % A 7J ( fg% h(/ J ( fg h( / % A ( ( ( flc m d g ( ( C ( flc m d ( ( ( (.4 ince fl i hrmoniclly convex on [] we hve flc m fl( ( fl ( (.5 ( If we e (.5 in (.4 we hve f J ( g h(/ J 7 % ( g h( / % A 7J ( fg% h(/ J ( fg h( / % A [ fl( ( fl( ] d g ( ( ( ( C ( [ fl( ( fl( ] d ( ( (.6 If we e (. nd (. in (.6 we hve (.. hi comlee he roof. Corollry. In heorem 6: ( If we e we hve he following Hermie Hdmrdejér ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of (.5: gx ( fxgx ( ( f dx dx x x g ( [ C( fl ( C( fl( ] ( If we e g(x we hve following HermieHdmrd ineliy for hrmoniclly convex fncion in frcionl inegrl form which i reled o he lefhnd ide of (.6: ( J C ( f h(/ % f * 4 J ( f h(/ % ( [ C ( f l ( C ( f l ( ] ( If we e nd g(x we hve he following HermieHdmrd ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of :(.4 fx ( f dx ( x 6 C( fl( C( Krelm en Müh. Derg. 6; 6(:879 89
4 Kn / n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl heorem 7. Le fi : ^ h " e differenile fncion on I o ch h fl! where I nd <. If fl $ i hrmoniclly convex on [] g: " i conino nd hrmoniclly ymmeric wih reec o hen he following ineliy for frcionl inegrl hold: f ( f ( 6J/ ( g% h(/ J/ ( g% h( 6J/ ( fg% h(/ J/ ( fg % h( C4( fl ( C ( g ( e C5( fl ( C( C7( fl ( ( e C8( fl ( where C ( C ( (.7 ( ( d ( ( d 4 C5( ( ( d ( ( ( ( ( d (.8 (.9 ( C7( d ( ( (. ( C8( d ( ( wih > nd h(x /x x! : Proof. Uing (.4 ower men ineliy nd he hrmoniclly convexiy of fl i follow h f J ( g h(/ J 7 % ( g h( / % A 7J ( fg% h(/ J ( fg h( / % A flc m d g ( ( ( ( C^ h ( flc m d ( ( ( d dn ( ( f d g ( ( ( ( d lc m n C^ h ( d dn ( ( ( d flc m dn ( ( ( d dn ( ( [ f ( ( f ( ] d g ( ( ( d l l n C^ h ( d dn ( ( ( d [ fl( ( fl( ] dn ( ( d dn ( ( J N d fl K ( ( ( K ( d f ( g ( ( ( K l L P C ^ h ( d dn ( ( J ( N K d fl ( ( ( K ( K d fl ( L ( ( P (. If we e (.8 (.9 nd (. in (. we hve (.7. hi comlee he roof. Corollry. In heorem 7: ( If we e we hve he following Hermie Hdmrdejér ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of (.5: gx ( fxgx ( ( f dx dx x x C4( fl ( C ( e C5( fl ( g ( C7( fl ( ( e C8( fl ( ( If we e g(x we hve following HermieHdmrd ineliy for hrmoniclly convex fncion in frcionl inegrl form which i reled o he lefhnd ide of (.6: ( J C ( f h(/ % f * 4 J ( f h(/ % ( ( ( ( C C4 fl e C5( fl ( C7( fl ( ( e C8( fl ( 9 Krelm en Müh. Derg. 6; 6(:879
5 Kn / n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl ( If we e nd g(x we hve he following HermieHdmrd ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of (.4: fx ( f dx ( x C4( fl ( C ( C5( fl (. C7( fl ( ( C8( fl ( e cn e noher ineliy for > follow: heorem 8. Le fi : ( " e differenile fncion on I o ch h fl! where I nd <. If fl i hrmoniclly convex on [ ] g: " i conino nd hrmoniclly ymmeric wih reec o hen he following ineliy for frcionl inegrl hold: f ( f ( 6J/ ( g% h(/ J/ ( g% h( 6J/ ( fg% h(/ J/ ( fg % h( fl( ( fl( C9 ( g ( < ( C( ( fl( fl( C( < ( where C ( d (. ( ( 9 dn C( ( d dn ( ( wih > h(x /x x! : D nd / /. (. Proof. Uing (.4 Hölder ineliy nd he hrmoniclly convexiy of fl i follow h f J ( g h(/ J 7 % ( g h( / % A 7J ( fg% h(/ J ( fg h( / % A flc m d g ( ( ( ( C^ h ( flc m d ( ( ( g ( C^ h < d d d f ( ( ( ( n d lc m dn ( ( ( d f ( ( n c lc m dm < d d g ( C ^ h ( ( ( ( ( dn c fl( ( fl( dm dn c fl( ( fl( dm (.4 Clcling following inegrl we hve fl( ( fl( d fl( ( fl( ( fl( ( fl( d ( fl( fl( ( (.5 (.6 If we e (. (.5 nd (.6 in (.4 we hve (.. hi comlee he roof. Corollry In heorem 8: ( If we e we hve he following Hermie Hdmrdejér ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of (.5: gx ( f( xgx ( f dx dx x x fl( ( fl( C ( < 9 ( ( ( fl( fl( C( < ( ( If we e g(x we hve following HermieHdmrd ineliy for hrmoniclly convex fncion in frcionl inegrl form which i reled o he lefhnd ide of (.6: ( J C ( f h(/ % f * 4 J ( f h(/ % ( ( ( ( ( C fl fl 9 < ( ( fl( fl( C( < ( ( If we e nd g(x we hve he following HermieHdmrd ineliy for hrmoniclly convex fncion which i reled o he lefhnd ide of (.4: Krelm en Müh. Derg. 6; 6(:879 9
6 Kn / n New Ineliie of HermieHdmrdejér ye for Hrmoniclly Convex ncion vi rcionl Inegrl fx ( f dx x fl( ( fl( C ( < 9 ( (. ( fl( fl( C( < (. eference Bomrdelli M. rošnec. 9. Proerie of hconvex fncion reled o he Hermie Hdmrd ejér ineliie. Com. Mh. Al. 58: Chen.. 4. ejér nd HermieHdmrd ye ineliie for hrmoniclly convex fncion. J. Al. Mh. volme 4 ricle id:8686. Dhmni Z.. n Minowi nd HermieHdmrd inegrl ineliie vi frcionl inegrion. Ann. nc. Anl. (: 558. ejér L. 96. Uer die orierreihen. II. Mh. Nrwie. Anz Ungr. Ad. i 4: 699 (in Hngrin. Hdmrd J. 89. Éde r le roriéé de foncion enière e en riclier d ne foncion conidérée r iemnn. J. Mh. Pre Al. 58: 75. İşcn İ.. New eime on generlizion of ome inegrl ineliie for convex fncion nd heir licion. In. J. Pre Al. Mh. 86(4: İşcn İ. 4. ome new generl inegrl ineliie for hconvex nd hconcve fncion. Adv. Pre Al. Mh. 5(: 9. İşcn İ.. enerlizion of differen ye inegrl ineliiefor convex fncion vi frcionl inegrl. Al. Anl. doi:.8/ İşcn İ. 4. n generlizion of differen ye inegrl ineliie for convex fncion vi frcionl inegrl. Mh. ci. Al. ENo. (: İşcn İ.. 4. HermieHdmrd ye ineliie for hrmoniclly convex fncion vi frcionl inegrl. Al. Mh. Com. 8: 744. İşcn İ. 4. HermieHdmrd ye ineliie for hrmoniclly convex fncion. Hce. J. Mh.. 4 (6: İşcn İ. 5. rowi ye ineliie for hrmoniclly convex fncion. Konr. J. Mh olme No : 674. Kil AA. rivv HM rjillo JJ. 6. heory nd licion of frcionl differenil eion. Elevier Amerdm. Kn M. İşcn İ. özüo NY. özüo U. 6. n new ineliie of HermieHdmrdejer ye for hrmoniclly convex fncion vi frcionl inegrl ringerl 5(65 9. Lif MA. Drgomir. Momoni E. 5. ome ejér ye ineliie for hrmonicllyconvex fncion wih licion o ecil men MIA e. e. Coll. 8 Aricle 4 7. rıy MZ.. n new Hermie Hdmrd ejér ye inegrl ineliie. d. Univ. BeBolyi Mh. 57(: rıy MZ. e E. Yldız H. Bş N.. Hermie Hdmrd ineliie for frcionl inegrl nd reled frcionl ineliie. Mh. Com. Mod. 57(9: 447. eng KL. Yng. H KC.. ome ineliie for differenile ming nd licion o ejér ineliy nd weighed rezoidl forml. iw. J. Mh. 5(4: ng J. Li. ecn M. Zho Y.. HermieHdmrdye ineliie for iemnnlioville frcionl inegrl vi wo ind of convexiy. Al. Anl. 9(: 45. ng J. Zh C. Zho Y.. New generlized Hermie Hdmrd ye ineliie nd licion o ecil men. J. Inel. Al. 5( 5. 9 Krelm en Müh. Derg. 6; 6(:879
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