Appl. Mth. In. Si. 8, No. 3, 085-0 0 085 Applied Mthemtis & Inormtion Sienes An Interntionl Journl http://.doi.org/0.785/mis/08038 On the Co-Ordinted Convex Funtions M. Emin Özdemir, Çetin Yıldız, nd Ahmet Ok Akdemir Deprtment o Mthemtis, K.K. Edution Fulty, Attürk University, 50, Cmpus, Erzurum, Turkey Deprtment o Mthemtis, Fulty o Siene nd Letters, Ağrı İbrhim Çeçen University, 000, Ağrı, Turkey Reeived: 0 My. 03, Revised: Sep. 03, Aepted: 5 Sep. 03 Published online: My. 0 Abstrt: In this pper, some new integrl ineulities re given or onvex untions on the o-ordintes. By using well-known lssil ineulities nd new integrl identity Lemm, we obtin some generl results or o-ordinted onvex untions. Keywords: Hdmrd s ineulity, o-ordintes, onvexity, Hölder s ineulity, Power men ineulity. Introdution Let us rell some known deinitions nd results whih we will use in this pper. A untion : I R, I R is n intervl, is sid to be onvex untion on I i tx ty t x t y holds or ll x,y I nd t 0,. I the reversed ineulity in holds, then is onve. Let : I R R be onvex untion deined on the intervl I o rel numbers nd <b. The ollowing double ineulity b b x b b is well known in the literture s Hdmrd s ineulity. Both ineulities hold in the reversed diretion i is onve. In, Drgomir deined onvex untions on the oordintes s ollowing; Deinition.Let us onsider the bidimensionl intervl =,b,d in R with < b, < d. A untion : R will be lled onvex on the o-ordintes i the prtil mppings y :,b R, y u = u,y nd x :,d R, x v = x,v re onvex deined or ll y,d nd x,b. Rell tht the mpping : R is onvex on i the ollowing ineulity holds, x z,w x, z,w or llx,y,z,w nd 0,. In, Drgomir estblished the ollowing ineulities o Hdmrd s type or o-ordinted onvex untions on retngle rom the plne R. Theorem.Suppose tht : =,b,d R is onvex on the o-ordintes on. Then one hs the ineulities; b, d b x, d b d b d,ydy b d x,ydy b d b x, b b d d,ydy d b d,,d b, b,d. The bove ineulities re shrp. x,d b,ydy In, Bkul nd Pečrić estblished severl Jensen type ineulities or o-ordinted onvex untions nd in 5, Hwng et l. gve mpping, disussed some properties o this mpping nd proved some Hdmrd-type ineulities or Lipshizin mpping in two vribles. In, Özdemir et l. estblished new Hdmrd-type ineulities or o-ordinted m onvex nd α, m onvex untions. Severl new results n be Corresponding uthor e-mil: etin@tuni.edu.tr 0 NSP Nturl Sienes Publishing Cor.
086 M. E. Özdemir et. l. : On the Co-Ordinted Convex Funtions ound relted to onvex untions on the oordintes in the ppers -3. In 3, Srıky et l. proved some Hdmrd-type ineulities or o-ordinted onvex untions s ollowings; Theorem.Let : R R be prtil dierentible mpping on :=,b,d inr with <b nd <d. I is onvex untion on the o-ordintes on, then one hs the ineulities:,,d b, b,d b d x,yda b d b d 6,,d b, b,d b d b d x, x,d,ydy b,ydy. Theorem 3.Let : R R be prtil dierentible mpping on :=,b,d inr with <b nd <d. I, >, is onvex untion on the o-ordintes on, then one hs the ineulities:,,d b, b,d b d x,yda b d b d p p,,d b, b,d b d nd p =. b d x, x,d,ydy b,ydy Theorem.Let : R R be prtil dierentible mpping on :=,b,d inr with <b nd <d. I,, is onvex untion on the o-ordintes on, then one hs the ineulities:,,d b, b,d b d x,yda b d b d 6,,d b, b,d 3 5 b d b d x, x,d,ydy b,ydy. In 6, Özdemir et l. proved ollowing ineulities or o-ordinted onvex untions. Theorem 5.Let : =,b,d R be prtil dierentible mpping on =,b,d. I is onvex untion on the o-ordintes on, then the ollowing ineulity holds; b, d d b d,y d b b b d x,ydy b d x, d b d 6,,d b, b,d. Theorem 6.Let : =,b,d R be prtil dierentible mpping on =,b,d. I, >, is onvex untion on the o-ordintes on, then the ollowing ineulity holds; b, d b d x,ydy 7 b d d b d,y d b x, d b b d p p, b,,d b,d. Theorem 7.Let : =,b,d R be prtil dierentible mpping on =,b,d. I,, is onvex untion on the o-ordintes on, then the ollowing ineulity holds; b, d b d x,ydy 8 b d d b d,y d b x, d b b d 6, b,,d b,d. In 7, Özdemir et l. proved the ollowing Theorem whih involves n ineulity o Simpson s type; Theorem 8.Let : R R be twie prtilly dierentible mpping on =,b,d. I is 6 0 NSP Nturl Sienes Publishing Cor.
Appl. Mth. In. Si. 8, No. 3, 085-0 0 / www.nturlspublishing.om/journls.sp 087 onvex untion on the o-ordintes on, then the ollowing ineulity holds:, d b, d b, d b, b,d, b,,d b,d b d x,yda b d 5b d 7,,d b, b,d 7 6b b 6d x, x, d d,y b,y x,d b,y dy. The min purpose o this pper is to estblish new lemm whih gives more generl results nd dierent type ineulities or speil vlues o nd to prove severl ineulities. Min Results For the simpliity, we will denote: Fx,y;, = b, d,,d b, b,d b b,,d, d b, d d b d,ydy b x, d b b x,d x, b d,y b,ydy. d In order to prove our min theorems, we need the ollowing lemm: Lemm.Let : R R be twie prtilly dierentible untion on < b, < d nd 0,. I holds: = L, then the ollowing eulity Fx,y;, b d nd b d Kx= My= b d x b b d x,ydy KxMy x,ydy,, x, b x b b, x b,b d, y, d d d., y d,d Proo.Integrting by prts, we obtin b d b = Kx d d KxMy x,ydy d b = Kx d b = d d x x,ydy d d d d x,ydy x,ydy d x x,y x x,y d d Kx d x, d x d x x, x x,d d d x x,ydy d x x,ydy. 0 NSP Nturl Sienes Publishing Cor.
088 M. E. Özdemir et. l. : On the Co-Ordinted Convex Funtions Integrting by prts gin, we obtin b d KxMy x,ydy = b b d, d,,d b, b,d b d b d b b,,d, d b, d d b b,ydy b d x, d d b x,d x, b d b d,y b,ydy x, ydy. Dividing both sides o the bove eulity byb d, we get the reuired result. Theorem.Let : =,b,d R be twie prtilly dierentible untion on nd 0,. I is onvex untion on the o-ordintes on, then one hs the ineulity: Fx,y;, 0 b d x,ydy b d b d 6,,d b, b,d. Proo.From Lemm nd property o the modulus, we n write Fx,y;, b d b d b d b d x,ydy KxMy x,ydy. Using the hnge o vribles y = sd s, d ds=dy, we obtin Fx,y;, b d x,ydy b d { d b Kx b 0 s x,sd s ds s x,sd s ds s x,sd s ds s x,sd s }. ds Sine is onvex untion on the o-ordintes on, we hve b d KxMy b d x,ydy { d b Kx s b s x,d ds s 0 s x, ds s s x,d ds s s x,d ds s s s s s s 0 x, ds x,d ds } x, ds. s s By lulting the bove integrls, we obtin x, ds Fx,y;, b d x,ydy b d d b { Kx b 8 x, } x,d. By similr rgument or other integrls nd using the hnge o vrible x = tb t, b dt = nd onvexity o x,y on the o-ordintes on, we dedue the result whih is the desired. Remrk.I we hoose = in 0, we hve the ineulity 3. Remrk.I we hoose = 0 in 0, we hve the ineulity 6. 0 NSP Nturl Sienes Publishing Cor.
Appl. Mth. In. Si. 8, No. 3, 085-0 0 / www.nturlspublishing.om/journls.sp 08 Remrk.I we hoose = 3 in 0, we hve the ineulity. Theorem 0.Let : =,b,d R be twie prtilly dierentible untion on nd 0,. I p p is onvex untion on the o-ordintes on, then one hs the ineulity: Fx,y;, b d x,ydy b d b d p p,,d b, b,d or >, = p p. Proo.Let p >. From Lemm nd using the Hölder ineulity see 8 or double integrls, we get Fx,y;, b b d b b d b d d d x,ydy KxMy p p dy x,y dy. Sine p p is onvex untion on the o-ordintes on, by tking into ount the hnge o vrible x = tb t, b dt = dt nd y = sd s, d ds=dy, we hve tb t,y nd t b,y t,y tb t,sd s ts b,dt s b, s t,d t s,. Thus, we obtin Fx,y;, b d x,ydy b d b d p p,,d b, this ompletes the proo. b,d Remrk.Under the ssumptions o Theorem 0, i we hoose = in, we hve the ineulity. Remrk.Under the ssumptions o Theorem 0, i we hoose = 0 in, we hve the ineulity 7. Corollry.Under the ssumptions o Theorem 0, i we hoose = 3 in, we hve the ineulity:, d b, d b, d b, b,d, b,,d b,d b d x,yda b d 5b d 3p p,,d 6b b, b 6d d b,d x, x, d,y, b,y x,d b,y dy. 0 NSP Nturl Sienes Publishing Cor.
00 M. E. Özdemir et. l. : On the Co-Ordinted Convex Funtions Remrk.In Corollry, sine < we hve the ollowing ineulity:, d p p b, d b, d b, b,d, b,,d b,d b d x,yda b d 5b d 3,,d 6b b, b 6d d b,d x, x, d,y b,y <, or p >, x,d b,y dy. Theorem.Let : =,b,d R be twie prtilly dierentible untion on nd 0,. I is onvex untion on the o-ordintes on nd, then one hs the ineulity: Fx,y;, b d x,ydy b d b d 6,,d b, b,d. Proo.From Lemm nd using the well-known Powermen ineulity see 8, we get Fx,y;, b d x,ydy b d b d b d b d KxMy x,y KxMy dy dy. Sine is onvex untion on the o-ordintes on, by tking into ount the hnge o vrible x= tb t,b dt = dt nd y=sd s,d ds= dy, we hve nd tb t,y t b,y t,y tb t,sd s ts b,dt s b, s t,d t s,. Hene, it ollows tht Fx,y;, b d x,ydy b d b d 6,,d b, this ompletes the proo. b,d Remrk.Under the ssumptions o Theorem, i we hoose = in we hve the ineulity 5. Remrk.Under the ssumptions o Theorem, i we hoose = 0 in, we hve the ineulity 8. 0 NSP Nturl Sienes Publishing Cor.
Appl. Mth. In. Si. 8, No. 3, 085-0 0 / www.nturlspublishing.om/journls.sp 0 Corollry.Under the ssumptions o Theorem, i we hoose = 3 in, we hve, d b, d b, d b, b,d, b,,d b,d b d x,yda b d 5b d,,d b, b,d b 6b d 6d Aknowledgement x, x, d,y b,y x,d b,y dy. The uthors re grteul to the nonymous reeree or reul heking o the detils nd or helpul omments tht improved this pper. Reerenes M. K. Bkul nd J. Pečrić, On the Jensen s ineulity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mth., 5, 7-006. M. E. Özdemir, E. Set, M. Z. Srıky, Some new Hdmrd s type ineulities or o-ordinted m onvex nd α,m onvex untions, Hettepe Journl o Mthemtis nd Sttistis, 0, - 0. 3 M. Z. Srıky, E. Set, M. Emin Özdemir nd S.S. Drgomir, New some Hdmrd s type ineulities or o-ordinted onvex untions, Tmsui Oxord Journl o Inormtion nd Mthemtil Sienes, 8, 37-5 0. S. S. Drgomir, On Hdmrd s ineulity or onvex untions on the o-ordintes in retngle rom the plne, Tiwnese Journl o Mth., 5, 775-788 00. 5 D. Y. Hwng, K. L. Tseng nd G. S. Yng, Some Hdmrd s ineulities or o-ordinted onvex untions in retngle rom the plne, Tiwnese Journl o Mthemtis,, 63-73 007. 6 M. E. Özdemir, H. Kvurmı, A. O. Akdemir nd M. Avı, Ineulities or onvex nd s onvex untions on =,b,d, Journl o Ineulities nd Applitions, Februry, 0, 0 0. 7 M. E. Özdemir, A. O. Akdemir nd H. Kvurmı, On the Simpson s ineulity or o-ordinted onvex untions, Submitted. 8 D. S. Mitrinovi, J. Pećrić nd A. M. Fink, Clssil nd new ineulities in nlysis, Kluwer Ademi, Dordreht, 3. M. E. Özdemir, M. A. Lti nd A. O. Akdemir, On some Hdmrd-type ineulities or produt o two s-onvex untions on the o-ordintes, Journl o Ineulities nd Applitions,, 0, 0 M. E. Özdemir, A. O. Akdemir nd Ç. Yıldız, On Co-ordinted Qusi-Convex Funtions, Czehoslovk Mthemtil Journl, 6, 88-00 0. M. E. Özdemir, Ç. Yıldız nd A.O. Akdemir, On Some New Hdmrd-type Ineulities or Co-ordinted Qusi-Convex Funtions, Hettepe Journl o Mthemtis nd Sttistis,, 67 707 0. M. A. Lti, S. Hussin nd S. S. Drgomir, New Owstroski type ineulites or o-ordinted onvex untions, Trnsylvnin Journl o Mthemtis nd Mehnis,, 5 0. 3 M. A. Lti nd S. S. Drgomir, On some new ineulities or dierentible o-ordinted onvex untions, Journl o Ineulities nd Applitions, 8, 0. M. Emin Özdemir reeived the PhD degree in Mthemtis. His reserh interests re in the res o integrl ineulities nd nlysis. He hs published reserh rtiles in reputed interntionl journls o mthemtil nd engineering sienes. He is reeree nd editor o mthemtil journls. Cetin Yıldız is reserh ssistnt t Attürk University. His min reserh interests re: integrl ineulities, nlysis. Ahmet Ok Akdemir reeived the PhD degree in Anlysis. His reserh interests re in the res o integrl ineulities nd nlysis. He hs published reserh rtiles in reputed interntionl journls o mthemtil nd engineering sienes. 0 NSP Nturl Sienes Publishing Cor.