Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 1938-9787 www.commun-mth-nl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKI-GRÜSS INEQUALITY QUÔ C ANH NGÔ Deprtment of Mthemtics College of Science, Việt Nm Ntionl University Hà Nội, Việt Nm. Deprtment of Mthemtics Ntionl University of Singpore Science Drive, Singpore 117543. WENJUN LIU College of Mthemtics nd Physics Nnjing University of Informtion Science nd Technology Nnjing 10044, Chin. Communicted by Mrtin Bohner) Abstrct In this pper we first derive shrp Grüss type inequlity on time scles nd then pply it to the shrp Ostrowski-Grüss inequlity on time scles which improves our recent result. AMS Subject Clssifiction: 6D15; 39A10; 39A1; 39A13. Keywords: Ostrowski inequlity, Grüss inequlity, Ostrowski-Grüss inequlity, time scles. 1 Introduction In 00, lmost t the sme time, by using similr somewht complicted methods, Cheng nd Sun in [7] s well s Mtić in [16] hve proved the following Grüss type inequlity, respectively. E-mil ddress: bookworm vn@yhoo.com E-mil ddress: wjliu@nuist.edu.cn
34 Quô c Anh Ngô nd Wenjun Liu Theorem 1.1. Let f, g : [, b] R be two integrble functions such tht for some constnts γ, Γ for ll x [,b]. Then f x)gx)dx 1 f x)dx b Γ γ γ gs) Γ 1.1) gx) dx f x) 1 b 1.) f y)dy dx. Moreover, Mtić hs proved tht there exists function g ttining the equlity in 1.), Cerone nd Drgomir in [8] hve proved tht the constnt 1 in 1.) is shrp. The result stted in Theorem 1.1 is of prticulr interest nd very useful in the cse when f x) 1 f y)dy b dx cn be evluted exctly. A gret del of shrp integrl inequlities cn be estblished by using this theorem. The development of the theory of time scles ws initited by Hilger [10] in 1988 s theory cpble to contin both difference nd differentil clculus in consistent wy. Since then, mny uthors hve studied certin integrl inequlities on time scles [1, 4, 5, 11, 17]). Recently, Liu nd Ngô [14] proved the following Ostrowski-Grüss type inequlity on time scles, which is combintion of both Grüss inequlity nd Ostrowski inequlity on time scles due to Bohner nd Mtthews [4, 5]). Theorem 1.. Let,b,s,t T, < b nd f : [,b] R be differentible. If f is rdcontinuous nd γ f t) Γ, t [,b]. Then we hve f t) 1 b f σ s) s for ll t [,b] where h i t,s) is s in Definition.1. f b) f ) b ) h t,) h t,b)) )Γ γ), 4 It is esy to see tht the foregoing inequlity is not shrp. In the first prt of this pper, we shll extend Theorem 1.1 to rbitrry time scle see Theorem 3.elow). We lso note tht, inspired by Liu in [13], our proof is very simple. As n ppliction, we shll slightly improve the bove Theorem 1. s follow by giving shrp bound. Theorem 1.3. Under the ssumptions of Theorem 1., we hve f t) 1 f σ f b) f ) s) s b b ) h t,) h t,b)) Γ γ b ) pt,x) h 1.3) t,) h t,b) b x, for ll t [,b], where pt,x) is defined s in 4.1). Moreover, the constnt 1 shrp. in 1.3) is
A Shrp Grüss Type Inequlity on Time Scles 35 In fct, we do not know the reltionship between the inequlities in Theorem 1.3 nd Theorem 1.. But we re sure tht when T R, Theorem 1.3 gives better result thn Theorem 1.. Time scles essentils Now we briefly introduce the time scles theory nd refer the reder to Hilger [10] nd the books [, 3, 1] for further detils. By time scle T we men ny closed subset of R with order nd topologicl structure present in cnonicl wy. For t T, we define the forwrd jump opertor σ : T T by σt) inf{s T : s > t}, while the bckwrd jump opertor ρ : T T is defined by ρt) sup{s T : s < t}. If σt) > t, then we sy tht t is right-scttered, while if ρt) < t then we sy tht t is left-scttered. If σt) t, the t is clled right-dense, nd if ρt) t then t is clled left-dense. If T hs left-scttered mximum t, then T κ : T {t}, otherwise T κ : T. Furthermore for function f : T R, we denote the function f σ : T R by f σ t) f σt)) for ll t T. Assume tht f : T R nd t T κ. Then we define f t) to be the number, if one exists, with the property tht for ny given ε > 0 there is neighborhood U of t such tht f σ t) f s) f t)σt) s) ε σt) s for ll s U. We sy tht f is -differentible on T κ provided f t) exists for ll t T κ. A mpping f : T R is clled rd-continuous denoted by C rd ) if: f is continuous t ech right-dense point or mximl element of T; the left-sided limit lim f s) f t ) s t exists t ech left-dense point t of T. A function F : T R is clled -ntiderivtive of f : T R provided F t) f t) holds for ll t T κ. Definition.1. Let h k : T R, k N 0 be defined by nd then recursively by h 0 t,s) 1 for ll s,t T t h k+1 t,s) h k τ,s) τ for ll s,t T. s Throughout this pper, we suppose tht T is time scle,,b T with < b nd n intervl mens the intersection of rel intervl with the given time scle. 3 The Grüss type inequlity on time scles We firstly stte our Grüss type inequlity for generl time scles. Theorem 3.1. Let,b,s T, f,g C rd nd f,g : [,b] R. Then for γ gs) Γ, 3.1)
36 Quô c Anh Ngô nd Wenjun Liu we hve f x)gx) x 1 b Γ γ f x) x gx) x f x) 1 b 3.) f y) y x. Moreover, the constnt 1 in 3.) is shrp. Proof. Since then it is cler tht f x)gx) x 1 b By ssumption 3.1), we get f x) 1 ) f y) y x 0, b f x) x gx) x f x) 1 b gx) γ + Γ sup <x<b gx) γ + Γ Γ γ nd the desired inequlity 3.) follows immeditely. To prove the shrpness of this inequlity, let us define Γ if f x) 1 f y) y 0, gx) b γ if f x) 1 f y) y < 0. b Then it is esy to verify tht 3.) is equlity. ) f y) y gx) γ + Γ ) x f x) 1 f y) y b x. Similrly to Theorem 3.1, we obtin the weighted Grüss type inequlity for generl time scles. Theorem 3.. Let,b,s T, f,g,w C rd nd f,g,w : [,b] R. Then for nd we hve γ gs) Γ, 3.3) wx) > 0 for.e. x [,b], 3.4) wx) f x)gx) x wx) x wx) f x) x wx)gx) x Γ γ wx) f x) wy) y wy) f y) y x. 3.5)
A Shrp Grüss Type Inequlity on Time Scles 37 Proof. Since ) wx) f x) wx) b wy) y wy) f y) y x 0, then wx) f x)gx) x b wx) x wx) f x) x wx)gx) x ) wx) f x) wx) b wx) x wy) f y) y gx) γ + Γ ) x sup gx) γ + Γ wx) f x) wy) y wy) f y) y x. <x<b By ssumption 3.3), we get gx) γ + Γ Γ γ nd the desired inequlity 3.5) follows immeditely. To prove the shrpness of this inequlity, let us define Γ gx) γ Then it is esy to verify tht 3.5) is equlity. if f x) wy) y wy) f y) y 0, if f x) wy) y wy) f y) y < 0. 4 The shrp Ostrowki-Grüss inequlity on time scles In this section, by pplying Theorem 3.1, we shll prove Theorem 1.3. Proof of Theorem 1.3. Let { s, s < t, pt,s) s b, t s b. 4.1) Applying Theorem 3.1 for the choices f x) : pt,x) nd gx) : f x) to get pt,x) f x) x 1 pt, x) x f x) x b Γ γ pt,x) 1 4.) pt, y) y b x. We obviously hve f x) x b f b) f ) b
38 Quô c Anh Ngô nd Wenjun Liu nd Hence, pt,x) 1 b b pt,x) x h t,) h t,b). b pt, y) y x pt,x) h t,) h t,b) b x. 4.3) By 4.) nd 4.3), we deduce tht pt,x) f x) x 1 pt, x) x f x) x b Γ γ pt,x) h 4.4) t,) h t,b) b x. By using Montgomery Identity, see [5], we deduce tht f t) 1 f σ x) x + 1 pt,x) f x) x, b b which helps us to deduce tht f t) 1 f σ f b) f ) s) s b b ) h t,) h t,b)) f t) 1 f σ s) s 1 b b 1 b pt,x) f x) x 1 b pt, x) x f x) x b pt, x) x f x) x. Hence f t) 1 b f σ f b) f ) s) s b ) h t,) h t,b)) Γ γ b ) pt,x) h t,) h t,b) b x, for ll t [,b]. It is obviously to see tht the foregoing inequlity is shrp. If we pply the Theorem 1.3 to different time scles, we will get some well-known nd some new results. Corollry 4.1 Continuous cse). Let T R, then inequlity 1.3) becomes f t) 1 f b) f ) f s)ds t + b ) )Γ γ) 4.5) b b 8 for ll t [,b], where γ f t) Γ. Actully, inequlity 4.5) is shrp, see [8, Theorem 3].
A Shrp Grüss Type Inequlity on Time Scles 39 Corollry 4. Discrete cse). Let T Z, 0, b 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, h t,0) t ) t t 1) t n, h t,n) ) t n)t n 1). Thus, we hve x i 1 n n x j x n x 0 i n + 1 ) j1 n Γ γ n for ll i 1,n, where γ x i Γ nd pi,0) 0, Moreover, the constnt 1 in 4.6) is shrp. n 1 j0 p1, j) j n, 1 j n 1, pn, j) j, { 0 j n 1, pi, j) j, 0 j < i, j n, i j n 1. pi, j) i n + 1 ) 4.6) Corollry 4.3 Quntum clculus cse). Let T q N 0, q > 1, q m,b q n with m < n. In this sitution, one hs h k t,s) k 1 ν0 t q ν s ν, q µ for ll t,s T. µ0 Therefore, Then where h t,q m ) t qm ) t q m+1) 1 + q f t) 1 q n q m qn q m f σ s) s Γ γ q n q m ) γ, h t,q n ) t qn ) t q n+1). 1 + q f q n ) f q m ) q n q m n 1 km f qt) f t) q 1)t) t qn+1 q m+1 q + 1 ) t,q p k) t qn+1 q m+1 ), q + 1 Γ, t [,b] 4.7)
40 Quô c Anh Ngô nd Wenjun Liu nd pt,q k ) { q k q m, q m q k < t, q k q n, t q k q n. Moreover, the constnt 1 in 4.7) is shrp. Acknowledgments The uthors thnk the referees for their creful reding of the mnuscript nd insightful comments. References [1] R. Agrwl, M. Bohner nd A. Peterson, Inequlities on time scles: A survey. Mth. Inequl. Appl. 4 001), no. 4, pp 535 557. [] M. Bohner nd A. Peterson, Dynmic equtions on time scles, Birkhäuser, Boston, 001. [3] M. Bohner nd A. Peterson, Advnces in dynmic equtions on time scles, Birkhäuser, Boston, 003. [4] M. Bohner nd T. Mtthews, The Grüss inequlity on time scles. Commun. Mth. Anl. 3 007), no. 1, pp 1 8. [5] M. Bohner nd T. Mtthews, Ostrowski inequlities on time scles. JIPAM. J. Inequl. Pure Appl. Mth. 9 008), no. 1, Art. 6, 8 pp. [6] X.L. Cheng, Improvement of some Ostrowski-Grüss type inequlities. Comput. Mth. Appl. 4 001), no. 1-, pp 109 114. [7] X.L. Cheng nd J. Sun, A note on the perturbed trpezoid inequlity. JIPAM. J. Inequl. Pure Appl. Mth. 3 00), no., Art. 9, 7 pp. [8] P. Cerone nd S.S. Drgomir, A refinement of the Grüss inequlity nd pplictions. RGMIA Res. Rep. Collect. 5 00), no., Art. 14. [9] S.S. Drgomir nd S. Wng, An inequlity of Ostrowski-Grüss type nd its pplictions to the estimtion of error bounds for some specil mens nd for some numericl qudrture rules. Comput. Mth. Appl. 33 1997), no. 11, pp 15 0. [10] S. Hilger, Ein Mβkettenklkül mit Anwendung uf Zentrumsmnnigfltigkeiten, PhD thesis, Univrsi. Würzburg, 1988. [11] R. Hilscher, A time scles version of Wirtinger-type inequlity nd pplictions. J. Comput. Appl. Mth. 141 00), no. 1-, pp 19 6. [1] V. Lkshmiknthm, S. Sivsundrm, nd B. Kymkcln, Dynmic systems on mesure chins, Kluwer Acdemic Publishers, 1996.
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