NEW INEQUALITIES OF OSTROWSKI TYPE FOR CO-ORDINATED s-convex FUNCTIONS VIA FRACTIONAL INTEGRALS

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1 Journl of Frtionl Clulus nd Applitions, Vol. 4() Jn. 3, pp ISSN: NEW INEQUALITIES OF OSTROWSKI TYPE FOR CO-ORDINATED s-convex FUNCTIONS VIA FRACTIONAL INTEGRALS M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK Abstrt. In this pper, using the identity proved 43in for frtionl integrls, some new Ostrowski type ineulities for Riemnn-Liouville frtionl integrls of funtions of two vribles re estblished. The estblished results in this pper generlize those results proved in 43.. Introdution Frtionl lulus hs been known sine the 7th entury. Reently, the interest in frtionl nlysis hs been growing ontinully due to its useful pplitions in mny fields of sienes. It hs been shown tht mthemtil expressions involved with frtionl derivtives n be elegntly desribed in interdisiplinry fields, for exmple, eletromgneti wves 34, viso-elsti systems 5, untum evolution of omplex systems 37 nd diffusion wves 9. Furthermore, pplitions of frtionl lulus hve been reported in mny res suh s physis 35, engineering 5, finne 44, soil sienes 8, 59, mthemtil biology 9, 3 nd hos theory,, 36. On the other hnd, in 938, Ostrowski 47 estblished n interesting integrl ineulity ssoited with differentible mppings. This Ostrowski ineulity hs powerful pplitions in numeril integrtion, probbility nd optimiztion theory, stohsti, sttistis, informtion nd integrl opertor theory. Thus, frtionl ineulities hve promising pplitions in ll fields of mthemtis nd pplied sienes. Theorem 47 Let f :, b R be differentible mpping on (, b) whose derivtive f : (, b) R is bounded on (, b), i.e., f := The we hve the ineulity f (x) b f (t) dt b for ll x, b. The onstnt 4 4 ( x b (b ) is the best possible. ) sup t (,b) f (t) <. (b ) f, () Mthemtis Subjet Clssifition. 6A33, 6A5, 6D7, 6D, 6D5. Key words nd phrses. strowski ineulity, o-ordinted onvex funtion,riemnn-liouville frtionl integrl. Submitted June 7,. Published Jn., 3.

2 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 3 The ineulity () n be rewritten in euivlent form s: f (x) b b f (t) dt (x ) (b x) f (b ). Sine 938 when A. Ostrowski proved his fmous ineulity, mny mthemtiins hve been working bout nd round it, in mny different diretions nd with lot of pplitions in Numeril Anlysis nd Probbility, et. Severl generliztions of the Ostrowski integrl ineulity for mppings of bounded vrition, Lipshitzin, monotoni, bsolutely ontinuous, onvex mppings, usi onvex mppings nd n-times differentible mppings with error estimtes for some speil mens nd for some numeril udrture rules re onsidered by mny uthors. For reent results nd generliztions onerning Ostrowski s ineulity see -4, 3, 6, -4, 38, 49-53, 58 nd 6 nd the referenes therein. Let us onsider now bidimensionl intervl =:, b, d in R with < b nd < d, mpping f : R is sid to be onvex on if the ineulity f(λx ( λ)z, λy ( λ)w) λf(x, y) ( λ)f(z, w), holds for ll (x, y), (z, w) nd λ,. The mpping f is sid to be onve on the o-ordintes on if the bove ineulity holds in reversed diretion, for ll (x, y), (z, w) nd λ,. A modifition for onvex (onve) funtions on, whih re lso known s o-ordinted onvex (onve) funtions, ws introdued by S. S. Drgomir 7 s follows: A funtion f : R is sid to be onvex (onve) on the o-ordintes on if the prtil mppings f y :, b R, f y (u) = f(u, y) nd f x :, d R, f x (v) = f(x, v) re onvex (onve) where defined for ll x, b, y, d. A forml definition for o-ordinted onvex (onve) funtions my be stted in: Definition 4 A mpping f : R is sid to be onvex on the o-ordintes on if the ineulity f(tx ( t)y, ru ( r)w) trf(x, u) t( r)f(x, w) r( t)f(y, u) ( t)( r)f(y, w), () holds for ll t, r, nd (x, u), (y, w). The mpping f is onve on the o-ordintes on if the ineulity () holds in reversed diretion for ll t, r, nd (x, u), (y, w). Clerly, every onvex (onve) mpping f : R is onvex (onve) on the o-ordintes. Furthermore, there exists o-ordinted onvex (onve) funtion whih is not onvex (onve), (see for instne 7). The following Hermite-Hdmrd type ineulities were proved in 7:

3 4 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Theorem 7 Suppose tht f : R is o-ordinted onvex on. Then one hs the ineulities: ( b f, d ) b ( f x, d ) dx d ( ) b f b d, y dy b d f (x, y) dydx b f (x, ) dx b f (x, d) dx 4 b b d f (, y) dy d f (b, y) dy d d f (, ) f (, d) f (b, ) f (b, d). (3) 4 The bove ineulities re shrp. The ineulities in (3) hold in reverse diretion if the mpping f is o-ordinted onve on. Alomri et l. 7 defined the o-ordinted s-onvexity in the seond sense s follows: Definition 7 Let =:, b, d, ) with < b nd < d. A mpping f : R is sid to be s-onvex in the seond sense on if the ineulity f(λx ( λ)z, λy ( λ)w) λ s f(x, y) ( λ) s f(z, w), holds for ll (x, y), (z, w), λ, nd for some fixed s (,. The mpping f is sid to be s-onve on the o-ordintes on if the bove ineulity holds in reversed diretion, for ll (x, y), (z, w), λ, nd for some fixed s (,. A funtion f : R is sid to be s-onvex (s-onve) in the seond senses on the o-ordintes on if the prtil mppings f y :, b R, f y (u) = f(u, y) nd f x :, d R, f x (v) = f(x, v) re s-onvex (s-onve) in the seond sense where defined for ll x, b, y, d for some fixed s (,. A forml definition for o-ordinted s-onvex (s-onve) funtions in the seond sense my be stted in: Definition 3 A mpping f : R is sid to be s-onvex in the seond sense on the o-ordintes on if the ineulity f(tx ( t)y, ru ( r)w) t s r s f(x, u) t s ( r) s f(x, w) r s ( t) s f(y, u) ( t) s ( r) s f(y, w), (4) holds for ll t, r,, (x, u), (y, w) nd for some fixed s (,. The mpping f is s-onve in the seond sense on the o-ordintes on if the ineulity (4) holds in reversed diretion for ll t, r,, (x, u), (y, w) for some fixed s (,. It is lso proved in 7 tht every s-onvex mpping f : R is s-onvex on the o-ordintes on. Furthermore, there exists o-ordinted s-onvex funtion whih is not s-onvex, (see for instne 7.

4 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 5 The following Hermite-Hdmrd type ineulities were proved in 7: Theorem 3 7 Suppose f : =:, b, d, ), ) with < b nd < d is s-onvex on the o-ordintes on. The one hs the ineulities: 4 s f ( b, d ) s b (s ) b f ( x, d ) dx d b b d d b d d f f (x, y) dydx f (x, ) dx b f (, y) dy d d b f (b, y) dy ( ) b, y dy f (x, d) dx f (, ) f (, d) f (b, ) f (b, d) (s ). (5) In reent yers, mny uthors hve proved severl ineulities for o-ordinted onvex funtions. These studies inlude, mong others, the works in 5-, 7, 8, 33, 39-43, 48 nd 56 (see lso the referenes therein). Alomri et l. 5-7, proved severl Hermite-Hdmrd type ineulities for o-ordinted s-onvex funtions. Bkul et. l, proved Jensen s ineulity for onvex funtions on the o-ordintes from the retngle from the pln. Drgomir 7, proved the Hermite-Hdmrd type ineulities for o-ordinted onvex funtions. Hwng et. l 33, lso proved some Hermite-Hdmrd type ineulities for o-ordinted onvex funtion of two vribles by onsidering some mppings diretly ssoited to the Hermite-Hdmrd type ineulity for o-ordinted onvex mppings of two vribles. Ltif et. l 39-43, proved some ineulities of Hermite-Hdmrd type for differentible o-ordinted onvex funtion, produt of two o-ordinted onvex mppings, for o-ordinted h-onvex mppings nd some Ostrowski type ineulities for o-ordinted onvex mppings. Özdemir et. l 48, proved Hdmrd s type ineulities for o-ordinted m-onvex nd (α, m)-onvex funtions. Sriky, et. l 56 proved Hermite-Hdmrd type ineulities for differentible o-ordinted onvex funtions. For more ineulities on o-ordinted onvex funtions see lso the referenes in the bove ited ppers. In the present pper, we estblish new Ostrowski type ineulities for o-ordinted s-onvex funtions similr to those from 43 but vi Riemnn-Liouville frtionl integrl nd hene generlizing those results from 43.. Min Results We give first some neessry definitions nd mthemtil preliminries of frtionl lulus theory whih re used in this setions.

5 6 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Definition 4 Let f L, b. The Riemnn-Liouville integrls J α f nd J α b f of order α > with re defined by J α f(x) = Γ (α) J α b f(x) = Γ (α) x b x (x t) α f(t)dt, x > (t x) α f(t)dt, x < b, where Γ (α) = e u u α du. It is to be noted tht J f(x) = Jb f(x) = f(x). In the se α =, the frtionl integrl redues to the lssil integrl. For further properties nd results onerning this opertor we refer the interested reder to, 4, 5-3, 43, 53 nd 54. For the ske of onveniene, we will use the following nottion throughout this setion: A = where Γ (α ) Γ (β ) Jx,y f α,β (, ) Jx,yf α,β (, d) Jx,y f α,β (b, ) Jx,yf α,β (b, d) Jy f β (x, ) Jyf β (x, d) (x )α (b x) α Γ (β ) J,f(x, α,β y) = Γ (α) Γ (β) J α,β b,d f(x, y) = Γ (α) Γ (β) J α,β,d f(x, y) = Γ (α) Γ (β) J α,β b, f(x, y) = Γ (α) Γ (β) (y ) β (d y) β Γ (α ) x y b d x y x d y b y x J α x f (, y) J α xf (b, y), (x u) α (y v) β f(u, v)dvdu, x >, y >, (u x) α (v y) β f(u, v)ddvdu, x < b, y < d, (x u) α (v y) β f(u, v)ddvdu, x >, y < d, (u x) α (y v) β f(u, v)ddvdu, x < b, y >, nd Γ is the Euler Gmm funtion. To estblish our min results we need the following identity: Lemm 43 Let f : :=, b, d R be twie prtil differentible mpping on with < b, < d. If f L ( ) nd α, β >,,, then the

6 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 7 following identity holds: (b x) α (x ) α (d y) β (y ) β f (x, y) A = (x )α (y ) β (x )α (d y) β (b x)α (y ) β (b x)α (d y) β r β t α f (tx ( t), ry ( r) ) r β t α f (tx ( t), ry ( r) d) r β t α f (tx ( t) b, ry ( r) ) r β t α f (tx ( t) b, ry ( r) d), (6) for ll (x, y). Theorem 4 Let f : :=, b, d R be twie prtil differentible mpping on with < b, < d,, suh tht f L ( ). If f is s-onvex on the o-ordintes on nd y x f(x, y) M, (x, y), then the following ineulity for frtionl integrls with α, β > holds: (b x) α (x ) α (d y) β (y ) β f (x, y) A (b x) α (x ) α (d y) β (y ) β K, (7) b d for ll (x, y), where M K = (α s ) (β s ) MΓ (s ) Γ (β ) Γ (α s ) Γ (α s ) Γ (β s ) MΓ (s ) Γ (α ) Γ (β s ) Γ (α s ) Γ (β s ) M (Γ (s )) Γ (β ) Γ (α ) Γ (α s ) Γ (β s ) nd Γ is the Euler Gmm funtion.

7 8 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Proof. From Lemm (), we hve tht the following ineulity holds for ll (x, y) : (b x) α (x ) α (d y) β (y ) β f (x, y) A (x )α (y ) β (x )α (d y) β (b x)α (y ) β (b x)α (d y) β r β t α r β t α r β t α r β t α f (tx ( t), ry ( r) ) f (tx ( t), ry ( r) d) f (tx ( t) b, ry ( r) ) f (tx ( t) b, ry ( r) d). (8) By the onvexity of f on the o-ordintes on nd y x f(x, y) M, (x, y), we get the following ineulities: M r β t α f (tx ( t), ry ( r) ) M = r βs t αs dsdt M r βs t α ( t) s M t αs r β ( r) s t α ( t) s r β ( r) s M MΓ (s ) Γ (β ) Γ (α s ) (α s ) (β s ) Γ (α s ) Γ (β s ) MΓ (s ) Γ (α ) Γ (β s ) Γ (α s ) Γ (β s ) M (Γ (s )) Γ (β ) Γ (α ). (9) Γ (α s ) Γ (β s ) Anlogously, we lso hve the following ineulities: r β t α f (tx ( t), ry ( r) d) M MΓ (s ) Γ (β ) Γ (α s ) (α s ) (β s ) Γ (α s ) Γ (β s ) MΓ (s ) Γ (α ) Γ (β s ) Γ (α s ) Γ (β s ) M (Γ (s )) Γ (β ) Γ (α ), () Γ (α s ) Γ (β s )

8 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 9 r β t α f (tx ( t) b, ry ( r) ) M (α s ) (β s ) MΓ (s ) Γ (β ) Γ (α s ) Γ (α s ) Γ (β s ) MΓ (s ) Γ (α ) Γ (β s ) Γ (α s ) Γ (β s ) M (Γ (s )) Γ (β ) Γ (α ) Γ (α s ) Γ (β s ) () nd r β t α f (tx ( t) b, ry ( r) d) M (α s ) (β s ) MΓ (s ) Γ (β ) Γ (α s ) Γ (α s ) Γ (β s ) MΓ (s ) Γ (α ) Γ (β s ) Γ (α s ) Γ (β s ) M (Γ (s )) Γ (β ) Γ (α ). () Γ (α s ) Γ (β s ) By using (9)-() in (8), we get the desired ineulity (7). This ompletes the proof of the theorem. Remrk In Theorem 4, if we tke α = β = nd s =, then the ineulity (7) redues to the ineulity estblished in 4, Theorem 3. The next result is bout the powers of the bsolute vlue of the prtil derivtives. Theorem 5 Let f : :=, b, d R be twie prtil differentible mpping on with < b, < d,, suh tht f L ( ). If f is s-onvex on the o-ordintes on, p, >, p = nd y x f(x, y) M, (x, y), then the following ineulity for frtionl integrls with α, β > holds: (b x) α (x ) α (d y) β (y ) β f (x, y) A ( ) (b x) α (x ) α (d y) β (y ) β M, (3) s (b ) (αp ) p (d ) (βp ) p for ll (x, y).

9 3 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Proof. From Lemm nd the Hölder ineulity, we hve tht the following ineulity holds, for ll (x, y) : (b x) α (x ) α (d y) β (y ) β (x ) α (y ) β ( f (x, y) A ) t αp r βp p ( ) f (tx ( t), ry ( r) ) (x )α (d y) β ( ) f (tx ( t), ry ( r) d) (b x)α (y ) β ( ) f (tx ( t) b, ry ( r) ) (b x)α (d y) β ( ) f (tx ( t) b, ry ( r) d). (4) By the o-ordinted onvexity of f nd y x f(x, y) M, for ll (x, y), we hve tht the following ineulity holds: f (tx ( t), ry ( r) ) 4M (s ). Similrly, we lso hve the following ineulities: f (tx ( t), ry ( r) d) 4M (s ), nd Using the ft f (tx ( t) b, ry ( r) ) 4M (s ) f (tx ( t) b, ry ( r) d) 4M (s ). t αp r αp = (αp ) (βp ) nd using the lst four ineulities in (4), we obtin (3). This ompletes the proof of the theorem. This ompletes the proof of the theorem. Remrk In Theorem 5, if we tke α = β =, then the ineulity (3) beomes the ineulity proved in 4, Theorem 4. A different pproh leds us to the following result: Theorem 6 Let f : :=, b, d R be twie prtil differentible mpping on with < b, < d,, suh tht f L ( ). If f is s-onvex on the o-ordintes on, nd y x f(x, y) M, (x, y), then

10 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 3 the following ineulity for frtionl integrls with α, β > holds: (b x) α (x ) α (d y) β (y ) β f (x, y) A K M (b x) α (x ) α (d y) β (y ) β, (α ) (β ) b d (5) for ll (x, y), where K is defined in Theorem 4. Proof. From Lemm nd the power men ineulity, we hve tht the following ineulity holds, for ll (x, y) : (b x) α (x ) α (d y) β (y ) β ( f (x, y) A (x ) α (y ) β ( t α r β (x )α (d y) β (b x)α (y ) β (b x)α (d y) β ( ( ( t α r β t α r β s t t α r β f (tx ( t), ry ( r) ) f (tx ( t), ry ( r) d) f (tx ( t) b, ry ( r) ) f (tx ( t) b, ry ( r) d) ) t α r β ) ) ) ). By the o-ordinted onvexity of f nd y x f(x, y) M, for ll (x, y), we hve tht the following ineulity holds: t α r β f (tx ( t), ry ( r) ) M M r sβ t α ( t) s M r sβ t sα M r β ( r) s t α ( t) s t sα r β ( r) s M K. In similrly wy, we lso hve the following ineulities: t α r β f (tx ( t), ry ( r) d) dsdt M K, t α r β f (tx ( t) b, ry ( r) ) dsdt M K nd t α r β f (tx ( t) b, ry ( r) d) dsdt M K. (6)

11 3 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Using the ft t α r β = (α ) (β ) nd the lst four ineulities, we obtin from (6) the ineulity (5). This ompletes the proof of the theorem. This ompletes the proof of the theorem. Remrk 3 In Theorem 6, if we tke α = β =, then the ineulity (5) beomes the ineulity proved in 4, Theorem 5. Now we drive some results with o-ordinted onvity property insted of oordinted onvexity. Theorem 7 Let f : R be twie prtil differentible mpping on suh tht f L ( ). If f is s-onve on the o-ordintes on nd p, >, p =, then the ineulity (b x) α (x ) α (d y) β (y ) β f (x, y) A 4 s ( αp) p ( βp) p (x ) α (y ) β ( x f, y ) (x ) α (d y) β ( x f, d y ) (b x) α (y ) β ( b x f, y ) (b x) α (d y) β f ( b x, d y ), (7) hods for ll (x, y), where. Proof. From Lemm nd using the Hölder ineulity for double integrls, we hve tht ineulity holds: (b x) α (x ) α (d y) β (y ) β ( ) f (x, y) A r βp t αp p (x ) α (y ) β (x )α (d y) β (b x)α (y ) β (b x)α (d y) β ( ( ( ( f (tx ( t), ry ( r) ) f (tx ( t), ry ( r) d) f (tx ( t) b, ry ( r) ) s t f (tx ( t) b, ry ( r) d) ) ) ) ), (8)

12 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 33 for ll (x, y). Sine f is onve on the o-ordintes on, so n pplition of (5) with ineulities in reversed diretion, gives us the following ineulities: f (tx ( t), ry ( r) ) s t s ( f tx ( t), y ) dt ( ) x f, ry ( r) dr 4 s ( x f, y ), (9) nd f (tx ( t), ry ( r) d) dsdt s ( f tx ( t), d y ( ) x f, ry ( r) f (tx ( t) b, ry ( r) ) s ( f tx ( t), y ( ) b x f, ry ( r) f (tx ( t) b, ry ( r) d) s ( f tx ( t) b, d y ( ) b x f, ry ( r) d ) dt dr 4 s ( x f, d y ), () ) dt dr 4 s ( b x f, y ) ) dt dr () 4 s ( b x f, d y ). () By mking use of (9)-() in (8), we obtin (7). Thus the proof of the theorem is omplete.

13 34 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 Remrk 4 If we tke α = β =, in Theorem 7, we get the ineulities proved in 4, Theorem 6. Aknowledgement. The uthors re very thnkful to the nonymous referee for his/her vluble omments whih hve improved the finl version of the pper. The first uthor would like to sy thnk to Professor Gston M. N Guérékt, Morgn Stte University, Bltimore, Mrylnd, USA, for his ontinuous enourgement nd support to write reserh rtiles. Referenes G. Anstssiou, M.R. Hooshmndsl, A. Ghsemi nd F. Moftkhrzdeh, Montogomery identities for frtionl integrls nd relted frtionl ineulities, J. Ine. Pure nd Appl. Mth., (4) (9), Art. 97. M. Alomri nd M. Drus, Some Ostrowski type ineulities for onvex funtions with pplitions, RGMIA 3 () () rtile No. 3. Preprint. 3 M. Alomri, M. Drus, S.S. Drgomir, P. Cerone, Ostrowski type ineulities for funtions whose derivtives re s-onvex in the seond sense, Appl. Mth. Lett. 3 () M. Alomri, M. Drus, Some Ostrowski type ineulities for usi-onvex funtions with pplitions to speil mens, RGMIA Res. Rep. Coll., 3 (),, Artile 3. 5 M. Alomri nd M. Drus, Hdmrd-type ineulities for s-onvex funtions, Interntionl Mthemtil Forum, 3 (8), no. 4, M. Alomri nd M. Drus, Co-ordinted s-onvex funtion in the first sense with some Hdmrd-type ineulities, Int. Journl Contemp. Mth. Sienes, 3 (8), no. 3, M. Alomri nd M. Drus, The Hdmrd s ineulity for s-onvex funtion of -vribles on the o-ordintes, Interntionl Journl of Mth. Anlysis, (8), no. 3, W.M. Ahmd, R. El-Khzli, Frtionl-order Dynmil Models of Love, Chos Solitons Frt., Vol.33(7),pp E. Ahmed, A.S. Elgzzr, On Frtionl Order Differentil Eutions Model for Nonlol Epidemis, Physi A, Vol.379(7),pp A.E. Mtouk, Stbility Conditions, Hyperhos nd Control in Novel Frtionl Order Hyperhoti System, Phys. Lett. A, Vol.373(9),pp A.E. Mtouk, Chos, Feedbk Control nd Synhroniztion of Frtionl-order Modified Autonomous Vn der Pol-Duffing Ciruit, Commun. Nonliner Si. Numer. Simult., Vol. 6(),pp M. K. Bkul nd J. Pečrić, On the Jensen s ineulity for onvex funtions on the oordintes in retngle from the plne, Tiwnese Journl of Mth., 5, 6, N.S. Brnett, P. Cerone, S.S. Drgomir, M.R. Pinheiro nd A. Sofo, Ostrowski type ineulities for funtions whose modulus of derivtives re onvex nd pplitions, RGMIA Res. Rep. Coll., 5() (), Artile. ONLINE: 4 S. Belrbi nd Z. Dhmni, On some new frtionl integrl ineulities, J. Ine. Pure nd Appl. Mth., (3) (9), Art R.L. Bgley, R.A. Clio, Frtionl order Stte Eutions for the Control of Visoelstilly Dmped Strutures, J. Guid. Control Dyn., Vol.4(99),pp P. Cerone nd S.S. Drgomir, Ostrowski type ineulities for funtions whose derivtives stisfy ertin onvexity ssumptions, Demonstrtio Mth., 37 (4), no., S. S. Drgomir, On the Hdmrd s ineulity for onvex funtions on the o-ordintes in retngle from the plne, Tiwnese Journl of Mthemtis, 5 (), no. 4, S. S. Drgomir nd C. E. M. Pere, Seleted Topis on Hermite-Hdmrd Type Ineulities nd Applitions, RGMIA (), Monogrphs. ONLINE : hdmrd.html. 9 S. S. Drgomir nd S. Fitzptrik, The Hdmrd s ineulity for s-onvex funtions in the seond sense, Demonstrtio Mth. 3(4), (999), S.S. Drgomir, On the Ostrowski s integrl ineulity for mppings with bounded vrition nd pplitions, Mth. Ine. &Appl., () (998). S.S. Drgomir, The Ostrowski integrl ineulity for Lipshitzin mppings nd pplitions, Comput. Mth. Appl., 38 (999),

14 JFCA-3/4 NEW INEQUALITIES VIA FRACTIONAL INTEGRALS 35 S. S. Drgomir, S. Wng, A new ineulity of Ostrowski s type in L -norm nd pplitions to some speil mens nd to some numeril udrture rules, Tmkng J. of Mth., 8 (997), S.S. Drgomir nd S. Wng, A new ineulity of Ostrowski type in L p -norm nd pplitions to some speil mens nd to some numeril udrture rules, Indin J. Mth., 4(3) (998), S.S. Drgomir, P. Cerone, J. Roumeliotis, A new generliztion of Ostrowski s integrl ineulity for mppings whose derivtives re bounded nd pplitions in numeril integrtion nd for speil mens, Appl. Mth. Lett. 3 () Z. Dhmni, New ineulities in frtionl integrls, Interntionl Journl of Nonliner Sinee, 9(4) (), Z. Dhmni, On Minkowski nd Hermite-Hdmrd integrl ineulities vi frtionl integrtion, Ann. Funt. Anl. () (), Z. Dhmni, L. Tbhrit, S. Tf, Some frtionl integrl ineulities, Nonl. Si. Lett. A, () (), Z. Dhmni, L. Tbhrit, S. Tf, New generliztions of Gruss ineulity usin Riemnn- Liouville frtionl integrls, Bull. Mth. Anl. Appl., (3) (), A.M.A El-Syed, Frtionl-order Diffusion-wve Eution, Int. J. Theor. Phys., Vol.35(996),pp A.M.A. El-Syed, A.E.M. El-Mesiry, H.A.A. El-Sk, On the Frtionl-order Logisti Eution, Applied Mthemtis Lett., Vol.(7),pp R. Gorenflo, F. Minrdi, Frtionl lulus: integrl nd differentil eutions of frtionl order, Springer Verlg, Wien (997), H. Hudzik nd L. Mligrnd, Some remrks on s-onvex funtions, Aeutiones Mth. 48 (994), D. Y. Hwng, K. L. Tseng nd G. S. Yng, Some Hdmrd s ineulities for o-ordinted onvex funtions in retngle from the plne, Tiwnese Journl of Mthemtis, (7), O. Heviside, Eletromgneti theory. New York: Chelse; R. Hilfer (Ed.), Applitions of frtionl lulus in physis. New Jersey: World Sientifi;. 36 A.S. Hegzi, E. Ahmed, A.E. Mtouk, The Effet of Frtionl order on Synhroniztion of Two Frtionl order Choti nd Hyperhoti Systems, Journl of Frtionl Clulus nd Applitions Vol. No. 3(),pp D. Kusnezov, A. Bulg, G.D. Dng, Quntum Levy Proesses nd Frtionl Kinetis, Phys. Rev. Lett., Vol.8(999),pp Z. Liu, Some ompnions of n Ostrowski type ineulity nd pplition, J. Ineul. in Pure nd Appl. Mth, (), 9, Art. 5, pp. 39 M. A. Ltif, nd M. Alomri, On Hdmrd-type ineulities for h-onvex funtions on the o-ordintes, Interntionl Journl of Mth. Anlysis, 3 (9), no. 33, M. A. Ltif nd M. Alomri, Hdmrd-type ineulities for produt two onvex funtions on the o-ordintes, Interntionl Mthemtil Forum, 4 (9), no. 47, M. A. ltif nd S. S. Drgomir, On some new ineulities for differentible o-ordinted onvex funtions, J. Ine. Appl., :8. 4 M. A. Ltif, S. Hussin nd S. S. Drgomir, New Ostrowski type ineulities for o-ordinted onvex funtions, RGMIA Reserh Report Colletion, 4(), Artile 49. ONLINE: 43 M. A. ltif nd S. Hussin, New ineulities of Ostroski type for o-ordinted onvex funtions vi frtionl integrls, Journl of Frtionl Clulus nd Applitions, Vol.. Jnury, No.9, pp N. Lskin, Frtionl Mrket Dynmis, Physi A, Vol.87(),pp S. Miller nd B. Ross, An introdution to the Frtionl Clulus nd Frtionl Differentil Eutions, John Wiley & Sons, USA, 993, p.. 46 D.S. Mitrinović, J. E. Pečrić nd A.M. Fink, Ineulities Involving Funtions nd Their Integrls nd Derivtives, Kluwer Ademi Publishers, Dortreht, A. Ostrowski, Über die Absolutbweihung einer differentiebren Funktion von ihrem Integrlmittelwert, Comment. Mth. Helv., (938), 6 7. Comp. Appl. Mth., Vol., N., 3.

15 36 M. A. LATIF, S. S. DRAGOMIR, A. E. MATOUK JFCA-3/4 48 M. E. Özdemir, E. Set, nd M. Z. Srıky, Some new Hdmrd s type ineulities for o-ordinted m-onvex nd (α, m)-onvex funtions, Aepted. 49 M.E. Özdemir, H. Kvurm nd E. Set, Ostrowski s type ineulities for (α, m)-onvex funtions, Kyungpook Mth. J., 5 (), B. G. Phptte, On n ineulity of Ostrowski type in three independent vribles, J. Mth.Anl. Appl., 49(), B. G. Phptte, On new Ostrowski type ineulity in two independent vribles, Tmkng J. Mth., 3(), (), I. Podlubny, Frtionl differentil eutions: n introdution to frtionl derivtives, frtionl differentil eutions, to methods of their solution nd some of their pplitions. New York: Ademi Press; A. Rfi, N.A. Mir nd F. Ahmd, Weighted Čebyšev-Ostrowski type ineulities, Applied Mth. Mehnis (English Edition), 7, 8(7), M.Z. Sriky, E. Set, H. Yldiz nd N. Bşk, Hermite-Hdmrd s ineulities for frtionl integrls nd relted frtionl ineulities, Submitted. 55 M.Z. Sriky nd H. Ogunmez, On new ineulities vi Riemnn-Liouville frtionl integrtion, rxiv:5.67v, submitted. 56 M. Z. Sriky, E. Set, M. E. Özdemir nd S. S. Drgomir, New some Hdmrd s type ineulities for o-ordinted onvex funtions, Aepted. 57 M. Z. Sriky, On the Ostrowski type integrl ineulity, At Mth. Univ. Comenine, Vol. LXXIX, (), pp E. Set, New ineulities of Ostrowski type for mppings whose derivtives re s-onvex in the seond sense vi frtionl integrls, Comput. Mth. Appl., 59 L. Song, S. Xu, J. Yng, Dynmil Models of Hppiness with Frtionl Order, Commun. Nonliner Si. Numer. Simult., Vol.5(),pp N. Ujević, Shrp ineulities of Simpson type nd Ostrowski type, Comput. Mth. Appl. 48 (4) L. Zhongxue, On shrp ineulities of Simpson type nd Ostrowski type in two independent vribles, Comput. Mth. Appl., 56 (8) M. A. Ltif College of Siene, Deprtment of Mthemtis, University of Hil, Hil-44, Sudi Arbi E-mil ddress: m mer ltif@hotmil.om S. S. Drgomir Shool of Engineering & Siene, Vitori University, PO Box 448, Melbourne City, MC 8, Austrli E-mil ddress: sever.drgomir@vu.edu.u A. E. Mtouk College of Siene, Deprtment of Mthemtis, University of Hil, Hil-44, Sudi Arbi E-mil ddress: emtouk@hotmil.om