Citation Abstract and Applied Analysis, 2013, v. 2013, article no

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1 Tile An Opil-Type Inequliy in Time Scle Auhor() Cheung, WS; Li, Q Ciion Arc nd Applied Anlyi, 13, v. 13, ricle no Iued De 13 URL hp://hdl.hndle.ne/17/ Righ Thi work i licened under Creive Common Ariuion- NonCommercil-NoDerivive 4. Inernionl Licene.

2 Arc nd Applied Anlyi Volume 13, Aricle ID 53483, 5 pge hp://d.doi.org/1.1155/13/53483 Reerch Aricle An Opil-Type Inequliy on Time Scle Qio-Lun Li 1 nd Wing-Sum Cheung 1 College of Mhemic nd Informion Science, Heei Norml Univeriy, Shijizhung 54, Chin Deprmen of Mhemic, The Univeriy of Hong Kong, Hong Kong Correpondence hould e ddreed o Qio-Lun Li; qll7115@163.com Received 4 Jnury 13; Acceped 16 Jnury 13 Acdemic Edior: Alln Peeron Copyrigh 13 Q.-L. Li nd W.-S. Cheung. Thi i n open cce ricle diriued under he Creive Common Ariuion Licene, which permi unrericed ue, diriuion, nd reproducion in ny medium, provided he originl work i properly cied. We elih ome new Opil-ype inequliie involving higher order del derivive on ime cle. Thee eend ome known reul in he coninuou ce in he lierure nd provide new eime in he eing of ime cle. 1. Inroducion Opil inequliy ppered for he fir ime in 196 in [1] nd h een receiving coninul enion hroughou he yer (cf., e.g., [ 7]). The inequliy ogeher wih i numerou generlizion, eenion, nd dicreizion h een plying fundmenl role in he udy of he eience nd uniquene properie of oluion of iniil nd oundry vlue prolem for differenil equion well difference equion [8, 9]. Two ecellen urvey on hee inequliie cn e found in [1, 11]. In196,Opilelihedhefollowinginegrlinequliy. Theorem A (ee [1]). If f C 1 [, h] ifie f() = f(h) = nd f() > for ll (, h),hen h f ()f () d h 4 h f () d. (1) Shorly fer he pulicion of Opil pper, Olech provided modified verion of Theorem A. Hi reul i ed in he following. Theorem B (ee [1]). If f i oluely coninuou on [, h] wih f() =,hen h f ()f () d h h f () d. () The equliy in () hold if nd only if f() = c, wherec i conn. The fir nurl eenion of Opil inequliy (1)involving higher order derivive (n) () (n 1) i emodied in he following. Theorem C (ee [1]). Le () C (n) [, ] e uch h (i) () =, i n 1 (n 1). Thenhefollowing inequliy hold: () (n) () d 1 n (n) () d. (3) In 1997, Alzer [13] conidered Opil-ype inequliie which involve higher-order derivive of wo funcion. Thee generlize erlier reul of Agrwl nd Png [14]. In hi pper, we conider he Opil-ype inequliy which involve higher-order del derivive of wo funcion on ime cle. Our reul in pecil ce yield ome of he recen reul on Opil inequliy nd provide ome new eime on uch ype of inequliie in hi generl eing.. Min Reul Le T e ime cle; h i, T i n rirry nonempy cloed ue of rel numer. Le, T.Weuppoeh he reder i fmilir wih he ic feure of clculu on ime cle for dynmic equion. Oherwie one cn conul

3 Arc nd Applied Anlyi Bohner nd Peeron ook [15] for mo of he meril needed. We fir quoe he following elemenry lemm nd he del ime cle Tylor formul. Lemm 1 (ee [16]). Le, p 1e rel conn. Then 1/p 1 p k(1 p)/p + p 1 p k1/p (4) for ny k>. Lemm (ee [17]). Le f C m rd (T) = he e of funcion h re m ime differenile wih rd-coninuou derivive on T, m N.Thenforny, T nd [,] T, f() = m 1 k= h k (, ) f Δk () + h m 1 (, σ (τ)) f Δm (τ) Δτ, where h (, ) := 1, h n+1 (, ) := h n(τ, )Δτ, n N. Our min reul re given in he following heorem. Theorem 3. Le r <, >, >1e rel numer, nd le m, k e ineger wih k m 1.Lep>nd q e meurle funcion on Υ := [, ] T.Furher,lef, g C m 1 rd (Υ) wih () = g fδi Δi () =, i =, 1,..., m 1, nd lef Δm 1,g Δm 1 e oluely coninuou on Υ uch h he inegrl () Δ nd () Δ ei. p() fδm p() gδm Then one h q () [ fδk () r gδm () + gδk () ( h() /( ) Δ) [βk β 1 F () G () +(1 β)k β (F () +G())], where := /, β := r/, P () := h /( 1) m k 1 (, σ (τ)) p(τ)1/(1 ) Δτ, h () := q () p() P() r( 1)/, F () := p (τ) fδm (τ) Δτ, G () := p (τ) gδm (τ) Δτ. Proof. Since f Δi () =, i =, 1,..., m 1,weoinfrom Tylor heorem h for ll Υ, f() = h m 1 (, σ (τ)) f Δm (τ) Δτ, (8) (5) (6) (7) nd hence f Δk () = h m k 1 (, σ (τ)) f Δm (τ) Δτ. (9) From (9)ndHölder inequliywe ge fδk () h m k 1 (, σ (τ)) fδm (τ) Δτ = h m k 1 (, σ (τ)) p(τ) 1/ p(τ) 1/ fδm (τ) Δτ 1 1/ [ h /( 1) m k 1 (, σ (τ)) p(τ)1/(1 ) Δτ] 1/ ( p (τ) fδm (τ) Δτ) =P() 1 1/ F() 1/, (1) where P() := h/( 1) m k 1 (, σ(τ))p(τ)1/(1 ) Δτ, F() := (τ) Δτ. p(τ) fδm Le G () := p (τ) gδm (τ) Δτ. (11) Then we hve gδm () =G Δ () / p() /. (1) So (1)ogeherwih(1)implie q () fδk () r gδm () h() F() r/ G Δ () /, (13) where h() := q()p() / P() r( 1)/. Inegring oh ide of (13) overυ nd mking ue of Hölder inequliy, we oin q () fδk () r gδm () Δ h () F() r/ G Δ () / Δ 1 / ( h() /( ) Δ) Similrly, we ge q () gδk () r fδm () Δ 1 / ( h() /( ) Δ) / ( F() r/ G Δ () Δ). (14) ( G() r/ F Δ () Δ) /. (15)

4 Arc nd Applied Anlyi 3 where Recll he elemenry inequliie c (A+B) A +B d (A+B), (A, B ), (16) 1, 1, c := {, 1, d := {, 1, 1, 1. (17) Le β = r/.since = / (, 1) nd F i nondecreing, from (14) (16), we hve q () [ fδk () r gδm () + gδk () ( h() /( ) Δ) [( F Δ () G β () Δ) ( h() /( ) Δ) +( G Δ () F β () Δ) ] ( [F Δ () G β () +G Δ () F β ()]Δ) ( h() /( ) Δ) ( [F Δ () G β () +G Δ () F β (σ ())]Δ). (18) From (18) nd(19), we conclude q () [ fδk () r gδm () + gδk () ( h() /( ) Δ) [βk β 1 F () G () +(1 β)k β (F () +G())]. () The proof i complee. Theorem 4. Le r, >, <, >1e rel numer, nd le m, k e ineger wih k m 1.Lep>,ndq e meurle funcion on Υ := [, ] T.Furher,lef, g C m 1 rd (Υ) wih le () = g fδi Δi () =, i =, 1,..., m 1, nd f Δm 1, g Δm 1 e oluely coninuou on Υ uch h he inegrl () Δ nd () Δ ei. p() fδm p() gδm Then one h q () [ fδk () r gδm () + gδk () ( h() /( ) Δ) [d β Γ (G+F) Γ(G) Γ(F)], (1) where Γ(H) := Hβ ΔH, β:=r/, :=/, h() :=q()p() P() r( 1)/, P() := h/( 1) m k 1 (, σ(τ))p(τ)1/(1 ) Δτ,nd By Lemm 1,wege [F Δ () G β () +G Δ () F β (σ ())]Δ [βk β 1 G () F Δ () +(1 β)k β F Δ () d β := { 1 β, β 1, 1, β 1. Proof. Following he proof of Theorem 3,weoin () +βk β 1 F (σ ()) G Δ () +(1 β)k β G Δ ()]Δ =βk β 1 [G () F Δ () +F(σ ()) G Δ ()]Δ +(1 β)k β [ F Δ () Δ + G Δ () Δ] =βk β 1 F () G () +(1 β)k β [F () +G()]. (19) q () [ fδk () r gδm () + gδk () ( h() /( ) Δ) ( [F Δ () G β +G Δ () F β ()]Δ). (3)

5 4 Arc nd Applied Anlyi Uing (16), [F Δ () G β () +G Δ () F β ()]Δ = (G β () +F β ())(G Δ () +F Δ ())Δ (G β () G Δ () +F β () F Δ ())Δ d β (G () +F()) β Δ (G () +F()) G β () ΔG () F β () ΔF () =d β Γ (G+F) Γ(G) Γ(F). The proof i complee. Remrk 5. In he pecil ce where T reduce o Theorem 1 of [13]. (4) = R, Theorem4 Theorem 6. Le f C m 1 rd (Υ), Υ := [, ] T e uch h f Δi () =, k m 1,lef Δm 1 () e oluely coninuou on Υ,ndle () Δ <.Then fδm fδk () f Δm () Δ 1/ ( h m k 1 (, σ (τ)) ΔτΔ) 1/ ( fδm (τ) Δτ). Proof. From he hypohee, we hve (5) fδk () h m k 1 (, σ (τ)) fδm (τ) Δτ. (6) Muliplying (6) y f Δm () nd uing Cuchy-Schwrz inequliy, we oin fδk () f Δm () fδm () h m k 1 (, σ (τ)) fδm (τ) Δτ fδm () ( h 1/ m k 1 (, σ (τ)) Δτ) 1/ ( fδm (τ) Δτ). (7) Inegring oh ide over from o nd uing Cuchy-Schwrz inequliy, we oerve fδk () f Δm () Δ [ 1/ h m k 1 (, σ (τ)) ΔτΔ] 1/ [ fδm () fδm (τ) ΔτΔ] 1/ =[ h m k 1 (, σ (τ)) ΔτΔ] 1/ [ fδm (τ) Δτ]. The proof i complee. (8) Theorem 7. Le p() >, q() e nonnegive nd meurle on Υ = [,] T, ndlef C m 1 rd (Υ) e uch h f Δk () =, k m 1.Iff Δm 1 () i oluely coninuou on Υ,henforr>1, r k >,ndny r m <r, rm r k q () fδm () fδk () Δ [ q() r/(r rm) p() r m/(r r m ) (r r m )/r Q() (r 1)r m/(r r ) m Δ] Φ(y) r m/r, (9) where Q() := hr/r 1 m k 1 (, σ(τ))p(τ) (r/r 1) Δτ, Φ(y) := yr k/r m Δy(), y() := p(τ) fδm (τ) r Δτ. Proof. Following he hypohee, i i ey o ee h (6) hold. By uing Hölder inequliy wih indice r nd r/(r 1), we oin fδk () h m k 1 (, σ (τ)) p(τ) 1/r p(τ) 1/r fδm (τ) Δτ [ h r/(r 1) (r 1)/r m k 1 (, σ (τ)) p(τ) r/(r 1) Δτ] r 1/r [ p (τ) fδm (τ) Δτ] =Q() (r 1)/r y() 1/r, (3) where Q() := hr/(r 1) m k 1 (, σ(τ))p(τ) r/(r 1) Δτ, y() := p(τ) fδm (τ) r Δτ.Sowege y Δ () =p() fδm () r, (31)

6 Arc nd Applied Anlyi 5 nd hence for ny r m, fδm () Thu for r k >, r m =(p()) r m/r (y Δ ()) r m/r. (3) r m r k q () fδm () fδk () q() (p ()) r m/r (y Δ ()) r m/r Q() (r 1)r k/r y() r k/r. (33) Inegring oh ide of (33) from o nd pplying Hölder inequliy wih indice r/r m nd r/(r r m ),weoin rm r k q () fδm () fδk () Δ q () p() rm/r (y Δ ()) r m/r Q() (r 1)r /r k y() rk/r Δ (r r m )/r [ q() r/(r rm) p() r m/(r r ) m Q() (r 1)r k/(r r ) m Δ] r m /r [ y Δ () y() r k/r m Δ] =[ q() r/(r rm) p() r m/(r r m ) (r r m )/r Q() (r 1)r k/(r r ) m Δ] Φ(y) r m/r, (34) where Φ(y) := y()r k/r m Δy().Theprooficomplee. Remrk 8. In he pecil ce where T = R,Theorem6 nd 7 reduce o Theorem 1 nd of [18]. Acknowledgmen The fir uhor reerch w uppored y NSF of Chin (117154), Nurl Science Foundion of Heei Province (A1151). The econd uhor reerch w prilly upporedynhkuurggrn. [4] W. S. Cheung, Some generlized opil-ype inequliie, Journl of Mhemicl Anlyi nd Applicion, vol.16,no., pp ,1991. [5] W. S. Cheung, Opil-ype inequliie wih m funcion in n vrile, Mhemik,vol.39,no.,pp , 199. [6] C.-J. Zho nd W.-S. Cheung, On ome opil-ype inequliie, Journl of Inequliie nd Applicion,vol.11,ricle7, 11. [7]C.J.ZhondW.S.Cheung, Onopil ypeinequliie for n inegrl operor wih homogeneou kernel, Journl of Inequliie nd Applicion,vol.1,ricle13,1. [8] R. P. Agrwl nd V. Lkhmiknhm, Uniquene nd Nonuniquene Crieri for Ordinry Differenil Equion, vol. 6 of Serie in Rel Anlyi, World Scienific Pulihing, Singpore, [9] J. D. Li, Opil-ype inegrl inequliie involving everl higher order derivive, Journl of Mhemicl Anlyi nd Applicion, vol. 167, no. 1, pp , 199. [1] R. P. Agrwl nd P. Y. H. Png, Opil Inequliie wih Applicion in Differenil nd Difference Equion,vol.3of MhemicndIApplicion, Kluwer Acdemic, Dordrech, The Neherlnd, [11] D. S. Mirinović, J. E. Pečrić, nda. M. Fink, Inequliie Involving Funcion nd Their Inegrl nd Derivive,vol.53of MhemicndIApplicion(EEuropenSerie),Kluwer Acdemic, Dordrech, The Neherlnd, [1] Z. Olech, A imple proof of cerin reul of Z. Opil, Annle Polonici Mhemici,vol.8,pp.61 63,196. [13] H. Alzer, An Opil-ype inequliy involving higher-order derivive of wo funcion, Applied Mhemic Leer, vol. 1,no.4,pp.13 18,1997. [14] R. P. Agrwl nd P. Y. H. Png, Shrp Opil-ype inequliie involving higher order derivive of wo funcion, Mhemiche Nchrichen,vol.174,pp.5,1995. [15] M. Bohner nd A. Peeron, Dynmic Equion on Time cle: An Inroducion wih Applicion, Birkhäuer, Boon, M, USA, 1. [16] W. N. Li, Some new dynmic inequliie on ime cle, Journl of Mhemicl Anlyi nd Applicion,vol.319,no.,pp.8 814,6. [17] M. Bohner nd G. Sh. Gueinov, The convoluion on ime cle, Arc nd Applied Anlyi, AricleID58373,4 pge, 7. [18] C. J. Zho nd W. S. Cheung, On opil-ype inequliie wih higher order pril derivive, Applied Mhemic Leer, vol. 5, pp , 1. Reference [1] Z.Opil, Sur une inéglié, Annle Polonici Mhemici,vol. 8, pp. 9 3, 196. [] G. A. Aniou, Opil ype inequliie involving Riemnn- Liouville frcionl derivive of wo funcion wih pplicion, Mhemicl nd Compuer Modelling, vol.48,no.3-4, pp ,8. [3] W.-S. Cheung, Z. Dndn, nd J. Pečrić, Opil-ype inequliie for differenil operor, Nonliner Anlyi: Theory, Mehod & Applicion,vol.66,no.9,pp.8 39,7.

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