GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE

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1 Journl of Alied Mthemtics nd Comuttionl Mechnics 6, 5(4), - wwwmcmczl -ISSN DOI: 75/jmcm64 e-issn GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE Hüseyin Budk, Mehmet Zeki Sriky 3, Erhn Set, Dertment of Mthemtics, Fculty of Science nd Arts, Düzce University, Düzce-Turkey 3 Dertment of Mthemtics, Fculty of Arts nd Sciences, Ordu University, 5, Ordu, Turkey hsynudk@gmilcom, srikymz@gmilcom, erhnset@yhoocom Received: 4 Mrch 6; cceted: 5 Setemer 6 Astrct In this er, we estlish some generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives re generlized s-convex in the second sense Keywords: generlized Hermite-Hdmrd ineulity, generlized Hölder ineulity, generlized convex functions Introduction In 938, Ostrowski estlished the following interesting integrl ineulity for differentile mings with ounded derivtives []: Theorem (Ostrowski ineulity) Let f : [, ] R e differentile, :,,, ie ming on ( ) whose derivtive f ( ) R is ounded on : = su < t (, ) f f t Then, we hve the ineulity for ll x [, ] ( x ) ( ) f ( x) ( ), f t f 4 The constnt is the est ossile 4 In recent yers, the frctl theory hs received significnt ttention The clculus on the frctl set cn led to etter comrehension for the vrious rel world models from science nd engineering [-9] The urose of this er is to estlish some locl frctionl integrl ineulities using generlized s-convex in the second sense on rel liner frctl set R (< < ) This er is divided into the following three sections In Section, we give the definitions of the locl frctionl derivtives nd locl frctionl ()

2 H Budk, MZ Sriky, E Set integrls nd introduce severl useful nottions on frctl sce which will e used our min results In Section 3, the min results re resented Preliminries Recll the set R of rel line numers nd use the Go-Yng-Kng's ide to descrie the definition of the locl frctionl derivtive nd locl frctionl integrl, see [4, 5] nd so on Recently, the theory of Yng s frctionl sets [yng] ws introduced s follows For <, we hve the following -tye set of element sets: Z : The -tye set of integer is defined s the set {, ±, ±,, ±n,} Q : The -tye set of the rtionl numers is defined s the set { = ( m ) :, Z, } J : ( ) { m : R : The -tye set of the irrtionl numers is defined s the set R = Q J, Z, } The -tye set of the rel line numers is defined s the set If, nd c elongs the set R of rel line numers, then () nd elongs the set ; R () = = ( ) = ( ) ; (3) ( c ) = ( ) c ; (4) = = ( ) = ( ) ; (5) ( c ) = ( ) c ; (6) ( c ) = c ; (7) = = nd = = The definition of the locl frctionl derivtive nd locl frctionl integrl cn e given s follows Definition [4] A non-differentile function f : R R, x f( x ) is clled to e locl frctionl continuous t x, if for ny ε >, there exists δ >, such tht f( x) f( x ) < ε holds for x x < δ, where ε, δ R If f( x ) is locl continuous on the intervl,, we denote f ( x) C (, )

3 Generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives 3 Definition [4] The locl frctionl derivtive of f ( x ) of order t x= x is defined y f ( ) where ( f x f x ) ( ) d f ( x) ( x ) = = lim, x x dx x= x x x ( f x f x ) ( )( f( x) f( x) ) Γ If there exists k times ( k ) = x x f ( x) D D f ( x) f D( k ) ( I), where k =,,, Definition 3 [4] Let f x C [ ] defined y, with = for ny x I R, then we denoted, Then the locl frctionl integrl is N I f x = = Γ ( ) f t f t j t j Γ ( ) t j = lim, tj tj t j nd = mx {,,, N }, = t < t< < t < t = is rtition of intervl [ ] nd N N t t t t where tj, tj, j=,, N, Here, it follows tht I f ( x) = if = nd I f ( x) = I f ( x) if < If for ny [ ] x,, there exists I f ( x), x then we denote y f x I [ ], Lemm [4] () (Locl frctionl integrtion is nti-differentition) Suose tht f ( x) = g ( x) C [, ], then we hve I f ( x) = g( ) g( ) () (Locl frctionl integrtion y rts) Suose tht f ( x), g( x) D [, ] ( f ) ( x), ( g ) ( x ) C [, ], Lemm [4] then we hve = I f ( x) g ( x) f ( x) g( x) I f ( x) g( x) x nd d x dx k Γ ( k) ( k ) = x ; Γ ( ) ( k ) k Γ ( k ) ( k ) ( k ) = ( ), Γ ( ) x dx Γ ( ) ( k ) k R

4 4 H Budk, MZ Sriky, E Set Lemm 3 (Generlized Hölder s ineulity) [4] Let f g C [ ] with =, then,,,, > Γ ( ) f x g x dx Γ ( ) f x dx Γ ( ) g x dx In [7], the uthors introduced two kinds of generlized s-convex functions on frctl sets R (< < ) s follows: Definition 4 Let [ ) R =, A function f : R R generlized s-convex (< s < ) in the first sense, if s s λ f( λ u λ v) λ f( u) f( v) is sid to e s s for ll u, v R nd λ, λ with λ λ = We denote this y f GK s Definition 5 A function f : R R is sid to e generlized s-convex (< s < ) in the second sense, if s s λ f( λ u λ v) λ f( u) f( v) for ll u, v R nd λ, λ with λ λ = We denote this y f GK If we hve the reverse ineulity, then f is clled s-concve Sriky nd Budk roved the following generlized Ostrowski ineulity in []: Theorem (Generlized Ostrowski ineulity) Let f : I R R [, ] f C for the identity ( I is the interior of I ) such tht, I with < Then, for ll [ ] s I R e n intervl, f D ( I ) nd x,, we hve ( ) Γ Γ x f ( x) I f ( t) ( ) f Γ 4 () In [8], Mo nd Sui estlished the following Hermite-Hdmrd ineulity for generlized s-convex functions on rel liner frctl set R (< < ) : Theorem 3 Suose tht f : R R in the second sense, where s (,) Let, [, ), < the following ineulities hold: is generlized s-convex function If f C [, ], then

5 Generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives 5 ( s ) Γ I f t s f f ( ) f ( ) Γ Γ ( s ) If f is generlized s-concve, then we hve the reverse ineulity 3 Min results We will strt with generlized identity for locl frctionl integrls: Theorem 4 Let I R e n intervl, f : I R R ( I is the interior of I ) such tht we hve the identity for ll x [ ], f D I nd f C [, ] Γ ( ) ( ) ( x ) ( ) Γ ( ) ( x) ( ) Γ ( ) for ( ) = t f tx t t f ( tx ( t) ), I with < Then, Proof Using the locl frctionl integrtion y rts (Lemm ), we hve (3) ( ) K = Γ t f tx t t f ( tx ( t) ) = ( x ) ( ) Γ ( ) x Γ f ( x) Γ ( ) ( x ) ( x ) Γ ( ) f ( x) Γ ( ) ( x ) ( x ) Γ ( ) f ( tx ( t) ) ( du) = f ( tx ( t) ) = x f ( u) (4)

6 6 H Budk, MZ Sriky, E Set Similrly, we hve K t f tx t = ( ( ) ) Γ ( ) f ( x) Γ ( ) ( ) ( ) Γ ( ) f u du x x x = (5) Using (4) nd (5), we otin which is the reuired result ( x ) Γ ( ) ( x) Γ ( ) ( ( ) ) t f tx t Γ ( ) ( ) ( ) t f tx t = x K x K = f ( x) f ( u) du Γ = f ( x) Γ I f ( u) Theorem 5 The ssumtions of Theorem 4 re stisfied If generlized s-convex in the second sense on [, ] we hve the ineulity f for some fixed s (,), then is Γ ( ) ( ) ( s) ( ( s ) ) Γ x Γ 4 ( ) f (6) for ll x [, ] where Proof By Theorem 4 nd since sense, then we hve f : = su f ( t) t (, ) f is generlized s-convex in the second

7 Generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives 7 ( x ) ( ) Γ ( ) Γ ( ) ( ) t f ( tx ( t) ) ( x) ( ) Γ ( ) t f ( tx ( t) ) ( x ) ( ) Γ ( ) s s ( ) t t f x t f ( x) ( ) Γ ( ) s s ( ) t t f x t f f ( ) ( ) ( ) s x x s Γ ( ) t t ( t) ( ) ( ) ( s) ( ( s ) ) ( ) Γ s x x = f Γ s Γ x = ( ) f Γ 4 Here, we used the fct ( ( s ) ) ( ( s ) ) ( s ) Γ = Γ t Γ nd Γ ( s) ( ( s ) ) ( ( s ) ) ( ( s ) ) Γ s ( ) = Γ t t Γ Γ This comletes the roof Remrk If we tke s = in (6), then (6) reduces to () Corollry Under ssumtion of Theorem 5 with x =, we hve the following midoint ineulity

8 8 H Budk, MZ Sriky, E Set ( ) ( ) ( s) Γ ( ( s ) ) Γ Γ f I f ( t) f Theorem 6 The ssumtions of Theorem 4 re stisfied If generlized s-convex in the second sense on [, ] we hve the ineulity f for some fixed s (,), then is Γ ( ) ( ) ( ) ( ( ) ) ( ) f ( s) ( ( s ) ) Γ Γ Γ Γ x 4 (7) for ll [, ] x where, > with = Proof Tking modulus in (3) nd using the generlized Hölder's ineulity (Lemm 3), we hve Γ ( ) ( ) ( x ) ( ) Γ ( ) ( x) ( ) Γ ( ) ( x ) ( ) ( ) t f tx t ( ) ( ( ) ) Γ t Γ ( ) f tx t ( x) ( ) t f tx t Γ t Γ f ( tx ( t) ) ( ) Since f is generlized s-convex in the second sense on [, ], then we hve

9 Generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives 9 Γ ( ) ( ) ( s) ( ( s ) ) f ( tx ( t) ) s s ( ) Γ t f x t f Γ = f ( x) f ( ) Γ Γ ( s) Γ ( ( s ) ) f, (8) nd similrly, Γ s ( ( ) ) Γ f tx t f Γ s ( ) (9) If we sustitute the ineulity (8) nd (9), then we otin the desired result Corollry Under ssumtion of Theorem 6 with x =, we hve the following midoint ineulity ( ) ( ( ) ) ( ( s ) ) Γ Γ Γ s f ( x) I f ( t) f Γ Γ Theorem 7 The ssumtions of Theorem 4 re stisfied If generlized s-concve on [, ] the ineulity f for some fixed s (,), then we hve is Γ ( ) ( ) s Γ ( ) Γ ( ) Γ ( ( ) ) ( ) x ( ) x ( x ) f ( x) f () for ll [, ] x where, > with =

10 H Budk, MZ Sriky, E Set Since Proof From Theorem 4 nd using generlized Hölder's ineulity, we hve Γ ( ) ( ) ( x ) ( ) Γ ( ) t f ( tx ( t) ) ( x) ( ) Γ ( ) t f ( tx ( t) ) ( ) Γ t Γ ( ) f tx t ( x ) ( ) ( x) ( ) ( ( ) ) Γ t Γ ( ) f tx t f is generlized s-concve on [ ],, lying Theorem 3, we hve nd I x f u ( ( ) ) = Γ ( ) f tx t x ( s ) x f Γ s x ( ( ) ) Γ f tx t f Γ () () If we sustitute the ineulity () nd (), then we otin the desired result Corollry 3 Under ssumtion of Theorem 7 with x =, we hve the following midoint ineulity Γ ( ) ( ) f I f ( t) s 4 ( ) ( ( ) ) Γ ( ) 3 ( ) 3 f f Γ Γ 4 4 where, > with =

11 Generlized Ostrowski tye ineulities for functions whose locl frctionl derivtives 4 Conclusions In this er, we resented some Ostrowski tye ineulities for function whose locl frctionl derivtives re generlized s-convex in the second sense A further study could e ssess similr ineulities y using different tyes of kernels or convexity References [] Budk H, Sriky MZ, Yildirim H, New ineulities for locl frctionl integrls, Irnin Journl of Science nd Technology (Sciences), in ress [] Chen G-S, Generliztions of Hölder's nd some relted integrl ineulities on frctl sce, Journl of Function Sces nd Alictions Volume 3, Article ID 9845 [3] Kılıçmn A, Sleh W, Notions of generlized s-convex functions on frctl sets, Journl of Ineulities nd Alictions 5, 3 DOI 86/s x [4] Mo H, Sui X, Yu D, Generlized convex functions on frctl sets nd two relted ineulities, Astrct nd Alied Anlysis 4, Article ID 63675, 7 ges [5] Mo H, Generlized Hermite-Hdmrd ineulities involving locl frctionl integrls, rxiv:46 [mthap] [6] Mo H, Sui X, Generlized s-convex function on frctl sets, rxiv:4565v [mthap] [7] Mo H, Sui X, Hermite-Hdmrd tye ineulities for generlized s-convex functions on rel liner frctl set R ( < < ), rxiv:56739v [mthca] [8] Ostrowski AM, Üer die solutweichung einer differentieren funktion von ihrem integrlmitelwert, Comment Mth Helv 938,, 6-7 [9] Sleh W, Kılıçmn A, On generlized s-convex functions on frctl set, JP Journl of Geometry nd Toology 5, 7(), 63-8 [] Sriky MZ, Budk H, Generlized Ostrowski tye ineulities for locl frctionl integrls, RGMIA Reserch Reort Collection 5, 8, Article 6, [] Sriky MZ, Erden S, Budk H, Some generlized Ostrowski tye ineulities involving locl frctionl integrls nd lictions, Advnces in Ineulities nd Alictions 6, 6 [] Sriky MZ, Erden S, Budk H, Some integrl ineulities for locl frctionl integrls, RGMIA Reserch Reort Collection 5, 8, Article 65, [3] Sriky MZ, Budk H, Erden S, On new ineulities of Simson's tye for generlized convex functions, RGMIA Reserch Reort Collection 5, 8, Article 66, 3 [4] Yng XJ, Advnced Locl Frctionl Clculus nd Its Alictions, World Science Pulisher, New York [5] Yng J, Blenu D, Yng XJ, Anlysis of frctl wve eutions y locl frctionl Fourier series method, Adv Mth Phys 3, Article ID 6339 [6] Yng XJ, Locl frctionl integrl eutions nd their lictions, Advnces in Comuter Science nd its Alictions (ACSA), (4) [7] Yng XJ, Generlized locl frctionl Tylor's formul with locl frctionl derivtive, Journl of Exert Systems, (), 6-3 [8] Yng XJ, Locl frctionl Fourier nlysis, Advnces in Mechnicl Engineering nd its Alictions, (), -6 [9] Yng XJ, Blenu D, Srivstv HM, Locl Frctionl Integrl Trnsforms nd their Alictions, Elsevier, 6