Research Article New General Integral Inequalities for Lipschitzian Functions via Hadamard Fractional Integrals

Transcription

1 Hindwi Pulishing orporion Inernionl Journl of Anlysis, Aricle ID 35394, 8 pges hp://d.doi.org/0.55/04/35394 Reserch Aricle New Generl Inegrl Inequliies for Lipschizin Funcions vi Hdmrd Frcionl Inegrls Em EGcn Deprmen of hemics, Fculy of Ars nd Sciences, Giresun Universiy, 800 Giresun, Turkey orrespondence should e ddressed o İm İşcn; im.iscn@giresun.edu.r Received 3 Jnury 04; Revised 8 rch 04; Acceped April 04; Pulished April 04 Acdemic Edior: Julien Slomon opyrigh 04 İmİşcn. This is n open ccess ricle disriued under he reive ommons Ariuion License, which permis unresriced use, disriuion, nd reproducion in ny medium, provided he originl work is properly cied. The uhor oins new esimes on generlizion of Hdmrd, Osrowski, nd Simpson ype inequliies for Lipschizin funcions vi Hdmrd frcionl inegrls. Some pplicions o specil mens of posiive rel numers re lso given.. Inroducion Le rel funcion f e defined on some nonempy inervl I of rel line R. Thefuncionf is sid o e conve on I if inequliy f(+( ) y) f () + ( ) f(y) () holds for ll, y I nd [0,]. Following re inequliies which re well known in he lierure s Hermie-Hdmrd inequliy, Osrowski inequliy, nd Simpson inequliy, respecively. Theorem. Le f:i R R e conve funcion defined on he inervl I of rel numers nd, I wih <.The following doule inequliy holds: f( + ) f () d f () +f(). () Theorem. Le f:i R R e mpping differenile in I,heineriorofI,ndle, I wih <.If f (), [,]; hen he following inequliy holds: f () for ll [,]. + ( ) [( ) ] (3) Theorem 3. Le f : [,] R e four imes coninuously differenile mpping on (, ) nd f (4) = sup (,) f (4) () <. Then he following inequliy holds: 3 [f () +f() +f( + )] 880 f(4) ( )4. f () d In recen yers, mny uhors hve sudied errors esimions for Hermie-Hdmrd, Osrowski, nd Simpson inequliies; for refinemens, counerprs, nd generlizion,see [ 9] nd references herein. The following definiions re well known in he lierure. Definiion 4. Afuncionf : I R R is clled n - Lipschizin funcion on he inervl I of rel numers wih 0if for ll, y I. (4) f () f(y) y (5) For some recen resuls conneced wih Hermie- Hdmrd ype inegrl inequliies for Lipschizin funcions, see [0 3].

2 Inernionl Journl of Anlysis Definiion 5 (see [4, 5]). A funcion f : I (0, ) R is sid o e GA-conve (geomeric rihmeiclly conve) if f( y )f() + ( ) f(y) (6) for ll, y I nd [0,]. We will now give definiions of he righ-sided nd lef-sided Hdmrd frcionl inegrls which re used hroughou his pper. Definiion 6. Le f L[, ]. The righ-sided nd lef-sided Hdmrd frcionl inegrls J +f nd J f of oder >0 wih > 0redefined y J + f () = J f () = Γ () Γ () (ln ), (ln ), <<, <<, respecively, where Γ() is Gmm funcion defined y Γ() = e (see [6]). 0 In [7], Iscn eslished Hermie-Hdmrd s inequliies for GA-conve funcions in Hdmrd frcionl inegrl forms s follows. Theorem 7. Le f : I (0, ) R e funcion such h f L[, ], where, I wih <.Iff is GA-conve funcion on [, ], hen he following inequliies for frcionl inegrls hold: wih >0. f( ) Γ (+) (ln (/)) {J + f () +J f ()} f () +f() In he inequliy (8), if we ke =,hen we hve he following inequliy: f( ) ln ln (7) (8) f () +f(). (9) orever, in [7], Iscn oined generlizion of Hdmrd, Osrowski, nd Simpson ype inequliies for qusigeomericlly conve funcions vi Hdmrd frcionl inegrls s reled o he inequliy (8). In his pper, he uhor oins new generl inequliies for Lipschizin funcions vi Hdmrd frcionl inegrls s reled o he inequliy (8).. in Resuls Le f : I (0, ) R e -Lipschizin funcion on I; hroughou his secion, we will ke I f (, λ,,, ) = ( λ) [ln + ln ]f() +λ[f() ln +f() ln ] Γ(+) [J f () +J + f ()], S f (,y,λ,,,)=λ f () + ( λ) f(y) Γ (+) ln (/) [J f () +J + f ()], (0) where, I wih <,, y [, ], λ [0,], = λ λ, >0nd Γ is Euler Gmm funcion. Theorem 8. Le f : I (0, ) R e -Lipschizin funcion on I nd, I wih <.Thenforll [,], λ [0,],nd > 0,wehvehefollowinginequliyfor Hdmrd frcionl inegrls: I f (, λ,,, ) {[( λ) λ] (ln ) +(λ ) (ln ) + [λ ( λ) ] (ln ) +( λ) (ln } ). () Proof. Using he hypohesis of f, we hve he following inequliy: I f (, λ,,, ) = ( λ) [ln + ln ]f() +λ[f() ln +f() ln ] [ (ln ) ( λ) f () ln +f () ln (ln ) ] (ln ) (ln )

3 Inernionl Journl of Anlysis 3 +λ f () ln +f () ln (ln ) (ln ) ( λ) [ (ln ) f () f() (ln ) f () f() ] +λ[ (ln ) f () f() (ln ) f () f() ] ( λ) [ (ln ) ( ) (ln ) ( ) ] +λ[ (ln ) ( ) (ln ) ( ) ] {[( λ) λ] (ln ) +(λ ) (ln ) + [λ ( λ) ] (ln ) + ( λ) (ln } ). orollry 9. In Theorem 8,Ifwekeλ=0,henwege (ln (/)) + (ln (/)) (ln (/)) [J f () +J + f ()] (ln (/)) f () Γ (+) (ln (/)) {[(ln ) (ln ) ] () In his inequliy, (i) if we ke =,hen f () ln ln { ln ln ln + (+ )}. (ii) If we ke =,hen f( ) Γ (+) (ln (/)) [J f () +J + f ()] (ln (/)) { (ln ) (ln ) }. (iii) If we ke = nd =,hen f( ) ln ln ln ln ( + ). orollry 0. In Theorem 8,ifwekeλ=,henwege () ln (/) +f() ln (/) [f (ln (/)) ] Γ (+) (ln (/)) (ln (/)) [J f () +J + f ()] {ln ln [ In his inequliy, if we ke =,hen f () +f() (ln (/)) Γ (+) (ln (/)) { (ln ) (4) (5) (6) (ln ) (ln ) ]}. (7) [J f () +J + f ()] +[ (ln ) (ln ) ]}. (3) [ (ln ) (ln ) ]}. (8)

4 4 Inernionl Journl of Anlysis Specilly if we ke =in his inequliy, hen we hve f () +f() ln ln ln ln { (ln ) (+ )}. (9) orollry. In Theorem 8, 3 () if we ke = nd λ=/3,hen [f () +f() Γ (+) (ln (/)) +f( )] [J f () +J + f ()] orollry. In Theorem 8,Ifweke=,hen () ln (/) +f() ln (/) ( λ) f () +λ[f ] ln ln ln ln ln ln {[( λ) λ] (ln ) + [λ ( λ) ] (ln ) + ( λ)(+ ) }. (4) 3 3(ln (/)) { (ln ) [ (ln ) (ln ) ]}. (0) Specilly, if we ke =in his inequliy, hen we hve [f () +f() +f( )] ln ln 3 (ln ln ) { (ln ) (+ )}. () () If we ke = nd λ=/,hen [f () +f() Γ (+) (ln (/)) ( ). 4 +f( )] [J f () +J + f ()] () Specilly, if we ke =in his inequliy, hen we hve [f () +f() ( ). 4 +f( )] ln ln (3) Specilly, if we ke = in his inequliy, hen we hve ( λ) f( ) + λ ( ln ln f () +f() ) ln ln λ ( ) [ ln + ( λ)(+ )]. (5) We noe h if we ke λ=0, λ=, λ=/3,ndλ=/in inequliy (5) we oin inequliies (6), (9), (), nd(3), respecively. Le f : I (0, ) R e n -Lipschizin funcion. In he ne heorem, le λ [0,], = λ λ,, y [, ] nd define,λ, >0s follows. () If y,hen,λ (, y) = λ (ln ) + y {( λ) (ln ) (ln } y ) y (ln y ) (ln ) (ln ). (6)

5 Inernionl Journl of Anlysis 5 () If y,hen () If y,hen,λ (, y) = {(ln ) λ (ln ) } + y {( λ) (ln ) (ln } y ) (ln ) (ln ) (7) (ln ) = λ (ln ) (ln ), y (ln ) (3) y (ln y ) (ln. ) = y {( λ) (ln ) (ln } y ) (3) If y,hen,λ (, y) = {(ln ) λ (ln ) } y ( λ) (ln ) (ln ) (ln ) (ln ). (8) Now, we shll give noher resul for Lipschizin funcions s follows. Theorem 3. Le, y,, λ,,,λ nd funcion f e defined s ove. Then we hve he following inequliy for Hdmrd frcionl inegrls: S f (,y,λ,,,),λ (, y) (ln (/)). (9) Proof. Using he hypohesis of f, wehvehefollowing inequliy: y () If y,hen (ln ) (ln y ) (ln. ) = {(ln ) λ (ln ) } (ln ) (ln ), y (ln ) = y {( λ) (ln ) (ln } y ) y (ln y ) (ln. ) (3) S f (,y,λ,,,) (3) If y,hen = (ln (/)) [f () f()] (ln ) (ln ) (ln (/)) [ [f (y) ] (ln ) f () f() (ln ) (30) = {(ln ) λ (ln ) } (ln ) (ln ), (33) (ln (/)) [ f (y) (ln ] ) (ln ) y (ln ) = (ln ) y ( λ) (ln. ) y (ln ] ). Now, using simple clculions, we oin he following ideniies ( /)(ln(/)) nd ( y /)(ln(/)). Using inequliy (30) nd he ove ideniies ( /)(ln(/)) nd ( y /)(ln(/)), we derive inequliy (9). This complees he proof. Under he ssumpions of Theorem 3, we hve he following corollries nd remrks.

6 6 Inernionl Journl of Anlysis orollry 4. In Theorem 3,ifweke=,heninequliy (9) reduces he following inequliy: ( λ) f () +λf(y) ln ln,λ (, y). ln (/) (34) orollry 5. In Theorem 3,leδ [/,], = δ δ,nd y= δ δ.then,wehveheinequliy (iii) If δλ,hen L (, λ, δ) = δ δ [( δ) λ ] δ δ ( λ) + (ln (/)) { (ln ) (ln ) δ δ δ δ (ln ) }. (38) s follows. λ f( δ δ )+( λ) f( δ δ ) Γ (+) (ln (/)) [J f () +J + f ()] L (, λ, δ), (i) If λ δ,hen (35) orollry 6. In Theorem 3, ifweke=y=,henwe hve he inequliy [λ + ( λ) ]f() Γ (+) (ln (/)) [J f () +J + f ()] {[λ ( λ) ]+ [ (ln (/)) (ln ) (ln ) ]}. (39) L (, λ, δ) = δ δ λ + δ δ [( λ) ( δ) ] + (ln (/)) { (ln δ δ ) δ δ (ii) If δλδ,hen L (, λ, δ) = δ δ [( δ) λ ]+ δ δ [( λ) ( δ) ]+ { (ln ) δ δ (ln ) (ln ) } (36) (ln (/)) δ δ (ln ) (ln δ δ δ δ ) (ln } ). (37) Remrk 7. In inequliy (39), if we choose λ=/, =, hen we ge inequliy (5). orollry 8. In inequliy (35), if we ke δ =,hen we hve he following weighed Hdmrd-ype inequliies for Lipschizin funcions vi Hdmrd frcionl inegrls: λ f () + ( λ) f () Γ (+) (ln (/)) [J f () +J + f ()] {( λ) λ + (ln (/)) [ (ln ) (ln ]} ). (40) Remrk 9. In inequliy (40), if we choose λ = /, =, hen we ge inequliy (8). 3. Applicion o Specil ens Leusrecllhefollowingspecilmensofwoposiive numers, wih >. () The rihmeic men: A=A(, ) := +. (4) () The geomeric men: G=G(, ) :=. (4)

7 Inernionl Journl of Anlysis 7 (3) The hrmonic men: (4) The logrihmic men: H=H(, ) := +. (43) L=L(, ) := (5) The idenric men: ln ln. (44) I=I(, ) = e ( /( ) ). (45) To prove he resuls of his secion, we need he following lemm. Lemm 0 (see []). Le f : [,] R e differenile wih f <.Thenf is n -Lipschizin funcion on [, ],where= f. Proposiion. For >>0, λ [0,]nd n,wehve ( λ) Gn (, ) +λa( n, n ) L( n, n ) n n ln ln λ ( ) [ ln +( λ)(a (, ) G(, ))]. (46) Proof. The proof follows y inequliy (5) ppliedforhe Lipschizin funcion f() = n on [, ]. Remrk. Le λ = 0 nd λ = in inequliy (46). Then, using inequliy (9), we hve he following inequliies, respecively, 0L( n, n ) G n (, ) (47) nn ln ln (A (, ) G(, )), 0A( n, n ) L( n, n ) n n ln ln [ ln (48) (A (, ) G(, ))]. Proposiion 3. For >>0nd λ [0,],wehve ( λ) G(e,e )+λa(e,e ) L (e,e )L(, ) e ( +) ( ) [λ ln ln ln +( λ) (A (, ) G(, )) ]. (49) Proof. The proof follows y inequliy (5) ppliedforhe Lipschizin funcion f() = e on [, ]. Remrk 4. Le λ = 0 nd λ = in inequliy (49). Then, using inequliy (9), we hve hefollowing inequliies, respecively, 0L(e,e )L(, ) G(e,e ) e (+) (A (, ) G(, )), ln ln 0A(e,e ) L(e,e )L(, ) e (+) ln ln [ ln (A (, ) G(, ))]. (50) Proposiion 5. For >>0, λ [0,],ndn,wehve ( λ) G (, ) +λh (, ) L(, ) G (, ) (ln ln ) λ ( ) [ ln +( λ)(a (, ) G(, ))]. (5) Proof. The proof follows y inequliy (5) ppliedforhe Lipschizin funcion f() = / on [, ]. Remrk 6. Le λ = 0 nd λ = in inequliy (5). Then, using inequliy (9), we hve he following inequliies, respecively, 0L(, ) G(, ) G (, ) (A (, ) G(, )), (ln ln ) 0G (, ) L(, ) H (, ) G (, ) H (, ) (ln ln ) [ ln (A (, ) G(, ))]. Proposiion 7. For >>0nd λ [0,],wehve ln G (, ) (ln ln ) λ ( ) [ ln +( λ) (A (, ) G(, )) ]. (5) (53) Proof. The proof follows y inequliy (5) ppliedforhe Lipschizin funcion f() = ln on [, ].

8 8 Inernionl Journl of Anlysis Proposiion 8. For >>e nd λ [0,],wehve ( λ) G (, ) ln G (, ) +λln G(, ) L (, ) ln I (, ) +ln ( ) [λ ln ln ln +( λ) (A (, ) G(, )) ]. (54) Proof. The proof follows y inequliy (5) ppliedforlipschizin funcion f() = ln on [, ]. onflic of Ineress The uhor declres h here is no conflic of ineress regrding he pulicion of his pper. [] K.-L. Tseng, S.-R. Hwng, nd K.-. Hsu, Hdmrd-ype nd Bullen-ype inequliies for Lipschizin funcions nd heir pplicions, ompuers nd hemics wih Applicions, vol.64,no.4,pp ,0. [3] G.-S. Yng nd K.-L. Tseng, Inequliies of hdmrd s ype for Lipschizin mppings, Journl of hemicl Anlysis nd Applicions,vol.60,no.,pp.30 38,00. [4]. P. Niculescu, onveiy ccording o he geomeric men, hemicl Inequliies nd Applicions,vol.3,no.,pp.55 67, 000. [5]. P. Niculescu, onveiy ccording o mens, hemicl Inequliies nd Applicions,vol.6,no.4,pp ,003. [6] A. A. Kils, H.. Srivsv, nd J. J. Trujillo, Theory nd Applicions of Frcionl Differenil Equions, ElsevierB.V., Amserdm, The Neherlnds, 006. [7] İ. İşcn, New generl inegrl inequliies for qusi-geomericlly conve funcions vi frcionl inegrls, Journl of Inequliies nd Applicions,vol.03,no.49,pp. 5,03. References []. Alomri,. Drus, S. S. Drgomir, nd P. erone, Osrowski ype inequliies for funcions whose derivives re s-conve in he second sense, Applied hemics Leers,vol. 3,no.9,pp ,00. []. Avci, H. Kvurmci, nd. E. Özdemir, New inequliies of Hermie-Hdmrd ype vi s-conve funcions in he second sense wih pplicions, Applied hemics nd ompuion,vol.7,no.,pp ,0. [3] Y. hu, G. Wng, nd X. Zhng, Schur conveiy nd hdmrd s inequliy, hemicl Inequliies nd Applicions,vol.3,no.4,pp.75 73,00. [4] İ. İşcn, A new generlizion of some inegrl inequliies for (, m)-conve funcions, hemicl Sciences, vol. 7, no., pp. 8, 03. [5] İ. İşcn, New esimes on generlizion of some inegrl inequliies for (, m)-conve funcions, Journl of onemporry Applied hemics,vol.,no.,pp.53 64,03. [6] İ. İşcn, New esimes on generlizion of some inegrl inequliies for s-conve funcions nd heir pplicions, Inernionl Journl of Pure nd Applied hemics,vol.86, no. 4, pp , 03. [7] E. Se,. E. Ozdemir, nd. Z. Srkıy, On new inequliies of Simpson s ype for qusiconve funcions wih pplicions, Tmkng Journl of hemics,vol.43,no.3,pp , 0. [8].Z.Srkıy,E.Se,nd.E.Ozdemir, Onnewinequliies of Simpson s ype for s-conve funcions, ompuers & hemics wih Applicions,vol.60,pp.9 99,00. [9] Y.-. hu, X.-. Zhng, nd X.-H. Zhng, The hermiehdmrd ype inequliy of GA-conve funcions nd is pplicion, Journl of Inequliies nd Applicions, vol.00, Aricle ID , pges, 00. [0] S. S. Drgomir, Y. J. ho, nd S. S. Kim, Inequliies of Hdmrd sypeforlipschizinmppingsnheirpplicions, Journl of hemicl Anlysis nd Applicions, vol. 45, no., pp , 000. [] S.-R. Hwng, K.-. Hsu, nd K.-L. Tseng, Hdmrd-ype inequliies for Lipschizin funcions in one nd wo vriles wih pplicions, Journl of hemicl Anlysis nd Applicions,vol.405,no.,pp ,03.

9 Advnces in Operions Reserch Hindwi Pulishing orporion hp:// Advnces in Decision Sciences Hindwi Pulishing orporion hp:// Journl of Applied hemics Alger Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp:// Journl of Proiliy nd Sisics The Scienific World Journl Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp:// Inernionl Journl of Differenil Equions Hindwi Pulishing orporion hp:// Sumi your mnuscrips hp:// Inernionl Journl of Advnces in ominorics Hindwi Pulishing orporion hp:// hemicl Physics Hindwi Pulishing orporion hp:// Journl of omple Anlysis Hindwi Pulishing orporion hp:// Inernionl Journl of hemics nd hemicl Sciences hemicl Prolems in Engineering Journl of hemics Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp:// Discree hemics Journl of Hindwi Pulishing orporion hp:// Discree Dynmics in Nure nd Sociey Journl of Funcion Spces Hindwi Pulishing orporion hp:// Asrc nd Applied Anlysis Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp:// Inernionl Journl of Journl of Sochsic Anlysis Opimizion Hindwi Pulishing orporion hp:// Hindwi Pulishing orporion hp://