SOME NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE HIGHER ORDER PARTIAL DERIVATIVES ARE CO-ORDINATED s-convex

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1 Krgujev Journl of Mthetis Volue 38 4, Pges 5 46 SOME NEW HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE HIGHER ORDER PARTIAL DERIVATIVES ARE CO-ORDINATED s-convex MUHAMMAD AMER LATIF Abstrt In this pper we point out soe ineulities of Herite-Hdrd type for double integrls of funtions whose prtil derivtives of higher order re o-ordinted s-onvex in the seond sense Our estblished results generlize the Herite-Hdrd type ineulities estblished for o-ordinted s-onvex funtions nd refine those results estblished for differentible funtions whose prtil derivtives of higher order re o-ordinted onvex proved in reent literture Introdution A funtion f : I R, I R, is sid to be onvex on I if the ineulity f λx λ y λf x λ f y, holds for ll x, y I nd λ, The ineulity holds in reverse diretion if f is onve The ost fous ineulity onerning the lss of onvex funtions, is the Herite- Hdrd s ineulity This double ineulity is stted s b f b f x dx f f b where f : I R, I R onvex funtion,, b I with < b The ineulities in re in reversed order if f onve funtion The ineulities hve beoe n iportnt ornerstone in thetil nlysis nd optiiztion nd ny uses of these ineulities hve been disovered in Key words nd phrses Convex funtion, Herite-Hdrd type ineulity, Co-ordinted onvex funtion, s-onvex funtion, Power-en integrl ineulity Mthetis Subjet Clssifition 6A33, 6A5, 6D7, 6D, 6D5 Reeived: June, 3 Revised: June, 4 5

2 6 M A LATIF vriety of settings Moreover, ny ineulities of speil ens n be obtined for prtiulr hoie of the funtion f Due to the rih geoetril signifine of Herite- Hdrd s ineulity, there is growing literture providing its new proofs, extensions, refineents nd generliztions, see for exple 8, 4, 9, 9, 3, 33 nd the referenes therein In the pper 5, Hudzik nd Mligrnd onsidered, ong others, the lss of funtions whih re s-onvex in the seond sense This lss is defined follows A funtion f :, R is sid to be s-onvex in the seond sense if f λx λ y λ s f x λ s f y holds for ll x, y,, λ, nd for soe fixed s, It n be esily seen tht for s, s-onvexity redues to ordinry onvexity of funtions defined on, In 9, Drgoir nd Fitzptrik proved vrint of Hdrd s ineulity whih holds for s-onvex funtions in the seond sense Theore 9 Suppose tht f :,, is n s-onvex funtion in the seond sense, where s, nd, b,, < b If f L, b, then the following ineulities hold b 3 s f The onstnt k s b f x dx f f b s is the best possible in the seond ineulity in 3 For ore bout properties nd Herite-Hdrd type ineulities of s-onvex funtions in the seond sense we refer the interested reders to 7, 9,, 5, Let us onsider now bidiensionl intervl :, b, d in R with < b nd < d A pping f : R is sid to be onvex on if the ineulity fλx λz, λy λw λfx, y λfz, w holds for ll x, y, z, w nd λ, A odifition for onvex funtions on, known s o-ordinted onvex funtions, ws introdued by S S Drgoir s follows A funtion f : R is sid to be onvex on the o-ordintes on if the prtil ppings f y :, b R, f y u fu, y nd f x :, d R, f x v fx, v re onvex where defined for ll x, b, y, d A forl definition for o-ordinted onvex funtions y be stted s follow Definition A funtion f : R is sid to be onvex on the o-ordintes on if the following ineulity holds for ll t, r, nd x, u, y, w ftx ty, ru rw trfx, u t rfx, w r tfy, u t rfy, w

3 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 7 Clerly, every onvex pping f : R is onvex on the o-ordintes but onverse y not be true The following Herite-Hdrd type ineulities for o-ordinted onvex funtions on the retngle fro the plne R were estblished in Theore Suppose tht f : R is o-ordinted onvex on, then b f, d b f x, d dx b f b d, y dy 4 4 b d b b f x, y dydx f x, f x, d dx f, y f b, y dy d f, f, d f b, f b, d 4 The bove ineulities re shrp The onept of s-onvex funtions on the o-ordintes in the seond sense ws introdued by Alori nd Drus in 3 s generliztion of the usul o-ordinted onvexity Definition 3 Consider the bidiensionl intervl, b, d in, with < b nd < d The pping f : R is s-onvex in the seond sense on if fλx λ z, λy λ w λ s fx, y λ s fz, w, holds for ll x, y, z, w, λ, with soe fixed s, A funtion f :, R is lled s-onvex in the seond sense on the o-ordintes on if the prtil ppings f y :, b R, f y u fu, y nd f x :, d R, f x v fx, v, re s-onvex in the seond sense for ll y, d, x, b nd s,, ie, the prtil ppings f y nd f x re s-onvex in the seond sense with soe fixed s, A forl definition of o-ordinted s-onvex funtion in seond sense y be stted s follows Definition 3 A funtion f :, R is lled s-onvex in the seond sense on the o-ordintes on if ftx ty, ru rw t s r s fx, u t s r s fx, w 5 r s t s fy, u t s r s fy, w holds for ll t, r, nd x, u, y, u, x, w, y, w, for soe fixed s, The pping f is onve on the o-ordintes on if the ineulity 5 holds in reversed diretion for ll t, r, nd x, y, u, w with soe fixed s,

4 8 M A LATIF Furtherore, Alori nd Drus 5 introdued new lss of s-onvex funtions on the o-ordintes on the retngle fro the plne s follows Definition 4 5 Consider the bidiensionl intervl :, b, d in, with < b nd < d The pping f : R is s-onvex in the seond sense on if there exist s, s, with s s s suh tht fλx λ z, λy λ w λ s fx, y λ s fz, w holds for ll x, y, z, w, λ, MW Os,s This lss of funtions is denoted by A funtion f :, R is lled s-onvex in the seond sense on the o-ordintes on if the prtil ppings f y :, b R, f y u fu, y nd f x :, d R, f x v fx, v, re s -onvex nd s -onvex in the seond sense for ll y, d, x, b nd s, s, with s s s, respetively, ie, the prtil ppings f y nd f x re s -onvex nd s -onvex in the seond sense, s, s, with s s s The definition 3 n be generlized s follows Definition 5 A funtion f : :, b, d, R is lled s-onvex in the seond sense on the o-ordintes on if ftx ty, ru rw t s r s fx, u t s r s fx, w 6 r s t s fy, u t s r s fy, w holds for ll t, r, nd x, u, y, u, x, w, y, w, s, s, with s s s The pping f is onve on the o-ordintes on if the ineulity 6 holds in reversed diretion for ll t, r, nd x, y, u, w, s, s, with s s s In 5, Alori et l lso proved vrint of ineulities given bove by 4 for s-onvex funtions in the seond sense on the o-ordintes on retngle fro the plne R Theore 3 5 Suppose f :,, is s-onvex funtion in the seond sense on the o-ordintes on Then one hs the ineulities 7 4 s 4 s f b, d s b s d f x, d b d dx b f, y dy fx, ydydx

5 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 9 s b s s f x, f x, d dx d f, fb, f, d fb, d f, y f b, y dy In reent yers, ny uthors hve proved severl ineulities for o-ordinted onvex funtions These studies inlude, ong others, the works in, 3, 4, 5, 6,, 3, -4, 5-8 nd 3 Alori et l, 3, 4, 5, 6, proved severl Herite-Hdrd type ineulities for o-ordinted s-onvex funtions nd oordinted log-onvex funtions Drgoir, proved the Herite-Hdrd type ineulities for o-ordinted onvex funtions Hwng et l 3, lso proved soe Herite-Hdrd type ineulities for o-ordinted onvex funtion of two vribles by onsidering soe ppings diretly ssoited to the Herite-Hdrd type ineulity for o-ordinted onvex ppings of two vribles Ltif et l -4, proved soe ineulities of Herite-Hdrd type for differentible o-ordinted onvex funtions, differentible funtions whose higher order prtil derivtives re oordinted onvex, produt of two o-ordinted onvex ppings nd for o-ordinted h-onvex ppings Özdeir et l 5-8, proved Hdrd s type ineulities for o-ordinted onvex funtions, o-ordinted s-onvex funtions nd o-ordinted -onvex nd α, -onvex funtions The in i of this pper is to estblish soe new Herite-Hdrd type ineulities for differentible funtions whose prtil derivtives of higher order re o-ordinted s-onvex in the seond sense on the retngle fro the plne R whih generlize the Herite-Hdrd type ineulities proved for o-ordinted s-onvex funtions in the seond sense nd refine those results estblished for differentible funtions whose prtil derivtives of higher order re o-ordinted onvex on the retngle fro the plne R see 4 Min Results In this setion we estblish new Herite-Hdrd type ineulities for double integrls of funtions whose prtil derivtives of higher order re o-ordinted s- onvex in the seond sense To ke the presenttion esier nd opt to understnd, we ke soe syboli representtions s follows A b l f x, f x, d dx d l d l l f, l f b, l! y l y l n k b k k! k f, y f b, y dy k f, k f, d x k x k

6 3 M A LATIF l d l b l! l k b k n d k k! l f x, dx y l k f, y dy x k n k l b k d l 4 k! l! k l kl f, x k y l, nd B n, n f, t n r, Cn, n f,d t n r, Dn, E n, n fb,d t n r, Fn, n f b, d, G n, H n, n f b,, J n, n f b,d, I n, n fb,, n f, d, n fb, d, where the sus bove tke, when n nd n nd hene A A b f x, f x, d dx f, y f b, y dy b d In wht follows is the interior of, b, d nd L is the spe of integrble funtions over The following two results will be very useful in the seuel of the pper Theore 8 Let f : R be ontinuous pping suh tht the prtil derivtives kl f,, k,,, n, l,,, exist on nd re x k y l ontinuous on, then n k f t, r dr n l n Y l y X k x Y l y kl f x, y x k y l l K n x, t nl f t, y, t n y l where, for x, y, we hve { t n, t, x K n x, t : n! t b n, t x, b n! S y, r : { r!, r, y r d!, r y, d K n x, t S y, r n f t, r t n r dr n X k x S y, r k f x, r x k r dr nd k X k x : b xk k x k k! Y l y : d yl l y l l!

7 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 3 Le 4 Let f : R, be ontinuous pping suh tht on nd n f x n y b n d 4n!! L for, n, then t n r n t r n f t t b, r r d dr A f, f, d f b, f b, d 4 b d Now we prove our in results n f x n y exists f x, y dydx Theore Let f :,,, < b, < d, be ontinuous pping suh tht n f exists on nd n f L If t n r n f t n r is s-onvex on the o-ordintes on in the seond sense, for, n N,, n, then we hve the following ineulity f, f, d f b, f b, d b f x, y dydx A 4 b d b n d LBn, MC n, ND n, RE n,, 4n!! where s, s, with s s s, n n s n s L, n s n s s s n n s n M B, s B, s, n s n s s N nb n, s B n, s, s s R nb n, s B n, s B, s B, s, nd Bx, y tx t y is the Euler Bet funtion Proof Suppose, n By Le, we hve f, f, d f b, f b, d 4 b d 3 b n d t n r n t r 4n!! n f t t b, r r d dr f x, y dydx A

8 3 M A LATIF By s-onvexity of n f t n s on the o-ordintes on, we get tht t n r n t r n f t t b, r r d dr 4 B n, t ns r s n t r dr Sine 5 C n, E n, D n, Anlogously, 6 7 nd 8 t ns r r s n t r dr t n t s n t r r s r dr t n r s t s n t r dr t ns r s n t r dr t ns n t n n s n n s n s r s r dr s s s t ns r r s n t r dr n n s n n s n s B, s B, s, t n r s t s n t r dr s s s nb n, s B n, s t n t s n t r r s r dr nb n, s B n, s B, s B, s Fro 4-8 in 3, we get the reuired ineulity This opletes the proof of the theore

9 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 33 Theore 3 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exists on nd n f L If t n r n f t n r,, is s-onvex on the o-ordintes on,, n N,, n, then f, f, d f b, f b, d 4 b d 9 b n d / n 4n!! n LB n, MD n, NC n, RE n,, f x, y dydx A where s, s, with s s s nd L, M, N, R nd Bx, y re s defined in Theore Proof The se is the Theore Suppose >, then by Le nd the power en ineulity, we hve f, f, d f b, f b, d b f x, y dydx A 4 b d b n d { } / t n r n t r dr 4n!! { t n r n t r n f t t b, r r d dr} / By the siilr rguents used to obtin nd the ft t n r n t r dr n n, we get 9 This opletes the proof of the theore Theore 4 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f t n s,, is s-onvex on the o-ordintes on, s, s, with s s s,, n N,, n

10 34 M A LATIF Then n k l b d n d! k l k n b n! l b k d l kl f b kl k! l! x k y l f t, r dr k b k k k! l d l l l! 4F n, n s s, d Qr k f b, r dr x k r P t nl f t, d t n y l 4 n b d 4n!! n B n, C n, D n, E n, B n, s B, s G n, I n, B n, s H n, J n, s n s, B, s where { t n, t {, P t : b t b n, t r, r, b, b nd Qr : d r d, r d, d Proof By letting x b nd y d in Theore nd using the properties of the bsolute vlue, we obtin n k l b k d l kl f b, d kl k! l! x k y l k l b d n d! k f t, r dr k b k k k! Qr k f b, r dr x k r

11 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 35 n l d l b n! l l! l b d!n! P t Qr dr P t Qr P t nl f t, d t n y l n f t, r By the power en ineulity for double integrls, we hve P t Qr n f t, r dr 3 b P t Qr dr P t Qr d b b d d b t n r n f t, r t n d r n f t, r b t n d r n f t, r dr n f t, r dr t n r n f t, r dr dr dr dr b Now we lulte eh integrl in 3 Sine t t t b b d r r r d d By the o-ordinted s-onvexity of n f d t n s, we hve t n r n f t, r s s 4 dr b d s s B b d n, t n r t r dr G n, H n, F n, s b t n t r s dr s t s n d r r dr t s n r s dr b nd

12 36 M A LATIF Now by the hnge of vribles u t, v r nd then by the hnge of vribles x u, y v, we get tht b d 5 s s s s b d t n r t r dr b d s s s s b u n d u du v v dv b d u n u s du v v s dv b d n d x n x s dx y y s dy n b d B n, s B, s Siilrly, 6 7 nd 8 s s b d b n d B n, s, s t n b t s r s dr s s s t s n d r r dr b d b n d B, s n s s s b d b n d n s s Using 5-8 in 4, we obtin 9 t n r n f t, r t s n r s dr n b d dr B n, B n, s B, s G n, B n, s s

13 Anlogously, INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 37 b H n, B, s n s b t n r n f t, r H n, B, s n s F n, n s s n b d dr D n, B n, s B, s F n, n s s I n, B n, s, s d t n d r n f t, r d dr n b d G n, B n, s s C n, B n, s B, s J n, B, s n s F n, n s s nd b d b t n d r n f t, r dr n b d F n, n s s I n, B n, s s J n, B, s n s It is not diffiult to observe tht 3 P t Qr dr E n, B n, s B, s 4 n n b d Fro -3, we get the desired ineulity The proof of the Theore for is the se This opletes the proof Soe results n be dedued fro the ineulities 9 nd s follows Letting s s in Theore 3 gives the following orollry

14 38 M A LATIF Corollry Let f :,,,, < b, < d, be ontinuous pping suh tht n f exists on nd n f L If t n r n f t n r,, is onvex on the o-ordintes on,, n N,, n, then f, f, d f b, f b, d 4 b d 4 f x, y dydx A b n d n / / 4 n!! n / / n B n, n C n, n D n, ne n, Corollry Under the ssuptions of Corollry with n, we hve f, f, d f b, f b, d 4 b d b d 4 f, 9 4 t r 4 f b, t r 4 f, d t r f x, y dydx A 4 f b, d t r The following orollry is speil se of Theore 4 for s s Corollry 3 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f t n s,, is onvex on the o-ordintes on,, n N,, n Then 5 n k l b d n d! k l k n b n! l b k d l kl f b kl k! l! x k y l f t, r dr k b k k k! l d l l l!, d Qr k f b, r dr x k r P t nl f t, d t n y l

15 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 39 b n d B n, C n, D n, E n, n n!! n G n, I n, n H n, J n, n n 4 n F n,, n where P t nd Qr re s defined in Theore 4 The following orollry is speil se of Theore 4 for s s nd n, whih gives tighter estite thn those fro 3, Theore 4, pge 8 Corollry 4 Under the ssuptions of Corollry 3 with n, we hve b b f t, r dr f b d, d d b f d, r b dr f t, d b b d B, C, D, E, G, I, 4 H, J, 8F,, where P t nd Qr re s defined in Theore 4 It is esy to see tht, when n f t n s,, is onvex on the o-ordintes on,, n N,, n, then G n, I n, B n, C n, D n, E n,, H n, J n, B n, C n, D n, E n, nd 4F n, B n, C n, D n, E n, Substituting these ineulities in Corollry 3, we get the following orollry whih is 4, Theore 3, pge Corollry 5 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f t n s,, is

16 4 M A LATIF onvex on the o-ordintes on,, n N,, n Then 7 n k l b d n d! k l b k d l kl f b kl k! l! x k y l f t, r dr k b k, d Qr k f b, r dr k k! k x k r n l d l P t nl f t, d b n! l l! l t n y l b n d B n n, C n, D n, E n,, n!! where P t nd Qr re s defined in Theore 4 A different pproh leds us to the following result Theore 5 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f t n s,, is s-onvex on the o-ordintes on, s, s, with s s s,, n N,, n Then 8 n k l b d n d! k l k n b n! l 4n!! b k d l kl f b kl k! l! x k y l f t, r dr k b k k k! l d l n l l! b, d Qr k f b, r dr x k r P t nl f t, d t n y l n d

17 { INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 4 B n, B n, s B, s G n, B n, s s F n, n s s H n, B, s n s H n, B, s D n, n s B n, s B, s F n, n s s I n, B n, s s G n, B n, s C n, s B n, s B, s J n, B, s F n, n s n s s F n, n s s I n, B n, s s E n, B n, s B, s }, J n, B, s n s where P t nd Qr re s defined in Theore 4 Proof By letting x b nd y d in Theore, using the properties of the bsolute vlue, we obtin n k l b d n d! k l k n b n! l b k d l kl f b kl k! l! x k y l f t, r dr k b k k k! l d l l l!, d Qr k f b, r dr x k r P t nl f t, d t n y l

18 4 M A LATIF 9 b d!n! b b d d b t n r n f t, r t n d r n f t, r b t n d r n f t, r t n r n f t, r dr dr dr dr Using the power-en ineulity for eh integrl on the right-side of 9 nd by the siilr rguents s in proving Theore 4, we get 8 Corollry 6 If the onditions of Theore 5 re stisfied nd if n nd s s, then we hve the ineulity b f t, r dr b d d b f d, r dr b b d 4 8 H, 36 D, 9 F, 8 I, 8 G, 36 C, 8 J, 9 F, 9 F, 8 I, 8 J, 36 E, b, d f t, d { 36 B, 8 G, 8 H, 9 F, } If we use the Hölder s ineulity insted of the power-en ineulity we get the following result Theore 6 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f p t n s, p >, is s-onvex on the o-ordintes on, s, s, with s s s,, n N,, n

19 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 43 Then for P t nd Qr defined s in Theore 4 nd p we hve 3 b f t, r dr b d n k l k l n d! k n b n! l b n d b k d l kl f b kl k! l! x k y l k b k k k! l d l l l! n n!! np p p B n, C n, D n, E n,, d Qr k f b, r dr x k r P t nl f t, d t n y l s s Proof The ineulity 3 follows fro the Hölder s ineulity nd 7 Corollry 7 Under the ssuptions of Theore 6, if n nd s s, then for we hve the ineulity p b f t, r dr f b d d b f d, r dr b d f, p p t r b, d b f t, d b f b, t r f, d t r Our lst result is for the s-onve funtions n be stted s follows f b, d t r Theore 7 Let f :,,,, < b, < d, be ontinuous pping suh tht n f exist on nd n f L If t n r n f p t n s, p >, is s- onve on the o-ordintes on, s, s, with s s s,, n N,, n Then for P t nd Qr defined s in Theore 4 nd we hve p

20 44 M A LATIF 3 b f t, r dr b d n k l k l n d! b k d l kl f b kl k! l! x k y l k b k, d Qr k f b, r dr k k! k x k r n l d l P t nl f t, d b n! l l! l t n y l b n d 4 s 4 s n f b, d n n!! np p p Proof The ineulity 3 follows fro the Hölder s ineulity nd the ineulity 7 with ineulities in reversed diretion Corollry 8 If the onditions of Theore 7 re stisfied nd if n nd s s, then for we hve the ineulity p b b f t, r dr f b d, d d b f d, r b dr f t, d b b d f b 4, d p p t r Referenes M Alori nd M Drus, Co-ordinted s-onvex Funtion in the First Sense with Soe Hdrd-type Ineulities, Int J Contep Mth Sienes, 3 3 8, M Alori, M Drus nd S S Drgoir, Ineulities Of Herite-hdrd s Type for Funtions Whose Derivtives Absolute Vlues Are Qusi-onvex, Rgi Reserh Report Colletion, Suppl M Alori nd M Drus, The Hdrd s Ineulity For s-onvex Funtion of -vribles on the Co-ordintes, Int Journl Mth Anlysis, 3 8, M Alori nd M Drus, The Hdrd s Ineulity For s-onvex Funtions, Int Journl Mth Anlysis, 3 8, M Alori nd M Drus, Hdrd-type Ineulities For s-onvex Funtions, Interntionl Mthetil Foru, 3 8, No 4, M Alori nd M Drus, on the Hdrd s Ineulity For Log-onvex Funtions on the Coordintes, J Of Ineul And Appl, Artile Id 8347, 9, 3 Pges

21 INEQUALITIES OF HERMITE-HADAMARD TYPE FOR DOUBLE INTEGRALS 45 7 M Avi, H Kvui nd M E Özdeir, New Ineulities of Herite hdrd Type vi s-onvex Funtions In The Seond Sense with Applitions, Applied Mthetis nd Coputtion, 7, S S Drgoir nd R P Agrwl, Two Ineulities For Differentible Mppings nd Applitions to Speil Mens of Rel Nubers And To Trpezoidl Forul, Appl Mth Lett 5 998, S S Drgoir, nd S Fitzptrik, the Hdrd s Ineulity for s-onvex Funtions in the Seond Sense, Deonstrtio Mth, , S S Drgoir, on Hdrd s Ineulity for Convex Funtions on the Co-ordintes in Retngle fro the Plne, Tiwnese Journl of Mthetis, 4, S S Drgoir nd C E M Pere, Seleted Topis On Herite-hdrd Ineulities nd Applitions, Rgi Monogrphs, Vitori University, Online: hdrdhtl W-D Jing, D-E Niu, Y Hu nd F Qi, Generliztions of Herite-Hdrd Ineulity to n-tie Differentible Funtions Whih Are s-onvex in the Seond Sense, Anlysis Munih 3, ; Avilble Online t 3 D Y Hwng, K L Tseng nd G S Yng, Soe Hdrd s Ineulities for Co-ordinted Convex Funtions in Retngle fro the Plne, Tiwnese Journl of Mthetis, 7, D Y Hwng, Soe Ineulities for n-ties Differentible Mppings nd Applitions, Kyungpook, Mth J 43 3, H Hudzik nd L Mligrnd, Soe Rerks on s-onvex Funtions, Aeutiones Mth, , 6 S-hong, B-y Xi nd F Qi, Soe New Ineulities Of Herite-Hdrd Type for n-ties Differentible Funtions Whih Are -onvex, Anlysis Munih 3, No 3, 47 6; Avilble Online t 7 G Hnn, Cubture Rule fro Generlized Tylor Perspetive, Phd Thesis 8 G Hnn, S S Drgoir nd P Cerone, Generl Ostrowski Type Ineulity for Double Integrls, Tkng J Mth Volue 33, Issue 4, 9 U S Kiri, Ineulities for Differentible Mppings And Applitions to Speil Mens of Rel Nubers to Midpoint Forul, Appl Mth Coput 47 4, U S Kiri, M K Bkul, M E Özdeir nd J PečArić, Hdrd-type Ineulities for s-onvex Funtions, Appl Mth And Copt, 93 7, 6 35 M A Ltif nd M Alori, Hdrd-type Ineulities For Produt Two Convex Funtions on the Co-ordinetes, Int Mth Foru, 447 9, M A Ltifnd nd M Alori On the Hdrd-type Ineulities for h-onvex Funtions on the Co-ordinetes, Int J Mth Anlysis, , M A Ltif nd S S Drgoir, on Soe New Ineulities For Differentible Co-ordinted Convex Funtions, Journl of Ineulities And Applitions, :8 Doi:86/9-4x--8 4 M A Ltif nd S Hussin, Soe New Herite-hdrd Type Ineulities for Funtions Whose Higher Order Prtil Derivtives Are Co-ordinted Convex Ft Universittis Niš Ser Mth Infor 73, M E Özdeir, E Set nd M Z Sriky, New Soe Hdrd s Type Ineulities for Coordinted -onvex nd α, -onvex Funtions, Rgi, Res Rep Coll 3, Suppleent, Artile 4 6 M E Özdeir, H Kvuri, A O Akdeir nd M Avi, Ineulities for Convex nd s-onvex Funtions On δ, b, d, Journl Of Ineulities nd Applitions :, doi:86/9-4x--

22 46 M A LATIF 7 M E Özdeir, M A Ltif nd A O Akdeir, On Soe Hdrd-type Ineulities for Produt of Two s-onvex Funtions on the Co-ordintes, Journl of Ineulities And Applitions, :, doi:86/9-4x-- 8 M E Özdeir, A O Akdeir nd M Tun, on the Hdrd-type Ineulities for Coordinted Convex Funtions, Arxiv:3437v 9 C M E Pere nd J E Pečrić, Ineulities For Differentible Mppings with Applitions to Speil Mens nd Qudrture Forul, Appl Mth Lett 3, J E Pečrić, F Proshn nd Y L Tong, Convex Funtions, Prtil Ordering nd Sttistil Applitions, Adei Press, New York, 99 3 M Z Sriky, E Set, M E Özdeir nd S S Drgoir, New Soe Hdrd s Type Ineulities for Co-ordinted Convex Funtions, Arxiv:57v MthC 3 B-Y Xi nd F Qi, Soe Herite-Hdrd Type Ineulities For Differentible Convex Funtions nd Applitions, Het J Mth Stt, 4: in Press, 3 33 B-Y Xi nd F Qi, Soe Herite-Hdrd Type Ineulities For Differentible Convex Funtions with Applitions to Mens, J Funt Spes Appl, : in Press, ; Avilble Online t Shool of Coputtionl nd Applied Mthetis, University of the Witwtersrnd, Privte Bg 3, Wits 5, Johnnesburg, South Afri E-il ddress: er ltif@hotilo

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