AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous nd discrete versions. We lso pply our result to the quntum clculus cse.. introduction In 997, Drgomir nd Wng [7] proved the following Ostrowski-Grüss type integrl inequlity. Theorem. Let I R e n open intervl,, I, <. If f : I R is differentile function such tht there exist constnts γ, Γ R, with γ f (x) Γ, x [, ]. Then we hve f(x) for ll x [, ]. f(t)dt f() f() ( x + ) ( )(Γ γ), () This inequlity is connection etween the Ostrowski inequlity [] nd the Grüss inequlity [3]. It cn e pplied to ound some specil men nd some numericl qudrture rules. For other relted results on the similr integrl inequlities plese see the ppers [6, 0,, ] nd the references therein. The im of this pper is to extend generliztions of Ostrowski-Grüss type integrl inequlity to n ritrry time scle.. Time scles essentils The development of the theory of time scles ws initited y Hilger [8] in 988 s theory cple to contin oth difference nd differentil clculus in consistent wy. Since then, mny uthors hve studied the theory of certin integrl inequlities on time scles. For exmple, we refer the reder to [,, 5, 5, 6]. Now we riefly introduce the time scles theory nd refer the reder to Hilger [8] nd the ooks [, 3, 9] for further detils. Definition. A time scle T is n ritrry nonempty closed suset of rel numers. Dte: April, 008. 000 Mthemtics Suject Clssifiction. 6D5; 39A0; 39A; 39A3. Key words nd phrses. Ostrowski inequlity, Grüss inequlity, Ostrowski-Grüss inequlity, time scles. This pper ws typeset using AMS-LATEX.
W. LIU AND Q. A. NGÔ Definition. For t T, we define the forwrd jump opertor σ : T T y σ(t) = inf {s T : s > t}, while the ckwrd jump opertor ρ : T T is defined y ρ(t) = sup {s T : s < t}. If σ(t) > t, then we sy tht t is right-scttered, while if ρ(t) < t then we sy tht t is left-scttered. Points tht re right-scttered nd left-scttered t the sme time re clled isolted. If σ(t) = t, the t is clled right-dense, nd if ρ(t) = t then t is clled left-dense. Points tht re right-dense nd left-dense t the sme time re clled dense. Definition 3. Let t T, then two mppings µ, ν : T [0, + ) stisfying re clled the grininess functions. µ (t) := σ(t) t, ν (t) := t ρ(t) We now introduce the set T κ which is derived from the time scles T s follows. If T hs left-scttered mximum t, then T κ := T {t}, otherwise T κ := T. Furthermore for function f : T R, we define the function f σ : T R y f σ (t) = f(σ(t)) for ll t T. Definition. Let f : T R e function on time scles. Then for t T κ, we define f (t) to e the numer, if one exists, such tht for ll ε > 0 there is neighorhood U of t such tht for ll s U f σ (t) f (s) f (t) (σ(t) s) ε σ(t) s. We sy tht f is -differentile on T κ provided f (t) exists for ll t T κ. Definition 5. A mpping f : T R is clled rd-continuous (denoted y C rd ) provided if it stisfies () f is continuous t ech right-dense point or mximl element of T. () The left-sided limit lim f (s) = f (t ) exists t ech left-dense point t of s t T. Remrk. It follows from Theorem.7 of Bohner nd Peterson [] tht every rd-continuous function hs n nti-derivtive. Definition 6. A function F : T R is clled -ntiderivtive of f : T R provided F (t) = f(t) holds for ll t T κ. Then the -integrl of f is defined y f (t) t = F () F (). Proposition. Let f, g e rd-continuous,,, c T nd α, β R. Then () () (3) () [αf(t) + βg(t)] t = α f(t) t = f(t) t, f(t) t = c f(t) t + c f(t) t + β f(t) t, f(t)g (t) t = (fg)() (fg)() g(t) t, f (t)g(σ(t)) t,
(5) AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES 3 f(t) t = 0. Definition 7. Let h k : T R, k N 0 e defined y nd then resursively y h k+ (t, s) = h 0 (t, s) = for ll s, t T s h k (τ, s) τ for ll s, t T. The present pper is motivted y the following results: Grüss inequlity on time scles nd Ostrowski inequlity on time scles due to Bohner nd Mtthews. More precisely, the following so-clled Grüss inequlity on time scles ws estlished in []. Theorem (See [], Theorem 3.). Let,, s T, f, g C rd nd f, g : [, ] R. Then for m f(s) M, m g(s) M, () we hve f σ (s)g σ (s) s f σ (s) s g σ (s) s (M m )(M m ). (3) The sme uthors lso proved the following so-clled Ostrowski inequlity on time scles in [5]. Theorem 3 (See [5], Theorem 3.5). Let,, s, t T, < nd f : [, ] R e differentile. Then f(t) f σ (s) s M (h (t, ) + h (t, )), () where M = sup <t< f (t). This inequlity is shrp in the sense tht the righthnd side of (3) cnnot e replced y smller one. In the present pper we shll first derive new inequlity of Ostrowski-Grüss type on time scles y using Theorem nd then unify corresponding continuous nd discrete versions. We lso pply our result to the quntum clculus cse. 3. The Ostrowski-Grüss type inequlity on time scles Similrly s in [7], the Ostrowski-Grüss type inequlity cn e shown for generl time scles. Theorem. Let,, s, t T, < nd f : [, ] R e differentile. If f is rd-continuous nd γ f (t) Γ, t [, ].
W. LIU AND Q. A. NGÔ Then we hve f(t) for ll t [, ]. f σ f() f() (s) s ( ) (h (t, ) h (t, )) ( )(Γ γ), (5) To prove Theorem, we need the following Montgomery Identity. Lemm (Montgomery Identity, see [5]). Let,, s, t T, < nd f : [, ] R e differentile. Then where f(t) = f σ (s) s + p(t, s)f (s) s, (6) p(t, s) = { s, s < t, s, t s. Proof of Theorem. By pplying Lemm, we get f(t) f σ (s) s = p(t, s)f (s) s, (7) for ll t [, ]. It is cler tht for ll t [, ] nd s [, ] we hve t p(t, s) t. Applying Theorem to the mpping p(t, ) nd f ( ), we get p(t, s)f (s) s p(t, s) s f (s) s [(t ) (t )](Γ γ) ( )(Γ γ). (8) By simple clcultion we get p(t, s) s = = (s ) s + (s ) s t = h (t, ) h (t, ) (s ) s (s ) s nd f f() f() (s) s =. By comining (7), (8) nd the ove two equlities, we otin (5).
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES 5 If we pply the Ostrowski-Grüss type inequlity to different time scles, we will get some well-known nd some new results. Corollry. (Continuous cse). Let T = R. Then our delt integrl is the usul Riemnn integrl from clculus. Hence, (t s) h (t, s) =, for ll t, s R. This leds us to stte the following inequlity f(t) ( f() f() f(s)ds t + ) ( )(Γ γ), (9) for ll t [, ], where γ f (t) Γ, which is exctly the Ostrowski-Grüss type inequlity shown in Theorem. Corollry. (Discrete cse). Let T = Z, = 0, = n, s = j, t = i nd f(k) = x k. With these, it is known tht ( ) t s h k (t, s) =, for ll t, s Z. k Therefore, ( ) t h (t, 0) = = Thus, we hve x i n t (t ) n j= x j x n n for ll i =, n, where γ x i Γ., h (t, n) = ( t n ) = (t n) (t n ). ( i n + ) n(γ γ), (0) Corollry 3. (Quntum clculus cse). Let T = q N0, q >, = q m, = q n with m < n. In this sitution, one hs Therefore, h k (t, s) = k ν=0 t q ν s ν, for ll t, s T. q µ µ=0 h (t, q m ) = (t qm ) ( t q m+) + q Then f(t) q n q m qn q m f σ (s) s, h (t, q n ) = (t qn ) ( t q n+). + q f(q n ) f(q m ) q n q m (qn q m )(Γ γ), (t qn+ q m+ q + ) () where γ f(qt) f(t) (q )(t) Γ, t [, ].
6 W. LIU AND Q. A. NGÔ If f is ounded on [, ] then we hve the following corollry. Corollry. With the ssumptions in Theorem, if f (t) M for ll t [, ] nd some positive constnt M, then we hve f(t) f σ f() f() (s) s ( ) (h (t, ) h (t, )) ( )M, () for ll t [, ]. Furthermore, choosing t = ( + )/ nd t =, respectively, in (5), we hve the following corollry. Corollry 5. With the ssumptions in Theorem, we hve ( ) [ ( ) ( )] + f f() f() + + ( ) h, h, ) (if ( )(Γ γ) + T nd f() f() f() ( ) h (, ) Acknowledgements f σ (s) s (3) f σ (s) s ( )(Γ γ). () This work ws supported y the Science Reserch Foundtion of Nnjing University of Informtion Science nd Technology nd the Nturl Science Foundtion of Jingsu Province Eduction Deprtment under Grnt No.07KJD5033. References [] R. Agrwl, M. Bohner nd A. Peterson, Inequlities on time scles: A survey, Mth. Inequl. Appl., () (00), 535-557. [] M. Bohner nd A. Peterson, Dynmic Equtions on Time Series, Birkhäuser, Boston, 00. [3] M. Bohner nd A. Peterson, Advnces in Dynmic Equtions on Time Series, Birkhäuser, Boston, 003. [] M. Bohner nd T. Mtthew, The Grüss inequlity on time scles, Communictions in Mthemticl Anlysis, 3 () (007), -8. [5] M. Bohner nd T. Mtthew, Ostrowski inequlities on time scles, J. Inequl. Pure Appl. Mth., 9 () (008), Art. 6, 8 pp. [6] X. L. Cheng, Improvement of some Ostrowski-Grüss type inequlities, Computers Mth. Applic, (00), 09-. [7] S. S. Drgomir nd S. Wng, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error ounds for some specil mens nd for some numericl qudrture rules, Computers Mth. Applic., 33(997), 5-0. [8] S. Hilger, Ein Mβkettenklkül mit Anwendung uf Zentrmsmnnigfltingkeiten, PhD thesis, Univrsi. Würzurg, 988. [9] V. Lkshmiknthm, S. Sivsundrm, nd B. Kymkcln, Dynmic Systems on Mesure Chins, Kluwer Acdemic Pulishers, 996. [0] W. J. Liu, Q. L. Xue nd S. F. Wng, Severl new pertured Ostrowski-like type inequlities, J. Inequl. Pure Appl. Mth., 8() (007), Art.0, 6 pp. [] W. J. Liu, C. C. Li nd Y. M. Ho, Further generliztion of some doule integrl inequlities nd pplictions, Act. Mth. Univ. Comenine, 77 ()(008), 7-5.
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES 7 [] D. S. Mitrinović, J. Pecrić nd A. M. Fink, Inequlities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic, Dordrecht, (99). [3] D. S. Mitrinović, J. Pecrić nd A. M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic, Dordrecht, (993). [] M. Mtić, J. Pecrić nd N. Ujević, Improvement nd further generliztion of inequlities of Ostrowski-Grüss type, Computers Mth. Applic, 39 (3/)(000), 6-75. [5] H. Romn, A time scles version of Wirtinger-type inequlity nd pplictions, Dynmic equtions on time scles, J. Comput. Appl. Mth., (/) (00), 9-6. [6] F.-H. Wong, S.-L. Yu, C.-C. Yeh, Andersons inequlity on time scles, Applied Mthemtics Letters, 9 (007), 93-935. (W. Liu) College of Mthemtics nd Physics, Nnjing University of Informtion Science nd Technology, Nnjing 00, Chin E-mil ddress: wjliu@nuist.edu.cn (Q. A. Ngô) Deprtment of Mthemtics, Mechnics nd Informtics, College of Science, Việt Nm Ntionl University, Hà Nội, Việt Nm E-mil ddress: ookworm vn@yhoo.com