On The Hermite- Hadamard-Fejér Type Integral Inequality for Convex Function

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1 Turkish Journl o Anlysis nd Numer Theory, 4, Vol., No. 3, Aville online h://us.scieu.com/jn//3/6 Science nd Educion Pulishing DOI:.69/jn--3-6 On The Hermie- Hdmrd-Fejér Tye Inegrl Ineuliy or Convex Funcion Mehme Zeki Sriky,*, Sme Erden Dermen o Mhemics, Fculy o Science nd Ars, Düzce Universiy, Konu-rl Cmus, Düzce-TURKEY Dermen o Mhemics, Fculy o Science, Brn Universiy, Konurl Cm-us, BARTIN-TURKEY *Corresonding uhor: srikymz@gmil.com Received My 4, 4; Revised June 5, 4; Acceed July 3, 4 Asrc In his er, we exend some esimes o he righ hnd side o Hermie- Hdmrd-Fejér ye ineuliy or uncions whose irs derivives solue vlues re convex. The resuls resened here would rovide exensions o hose given in erlier works. Keywords: Osrowski s ineuliy, Mongomery s ideniies, convex uncion, Hölder ineuliy Cie This Aricle: Mehme Zeki Sriky, nd Sme Erden, On The Hermie- Hdmrd-Fejér Tye Inegrl Ineuliy or Convex Funcion. Turkish Journl o Anlysis nd Numer Theory, vol., no. 3 (4: doi:.69/jn Inroducion Deiniion. The uncion :[, ], is sid o e convex i he ollowing ineuliy holds λx+ λ y λ x + λ y ( or ll x,y [,] nd λ [,]. We sy h is concve i (- is convex. The ollowing ineuliy is well known in he lierure s he Hermie-Hdmrd inegrl ineuliy (see, [4,]: + + ( x dx (. where : I is convex uncion on he inervl I o rel numers nd, I wih <. In [3], Drgomir nd Agrwl roved he ollowing resuls conneced wih he righ r o (.. Lemm. Le : I e dierenile ming on I,, I wih <. I L[,], hen he ollowing euliy holds: + = ( '( + ( (. Theorem. Le : I e dierenile ming on I,, I wih <. I is convex on [,], hen he ollowing ineuliy holds: + ( ( ' ' + 8 (.3 Theorem. Le : I e dierenile ming on I,, I wih < ; L(, nd >. I he ming /(- is convex on [,], hen he ollowing ineuliy holds: + ( / (.4 /( /( ' + ' / ( + The mos well-known ineuliies reled o he inegrl men o convex uncion re he Hermie Hdmrd ineuliies or is weighed versions, he so-clled Hermie- Hdmrd- Fejér ineuliies (see, [8,3,4,5,6,9,]. In [7], Fejer gve weighed generlizinon o he ineuliies (. s he ollowing: Theorem 3. : [, ] he ineuliy +, e convex uncion, hen + w( ( x w( x dx holds, where w: [, ] w (.5 is nonnegive, inegrle, + nd symmeric ou x =. In [3], some ineuliies o Hermie-Hdmrd-Fejer ye or dierenile convex mings were roved using he ollowing lemm. Lemm. Le : I e dierenile ming on I,, I wih <, nd w: [,] [, e dierenile ming. I L[,], hen he ollowing euliy holds:

2 Turkish Journl o Anlysis nd Numer Theory 86 + ( or ech [,]; where '( + ( w x w = w s s ds w s s ds = ( + ( ( + ( The min resul in [3] is s ollows: (.6 Theorem 4. Le : I e dierenile ming on I,, I wih <, nd w: [,] [, e + dierenile ming nd symmeric o. I is convex on [,] ; hen he ollowing ineuliy holds: + ' ' + ( g where g w x w ( = w or [,]. + ( (.7 Deiniion. Le L [,]. The Riemnn-Liouville inegrls J+ nd J o order > wih re deined y And x J+ ( x = ( x, x > Γ ( x J ( x = ( x, x > Γ ( resecively. Here, Γ ( is he Gmm uncion nd + J x = J x = x. Menwhile, Sriky e l. [] resened he ollowing imorn inegrl ideniy including he irs-order derivive o o eslish mny ineresing Hermie- Hdmrd ye ineuliies or convexiy uncions vi Riemnn-Liouville rcionl inegrls o he order >. :, e dierenile Lemm 3. Le [ ] ming on (, wih <. I L[,], hen he ollowing euliy or rcionl inegrls holds: + Γ ( + J J + + ( ( '( ( = + (.8 I is remrkle h Sriky e l. [] irs give he ollowing ineresing inegrl ineuliies o Hermie- Hdmrd ye involving Riemnn-Liouville rcionl inegrls. Theorem 5. Le : [, ] e osiive uncion wih < nd L [,]. I is convex uncion on [,], hen he ollowing ineuliies or rcionl inegrls hold: + Γ ( + ( + J J + + (.9 wih > : For some recen resuls conneced wih rcionl inegrl ineuliies see [,,8,7,8]. In his ricle, using uncions whose derivives solue vlues re convex, we oined new ineuliies o Hermie-Hdmrd-Fejer ye nd Hermie-Hdmrd ye involving rcionl inegrls. The resuls resened here would rovide exensions o hose given in erlier works.. Min Resuls We will eslish some new resuls conneced wih he righ-hnd side o (.5 nd (. involving rcionl inegrls used he ollowing Lemm. Now, we give he ollowing new Lemm or our resuls: Lemm 4. Le : I e dierenile ming on I,, I wih < nd le w: [, ]. I, w L[,], hen, or ll x [,], he ollowing euliy holds: ' ' w s ds w s ds = w s ds + [ ] w s ds w w s ds w (. where > : Proo. By inegrion y rs, we hve he ollowing euliies: w( s ds ' = w( s ds w( s ds w = w( s ds w( s ds w nd (.

3 87 Turkish Journl o Anlysis nd Numer Theory ' w s ds = w s ds + w s ds w = w s ds + w s ds w (.3 Surcing (.3 rom (., we oin (.. This comlees he roo. Remrk. I we ke w(s = in.; he ideniy (. reduces o he ideniy (.8. Corollry. Under he sme ssumions o Lemm 4 wih = ; hen he ollowing ideniy holds: w( s ds w( s ds ' + = w( s ds w (.4 Remrk. I we ke w(s = in (.4, he ideniy (.4 reduces o he ideniy (.. Now, y using he ove lemm, we rove our min heorems: Theorem 6. Le : I e dierenile ming on I,, I wih < nd le w: [, ] e coninuous on [,]. I is convex on [,], hen he ollowing ineuliy holds: w( s ds + ( + ( + w( s ds w w( s ds w w ' + ' su where > nd w = w. [, ] Proo. We ke solue vlue o (., we ind h w( s ds + w s ds w w s ds w w( s ds ' + w( s ds ' w [, ], ' [ ] ',, + w = w [, ], ( ' + + ( ' + Since is convex on [,], i ollows h ( + ( + w = ' ' +. w( s ds + w s ds w w s ds w w ( ' + ' + ( ' + ' Hence, he roo o heorem is comleed. Corollry. Under he sme ssumions o Theorem 6 wih w(s =, hen he ollowing ineuliy holds: + Γ ( + ( J J + + (.5 ( ' + ' ( + Proo. This roo is given y Sriky e. l in []. Remrk 3. I we ke = in (.5; he ineuliy (.5 reduces o (.3. Corollry 3. Under he sme ssumions o Theorem 6 wih =, hen he ollowing ineuliy holds: ( + w( s ds w w + 4

4 Turkish Journl o Anlysis nd Numer Theory 88 Theorem 7. Le : I e dierenile ming on I,, I wih < nd le w: [, ] e coninuous on [,]. I is convex on [,], >, hen he ollowing ineuliy holds: w( s ds + ( w( s ds w w( s ds w + w ( ' + '( + su (.6 where >, + =, nd w = w. [, ] Proo. We ke solue vlue o (.. Using Holder s ineuliy, we ind h w( s ds + w( s ds w w( s ds w ' ' w s ds + w s ds w( s ds ' + w( s ds ' w ' + Since ' is convex on [,] ' + ' + ' (.7 From (.7, i ollows h w( s ds + ( w( s ds w w( s ds w + w ( ' + '( + which his comlees he roo. Corollry 4. Under he sme ssumions o Theorem 6 wih w(s =, hen he ollowing ineuliy holds: + Γ ( + ( ( ( ' + ' + J J x + + (.8 Corollry 5. Le he condiions o Theorem 7 hold. I we ke = in (.6, hen he ollowing ineuliy holds: ( + w( s ds w w ( ' + '( + Remrk 4. I we ke w(s = in (.9, we hve + ( ( ' + ' + which is roved y Drgomir nd Agrwl in [3]. Reerences [] Z. Dhmni, On Minkowski nd Hermie-Hdmrd inegrl ineuliies vi rcionl inegrion, Ann. Func. Anl. ( (, [] J. Deng nd J. Wng, Frcionl Hermie-Hdmrd ineuliies or (; m-logrihmiclly convex uncions. [3] S. S. Drgomir nd R.P. Agrwl, Two ineuliies or dierenile mings nd licions o secil mens o rel numers nd o rezoidl ormul, Al. Mh. le., (5 (998, 9-95.

5 89 Turkish Journl o Anlysis nd Numer Theory [4] S. S. Drgomir nd C. E. M. Perce, Seleced Toics on Hermie- Hdmrd Ineuliies nd Alicions, RGMIA Monogrhs, Vicori Universiy,. [5] S. Hussin, M.A. Li nd M. Alomri, Generlized duleinegrl Osrowski ye ineuliies on ime scles, Al. Mh. Leers, 4 (, [6] M. E. Kiris nd M. Z. Sriky, On he new generlizion o Osrowski ye ineuliy or doule inegrls, Inernionl Journl o Modern Mhemicl Sciences, 4, 9 (3: -9. [7] L. Fejer, Üer die Fourierreihen, II. Mh. Nurwiss. Anz Ungr. Akd. Wiss., 4 (96, (Hungrin. [8] I. I, scn, Hermie-Hdmrd-Fejer ye ineuliies or convex uncions vi rcionl inegrls, rxiv rerin rxiv: (4. [9] U.S. Krmc, Ineuliies or di erenile mings nd licions o secil mens o rel numers nd o midoin ormul, Al. Mh. Com., 47 (4, [] J. Peµcri c, F. Proschn nd Y.L. Tong, Convex uncions, ril ordering nd sisicl licions, Acdemic Press, New York, 99. [] M. Z. Sriky, E. Se, H. Yldiz nd N., Bsk, Hermie - Hdmrd.s ineuliies or rcionl inegrls nd reled rcionl ineuliies, Mhemicl nd Comuer Modelling, 57 ( [] M. Z. Sriky nd H. Yildirim, On Hermie-Hdmrd ye ineuliies or Riemnn- Liouville rcionl inegrls, Sumied [3] M. Z. Sriky, On new Hermie Hdmrd Fejer Tye inegrl ineuliies, Sudi Universiis Bes-Bolyi Mhemic., 57 (, No. 3, [4] K-L. Tseng, G-S. Yng nd K-C. Hsu, Some ineuliies or dierenile mings nd licions o Fejer ineuliy nd weighed rozidl ormul, Tiwnese J. Mh. 5 (4, : ,. [5] C.-L. Wng, X.-H. Wng, On n exension o Hdmrd ineuliy or convex uncions, Chin. Ann. Mh. 3 ( [6] S.-H. Wu, On he weighed generlizion o he Hermie- Hdmrd ineuliy nd is licions, The Rocky Mounin J. o Mh., vol. 39, no. 5, , 9. [7] M. Tunc, On new ineuliies or h-convex uncions vi Riemnn-Liouville rcionl inegrion, Filom 7: 4 (3, [8] J. Wng, X. Li, M. Feckn nd Y. Zhou, Hermie-Hdmrd-ye ineuliies or Riemnn-Liouville rcionl inegrls vi wo kinds o convexiy, Al. Anl. (. [9] B-Y, Xi nd F. Qi, Some Hermie-Hdmrd ye ineuliies or dierenile convex unc- ions nd licions, Hce. J. Mh. S. 4 (3, (3. [] B-Y, Xi nd F. Qi, Hermie-Hdmrd ye ineuliies or uncions whose derivives re o convexiies, Nonliner Func. Anl. Al. 8 (, (3. [] Y. Zhng nd J-R. Wng, On some new Hermie-Hdmrd ineuliies involving Riemnn-Liouville rcionl inegrls, Journl o Ineuliies nd Alicions 3, 3:.

Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals

$Hermite-Hadamard-Fejér type inequalities for convex functions via fractional integrals$ Sud. Univ. Beş-Bolyi Mh. 6(5, No. 3, 355 366 Hermie-Hdmrd-Fejér ype inequliies for convex funcions vi frcionl inegrls İmd İşcn Asrc. In his pper, firsly we hve eslished Hermie Hdmrd-Fejér inequliy for