Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// RESEARCH Open Aess Ineulities or onvex nd s-onvex untions on Δ =, b], d] Muhmet Emin Özdemir, Hvv Kvurmi *, Ahmet Ok Akdemir nd Merve Avi 3 * Correspondene: hkvurmi@tuni.edu.tr Deprtment o Mthemtis, K.K. Edution Fulty, Atturk University, Erzurum 5, Turkey Full list o uthor inormtion is vilble t the end o the rtile Abstrt In this rtile, two new lemms re proved nd ineulities re estblished or oordinted onvex untions nd o-ordinted s-onvex untions. Mthemtis Subjet Clssiition ): 6D; 6D5. Keywords: Hdmrd-type ineulity, o-ordintes, s-onvex untions.. Introdution Let : I R R be onvex untion deined on the intervl I o rel numbers nd <b. The ollowing double ineulity; ) b )b) x)dx b is well known in the literture s Hermite-Hdmrd ineulity. Both ineulities hold in the reversed diretion i is onve. In ], Orliz deined s-onvex untion in the seond sense s ollowing: Deinition. A untion : R R, where R =, ), is sid to be s-onvex in the seond sense i αx βy) α s x)β s y) or ll x, y Î, ),, b with b =nd or some ixed s Î, ]. We denote by Ks the lss o ll s-onvex untions. Obviously one n see tht i we hoose s =, the bove deinition redues to ordinry onept o onvexity. For severl results relted to bove deinition we reer reders to -]. In ], Drgomir deined onvex untions on the o-ordintes s ollowing: Deinition. Let us onsider the bidimensionl intervl Δ =, b], d] in R with <b, <d. Auntion: Δ R will be lled onvex on the oordintes i the prtil mppings y :, b] R, y u) =u, y) nd x :, d] R, x v) =x, v) re onvex Ozdemir et l.; liensee Springer. This is n open ess rtile distributed under the terms o the Cretive Commons Attribution Liense http://retiveommons.org/lienses/by/.), whih permits unrestrited use, distribution, nd reprodution in ny medium, provided the originl work is properly ited.
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 where deined or ll y Î, d] nd x Î, b]. Rell tht the mpping : Δ R is onvex on Δ i the ollowing ineulity holds, λx λ)z, λy λ)w) λ x, y) λ) z, w) or ll x, y), z, w) Î Δ nd l Î, ]. In ], Drgomir estblished the ollowing ineulities o Hdmrd-type or oordinted onvex untions on retngle rom the plne R. Theorem. Suppose tht : Δ =, b], d] R is onvex on the o-ordintes on Δ. Then one hs the ineulities; b, d ) b x, d ) d ) ] b dx b d, y dy x, y)dxdy b )d ) b x, )dx b ) b ) d ), y)dy d ), ), d) b, ) b, d). x, d)dx b, y)dy ] :) The bove ineulities re shrp. Similr results n be ound in -]. In 3], Alomri nd Drus deined o-ordinted s-onvex untions nd proved some ineulities bsed on this deinition. Another deinition or o-ordinted s-onvex untions o seond sense n be ound in 5]. Deinition 3. Consider the bidimensionl intervl Δ =, b], d] in, ) with <b nd <d. The mpping : Δ R is s-onvex on Δ i λx λ)z, λy λ)w) λ s x, y) λ) s z, w) holds or ll x, y), z, w) Î Δ with l Î, ] nd or some ixed s Î, ]. In 6], Srıky et l. proved some Hdmrd-type ineulities or o-ordinted onvex untions s ollowing: Theorem. Let : Δ R R be prtil dierentible mpping on Δ :=, b], d] in R with <b nd<d. I t s is onvex untion on the o-ordintes on Δ, then one hs the ineulities: b )d ) J 6 t s, ) t s, d) t s b, ) t s b, d) :)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 3 o 9 where J =, ), d) b, ) b, d) nd A = b ) x, ) x, d)]dx b )d ) d ) x, y)dxdy A, y) b, y)]dy Theorem 3. Let : Δ R R be prtil dier entible mpping on Δ :=, b], d] in R with <b nd<d. I t s, >,is onvex untion on the o-ordintes on Δ, then one hs the ineulities: b )d ) J p ) p t s, ) t s, d) t s b, ) :3) t s b, d) ]. where A, J re s in Theorem nd p =. Theorem. Let : Δ R R be prtil dierentible mpping on Δ :=, b], d] in R with <b nd<d. I t s,, is onvex untion on the o-ordintes on Δ, then one hs the ineulities: b )d ) J 6 t s, ) t s, d) t s b, ) t s b, d) :) where A, J re s in Theorem. In 7], Brnett nd Drgomir proved n Ostrowski-type ineulity or double integrls s ollowing: Theorem 5. Let :, b], d] R be ontinuous on, b], d], x,y = x y exists on, b), d) nd is bounded, tht is x,y = sup x, y) x y <, x,y),b),d)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 then we hve the ineulity; s, t)dtds b ) x, t)dt d ) s, y)ds b )d ) x, y) x b ) ]d ) y d ) ] :5) x,y b ) or ll x, y) Î, b], d]. In 8], Srıky proved n Ostrowski-type ineulity or double integrls nd gve orollry s ollowing: Theorem 6. Let :, b], d] R be n bsolutely ontinuous untion suh tht the prtil derivtive o order exists nd is bounded, i.e., t, s) t s = sup x,y),b),d) t, s) t s < or ll t, s) Î, b], d]. Then we hve, b β α)β α), d β α) t, d ) dt β α) α ) t, )d β) t, d)]dt α ), s)b β) b, s)]ds t, s)dsdt α ) b β) α ) d β) ) Hα, α, β, β)gα, α, β, β) ) b, s ds ] b α) b β) 8 d α) d β) 8 ] t, s) t s :6) or ll, ), b, b ) Î, b], d] with <b, <b where nd Hα, α, β, β ) =α )α ), )d β ), d)] b β )α ) b, )d β ) b, d)] Gα, α, β, β ) ) )] b b =β α ) α ), d β ), d ) β α ) α ) b β ), d b, d )]. Corollry. Under the ssumptions o Theorem 6, we hve b b )d ), d ) d ) t, d ) dt b ) t, s) 6 t s b ) d ). t, s)dsdt ) b, s ds :7)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 5 o 9 In 9], Phptte estblished new Ostrowski type ineulity similr to ineulity.5) by using elementry nlysis. The min purpose o this rtile is to estblish ineulities o Hdmrd-type or oordinted onvex untions by using Lemm nd to estblish some new Hdmrdtype ineulities or o-ordinted s-onvex untions by using Lemm.. Ineulities or o-ordinted onvex untions To prove our min results, we need the ollowing lemm whih ontins kernels similr to Brnett nd Drgomir s kernels in 7], see the rtile 7, proo o Theorem.]). Lemm. Let : Δ =, b], d] R be prtil dierentible mpping on Δ =, b], d]. I t s LΔ), then the ollowing eulity holds: = b, d ) b d ), y b b )d ) b )d ) ) dy b ) x, y)dydx px, t)y, s) t s x, d ) dx b t b t b b, d s d s ) d d dsdt where t ), t, b ] px, t) = ] b t b), t, b nd s ), s, d ] y, s) = ] d s d), s, d or eh x Î, b] nd y Î, d]. Proo. We note tht B = px, t)y, s) b t t s b t b b, d s d s ) d d dsdt.
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 6 o 9 Integrtion by prts, we n write B = = b y, s) t ) b t t s b t b b, d s d s ) d d dt b t b) b t t s b t b b, d s d s ) d d dt ds { y, s) t ) b t s b t b b, d s d s )] b d d b b t dt s t b) s b =b ) s ) d d b t b b, d s d s b t b t b b, d s d s d d b t s b t b b, d s d s ) d d { y, s) s d =b ) d b, d s d s ) d d ) b t b t b b, d s d s d d )] b b dt ds } dt ds s ) b s, d s d s ) d d ds s d) b s, d s d s ) d d ds d s ) b t s b t b b, d s d s ) d d ds d s d) b t s b t b b, d s d s ) d d ds dt. By lulting the bove integrls, we hve b B =b )d ), d ) b b ), d s d s ) d d ds b t d ) b t b b, d ) dt b t b t b b, d s d s ) d d dsdt.
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 7 o 9 Using the hnge o the vrible x = b t b t b b nd y = d s d s d d, then dividing both sides with b - ) d- ), this ompletes the proo. Theorem 7. Let : Δ =, b], d] R be prtil dierentible mpping on Δ =, b], d]. I t s is onvex untion on the o-ordintes on Δ, then the ollowing ineulity holds; b, d ) ) b d ), y dy x, d ) dx b ) b x, y)dydx b )d ) b )d ), ) 6 t s b, ) t s, d) t s ] b, d) t s. Proo. We note tht C = b, d ) d ) b )d ) ) b, y dy b ) x, y)dydx. x, d ) dx From Lemm nd using the property o modulus, we hve C b )d ) px, t)y, s) t s Sine t s is o-ordinted onvex, we n write C b )d ) y, s) b b b t t ) b b t) b b t) b t b t b b t b t b b, d s d s ) d d dsdt b t t ) b t s t s t s b, d s t s, d s d s )] d d dt )] d d )] d s, d s d s d d b, d s d s )] d d dt dt dt ds.
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 8 o 9 By omputing these integrls, we obtin b ) d C y, s) 8d ), d s t s d s d) d y, s) b, d s t s d s )] d d ds. Using o-ordinted onvexity o t s gin, we get C b 8d ) d d s s ) ] d, ) r s d s d d s) ] d, ) t s ds d d s s ) ] d b, ) t s ds d s d d s) ] d b, ) t s ds d ds s s ) d s d d s) d d d d d s) s s ) d ], d) t s ds ], d) t s ds ] b, d) t s ds s ] d b, d) t s ds. By simple omputtion, we get the reuired result. Remrk. Suppose tht ll the ssumptions o Theorem 7 re stisied. I we hoose t s is bounded, i.e., we get t, s) t s = C b )d ) 6 sup t,s),b),d) t, s) t s t, s) t s <, :) whih is the ineulity in.7). Theorem 8. Let : Δ =, b], d] R be prtil dierentible mpping on Δ =, b], d]. I t s, >,is onvex untion on the o-ordintes on Δ, then the ollowing ineulity holds; b )d ) C p ) p, ) t s b, ) t s, d) t s b, d) :) t s
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 9 o 9 where C is in the proo o Theorem 7. Proo. From Lemm, we hve C b )d ) x px, t)y, s) t s b t b t b b, d s d s ) d d dsdt. By pplying the well-known Hölder ineulity or double integrls, then one hs { b C px, t)y, s) ) p dtds b )d ) p t s b t b t b b, d s d s ) ) :3) d d dsdt. Sine t s is o-ordinted onvex untion on Δ, we n write b t t s b t b b, d s d s ) d d ) ) b t d s, ) b d t s ) ) b t s, d) b d t s ) ) t d s b, ) b d t s ) ) t s b, d) b d t s. :) Using the ineulity.) in.3), we get b )d ) C p ) p, ) t s b, ) t s, d) t s b, d) t s where we hve used the t tht This ompletes the proo. ) px, t)y, s) p p b )d )] p dtds =. p ) p
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 Remrk. Suppose tht ll the ssumptions o Theorem 8 re stisied. I we hoose t s is bounded, i.e., t, s) t s = sup t,s),b),d) t, s) t s <, we get b )d ) C t, s) t s :5) p ) p whih is the ineulity in.3) with t, s) t s. Theorem 9. Let : Δ =, b], d] R be prtil dierentible mpping on Δ =, b], d]. I t s, >,is onvex untion on the o-ordintes on Δ, then the ollowing ineulity holds; b )d ) C 6, ) t s b, ) t s, d) t s b, d) :6) t s where C is in the proo o Theorem 7. Proo. From Lemm nd pplying the well-known Power men ineulity or double integrls, then one hs C b )d ) px, t)y, s) t s b b )d ) px, t)y, s) t s b t b t b b, d s d s ) d d dsdt px, t)y, s) ) dsdt b t b t b b, d s d s ) d d dsdt. :7)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 Sine t s is o-ordinted onvex untion on Δ, we n write b t t s b t b b, d s d s ) d d ) ) b t d s, ) b d t s ) ) b t s, d) b d t s ) ) t d s b, ) b d t s ) ) t s b, d) b d t s. :8) I we use.8) in.7), we get { b C b )d ) px, t)y, s) px, t)y, s) ) dsdt b t b ) ) t d s b, ) b d t s ) ) d s d t s t b ), ) b t s b d ]) b, d). ) ) s d t s ) t s, d) ) Computing the bove integrls nd using the t tht px, t)y, s) ) ) b ) d ) dtds =, 6 we obtined the desired result. 3. Ineulities or o-ordinted s-onvex untions To prove our min results we need the ollowing lemm: Lemm. Let : Δ R R be n bsolutely ontinuous untion on Δ where <b, <d nd t, l Î, ], i where D = b )d ) r )r ) E t λ LΔ), then the ollowing eulity holds: D =, )r, d)r b, )r r b, d) r r ) b )d ) ) r r d ) r r b x, y)dxdy ) b, y)dy r d ) x, d)dx r b, y)dy x, )dx
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 nd E = r )t )r )λ ) tb t), λd λ))dtdλ t λ or some ixed r, r Î, ]. Proo. Integrtion by prts, we get E = = r )λ ) ] r )t ) tb t), λd λ))dt dλ t λ r )t ) r )λ ) b ) λ tb t), λd λ)) r ] tb t), λd λ))dt dλ b λ r = r )λ b λ b, λd λ)), λd λ)) b λ r ] tb t), λd λ))dt dλ b λ = r r )λ b, λd λ)) b d r r ) b, λd λ))dλ b )d ) b r b r )λ d Computing these integrls, we obtin, λd λ)) r ) b )d ) r )λ ) tb t), λd λ))dλ λ E = b )d ), )r, d)r b, )r r b, d) r r ) r r ) r )r ) b, λd λ))dλ r ) tb t), d)dt r ), λd λ))dλ ] dt., λd λ))dλ tb t), )dt ] tb t), λd λ))dtdλ. Using the hnge o the vrible x = tb -t) nd y = ld -l) or t, l Î, ] nd multiplying the both sides by b )d ), we get the reuired result. r r ) Theorem. Let : Δ =, b], d], ), ) be n bsolutely ontinuous untion on Δ. I t λ is s-onvex untion on the o-ordintes on Δ, then one hs the ineulity: b )d ) D r )r )s ) s ) MS, ) t λ ML, d) t λ KR b, ) t λ KN ] b, d) t λ 3:)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 3 o 9 where M = s r ) r r ) s r ) N = r ) s L = r s ) r ) s ) s r R = s r r r ) s r S = r r Proo. From Lemm nd by using o-ordinted s-onvexity o t λ, we hve; b )d ) D r )r ) r )t )r )λ ) tb t), λd λ)) t λ dtdλ b )d ) r )r ) r )t )r )λ ) { t s b, λd λ)) t λ } ] t)s, λd λ)) t λ dt dλ. By lulting the bove integrls, we hve r )t ) { t s b, λd λ)) t λ t) s }, λd λ)) t λ dt { = r r )t) t s b, λd λ)) t λ t) s }, λd λ)) t λ dt r )t ) { t b, λd λ)) t λ r t) s }, λd λ)) t λ dt ) ) s = r s ) s )s ) r b, λd λ)) t λ ) ) s r ] s r ) r, λd λ)) r t λ. 3:)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge o 9 By similr rgument or other integrls, by using o-ordinted s-onvexity o t λ, we get r )λ ) { b, λd λ)) t λ }, λd λ)) t λ dλ { r r )λ) λ s b, d) t λ } λ)s b, ) t λ dλ { r )λ ) λ s, d) t λ } λ)s, ) t λ dλ r { = s )s ) r ) s b, d) t λ ] r s ) r ) s, d) t ) ] s r s r r b, ) r t ) s r } r, ) r t. By using these in 3.), we obtin the ineulity 3.). Corollry ) I we hoose r = r = in 3.), we hve, ), d) b, ) b, d) ] d b, y), y)]dy d ] b b x, d) x, )]dx x, y)dxdy b b )d ) b )d ) s s ) s ) ) s s, ) t λ ) b, d) t λ s ) s, d) t λ )] b, ) t λ. 3:3) ) I we hoose r = r = in 3.), we hve d, ), y)dy x, )dx d b b x, y)dxdy b )d ) b )d ) s ) s ) s ) b, ) t λ ] b, d) t λ. Theorem. Let : Δ =, b], d], ), ) be n bsolutely ontinuous untion on Δ. I p p is s-onvex untion on the o-ordintes on Δ, or t λ
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 5 o 9 some ixed s Î, ] nd p >,then one hs the ineulity: b )d ) r )r ) r p ) p ) r p p D r ) p r ) p p ) p t λ, ) t λ, d) t λ b, ) t λ b, d) s ) 3:) or some ixed r, r Î, ], where = p p. Proo. From Lemm nd using the Hölder ineulity or double integrls, we n write D b )d ) r )r ) r )t )r )λ ) dtdλ) p p dtdλ) tb t), λd λ) t λ. Inboveineulityusingthes-onvexity on the o-ordintes o t λ on Δ nd lulting the integrls, then we get the desired result. Corollry 3 ) Under the ssumptions o Theorem, i we hoose r = r = in 3.), we hve, ), d) b, ) b, d) d b )d ) b, y), y)]dy x, y)dxdy b x, d) x, )]dx b )d ) p ) p t λ, ) t λ, d) t λ b, ) t λ b, d) s ). ] 3:5)
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 6 o 9 ) Under the ssumptions o Theorem, i we hoose r = r = in 3.), we hve, ) d b )d ), y)dy b x, y)dxdy x, )dx b )d ) = p ) p t λ, ) t λ, d) t λ b, ) t λ b, d) s ). Remrk. I we hoose s =in 3.5), we obtin the ineulity in.3) Theorem. Let : Δ =, b], d], ), ) be n bsolutely ontinuous untion on Δ. I t λ is s-onvex untion on the o-ordintes on Δ, or some ixed s Î, ] nd, then one hs the ineulity: b )d ) r D ) r ) ) r )r ) r )r ) MS, ) t λ ML, d) t λ KR b, ) t λ KN b, d) t λ s ) s ) or some ixed r, r Î, ]. Proo. From Lemm nd using the well-known Power-men ineulity, we n write ) b )b ) D r )t )r )λ ) dtdλ r )r ) r )t )r )λ ) tb t), λd λ)) t λ dtdλ ]. Sine t λ is s-onvex untion on the o-ordintes on Δ, we hve tb t), λd λ) t λ t s b, λd λ)) t λ t) s, λd λ)) t λ
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 7 o 9 nd tb t), λd λ) t λ t s λ s t λ b, d)t s λ) s t λ b, ) λ s t) s t λ, d) λ) s t) s t λ, ) hene, it ollows tht b )d ) r D ) r ) ) r )r ) r )r ) r )t )r )λ ) { t s λ s t λ t s λ) s t λ b, )λ s t) s t λ, d) λ) s t) s } ) t λ, ) dtdλ b, d) 3:6) By simple omputtion, one n see tht r )t )r )λ ) { t s λ s t λ b, d) t s λ) s t λ b, )λ s t) s t λ, d) λ) s t) s } ) t λ, ) dtdλ MS, ) t λ ML, d) t λ KR b, ) t λ KN b, d) t λ = s ) s ) where K, L, M, N, R, nd S s in Theorem. By substituting these in 3.6) nd simpliying we obtin the reuired result. Corollry ) Under the ssumptions o Theorem, i we hoose r = r =, we hve, ), d) b, ) b, d) d ] ] b ] b, y), y) dy x, d) x, ) dx d b b x, y)dxdy b )d ) ) b )d ) s ) s s, ) t λ b, d) t λ s ) s, d) t λ b, ) t λ s ) s )
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 8 o 9 ) Under the ssumptions o Theorem, i we hoose r = r =, we hve, ) b x, y)dxdy b )d ), y)dy x, )dx d b ) b )d ) s ) b, ) t λ b, d) t λ s ) s ) Remrk 5. Under the ssumptions o Theorem.., i we hoose r = r =nd s =, we get the ineulity in.). Author detils Deprtment o Mthemtis, K.K. Edution Fulty, Atturk University, Erzurum 5, Turkey Deprtment o Mthemtis, Fulty o Siene nd Arts, Ağri İbrhim Çeçen University, Ağri, Turkey 3 Deprtment o Mthemtis, Fulty o Siene nd Arts, Adiymn University, Adiymn, Turkey Authors ontributions HK, AOA nd MA rried out the design o the study nd perormed the nlysis. MEO dviser) prtiipted in its design nd oordintion. All uthors red nd pproved the inl mnusript. Competing interests The uthors delre tht they hve no ompeting interests. Reeived: My Aepted: Februry Published: Februry Reerenes. Orliz, W: A note on modulr spes-i. Bull Ad Polon Si Mth Astronom Phys. 9, 57 6 96). Hudzik, H, Mligrnd, L: Some remrks on s-onvex untions. Aeutiones Mth. 8, 99). doi:.7/ BF83798 3. Drgomir, SS, Fitzptrik, S: The Hdmrd s ineulity or s-onvex untions in the seond sense. Demonstrtio Mth. 3):687 696 999). Kırmı, US, Bkul, MK, Özdemir, ME, Pečrić, J: Hdmrd-type ineulities or s-onvex untions. Appl Mth Comput. 93, 6 35 7). doi:.6/j.m.7.3.3 5. Buri, P, Házy, A, Juhász, T: Bernstein-Doetsh type results or s-onvex untions. Publ Mth Debreen. 75-):3 3 9) 6. Buri, P, Házy, A, Juhász, T: On pproximtely Brekner s-onvex untions. Control Cybern. in press) 7. Brekner, WW: Stetigkeitsussgen ür eine klsse verllgemeinerter konvexer unktionen in topologishen lineren räumen. In Publ Inst Mth, vol. 3, pp. 3.Beogrd 978) 8. Brekner, WW, Orbán, G: Continuity Properties o Rtionlly s-onvex Mppings with Vlues in Ordered Topologil Liner Spe. Bbes-Bolyi University, Kolozsvr 978) 9. Pinheiro, MR: Exploring the onept o s-onvexity. Aeutiones Mth. 73): 9 7). doi:.7/s-7-89-9. Pyi, M: A diret proo o the s-hölder ontinuity o Brekner s-onvex untions. Aeutiones Mth. 6, 8 3 ). doi:.7/s565. Drgomir, SS: On Hdmrd s ineulity or onvex untions on the o-ordintes in retngle rom the plne. Tiwnese J Mth. 5, 775 788 ). Bkul, MK, Pečrić, J: On the Jensen s ineulity or onvex untions on the o-ordintes in retngle rom the plne. Tiwnese J Mth. 5, 7 9 6) 3. Alomri, M, Drus, M: The Hdmrd s ineulity or s-onvex untions o -vribles. Int J Mth Anl. 3):69 638 8). Özdemir, ME, Set, E, Srıky, MZ: Some new Hdmrd s type ineulities or o-ordinted m-onvex nd α, m)- onvex untions. Hettepe J Mth Ist., 9 9 ) 5. Alomri, M, Drus, M: Hdmrd-type ineulities or s-onvex untions. Int Mth Forum. 3):965 975 8) 6. Srıky, MZ, Set, E, EminÖzdemir, M, Drgomir, SS: New some Hdmrd s type ineulities or o-ordinted onvex untions. Tmsui Oxord J Mth Si. )
Özdemir et l. Journl o Ineulities nd Applitions, : http://www.journloineulitiesndpplitions.om/ontent/// Pge 9 o 9 7. Brnett, NS, Drgomir, SS: An Ostrowski type ineulity or double integrls nd pplitions or ubture ormule. Soohow J Mth. 7): ) 8. Srıky, MZ: On the Ostrowski type integrl ineulity or double integrl.http://rxiv.org/bs/5.5v 9. Phptte, BG: A new Ostrowski type ineulity or double integrls. Soohow J Mth. 3):37 3 6) doi:.86/9-x-- Cite this rtile s: Özdemir et l.: Ineulities or onvex nd s-onvex untions on Δ =, b], d]. Journl o Ineulities nd Applitions :. Submit your mnusript to journl nd beneit rom: 7 Convenient online submission 7 Rigorous peer review 7 Immedite publition on eptne 7 Open ess: rtiles reely vilble online 7 High visibility within the ield 7 Retining the opyright to your rtile Submit your next mnusript t 7 springeropen.om