DSpace at VNU: A generalization of Ostrowski inequality on time scales for k points

Applied Mathematics and Computation 203 (2008) 754–760 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc A generalization of Ostrowski inequality on time scales for k points Wenjun Liu a,*, Qũc-Anh Ngơ b a b College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam a r t i c l e i n f o Keywords: Ostrowski inequality Time scales Simpson inequality Trapezoid inequality Mid-point inequality a b s t r a c t In this paper we first generalize the Ostrowski inequality on time scales for k points and then unify corresponding continuous and discrete versions We also point out some particular Ostrowski type inequalities on time scales as special cases Ó 2008 Elsevier Inc All rights reserved Introduction In 1938, Ostrowski proved the following interesting integral inequality which has received considerable attention from many researchers [10–12,14,15] Theorem Let f : ½a; bŠ ! R be continuous on ½a; bŠ and differentiable in ða; bÞ and its derivative f : ða; bÞ ! R is bounded in ða; bÞ, that is, kf k1 :ẳ supt2a;bị jf xịj < Then for any x ½a; bŠ, we have the inequality  Z   b   f ðtÞdt À f ðxÞðb À aÞ    a  2 ! b aị2 aỵb kf k1 : ỵ xÀ The inequality is sharp in the sense that the constant cannot be replaced by a smaller one The development of the theory of time scales was initiated by Hilger [8] in 1988 as a theory capable to contain both difference and differential calculus in a consistent way Since then, many authors have studied the theory of certain integral inequalities or dynamic equations on time scales For example, we refer the reader to [1,4,5,7,13,16–18] In [5], Bohner and Matthews established the following so-called Ostrowski inequality on time scales Theorem (See [5], Theorem 3.5) Let a; b; x; t T, a < b and f : ½a; bŠ ! R be differentiable Then  Z   b   r f ðtÞDt À f xịb aị M h2 x; aị ỵ h2 ðx; bÞÞ;    a ð1Þ where h2 ðÁ; ị is dened by Denition and M ẳ supa t, then we say that t is right-scattered, while if qðtÞ < t then we say that t is left-scattered Points that are right-scattered and left-scattered at the same time are called isolated If rtị ẳ t, the t is called right-dense, and if qtị ẳ t then t is called left-dense Points that are both right-dense and left-dense are called dense Definition Let t T, then two mappings l; m : T ! ½0; ỵ1ị satisfying ltị :ẳ rtị t; mtị :ẳ t À qðtÞ are called the graininess functions We now introduce the set Tj which is derived from the time scales T as follows If T has a left-scattered maximum t, then T :¼ T À ftg, otherwise Tj :¼ T Furthermore for a function f : T ! R, we define the function f r : T ! R by f r tị ẳ f rtịị for all t T j Definition Let f : T ! R be a function on time scales Then for t Tj , we define f D ðtÞ to be the number, if one exists, such that for all e > there is a neighborhood U of t such that for all s U  r  f ðtÞ À f ðsÞ À f D ðtÞðrðtÞ À sÞ ejrðtÞ À sj: We say that f is D-differentiable on Tj provided f D ðtÞ exists for all t Tj Definition A mapping f : T ! R is called rd-continuous (denoted by C rd ) provided if it satisfies (1) f is continuous at each right-dense point or maximal element of T (2) The left-sided limit lims!t f sị ẳ f tị exists at each left-dense point t of T Remark It follows from Theorem 1.74 of Bohner and Peterson [2] that every rd-continuous function has an antiderivative Definition A function F : T ! R is called a D-antiderivative of f : T ! R provided F D tị ẳ f tị holds for all t Tj Then the D-integral of f is dened by Z b f tịDt ẳ Fbị À FðaÞ: a Proposition Let f ; g be rd-continuous, a; b; c T and a; b R Then (1) (2) (3) (4) (5) Rb Rb Rb af tị ỵ bgtịịDt ẳ a a f tịDt ỵ b a gtịDt, Ra Rab f tịDt ẳ b f tịDt, Rab Rc Rb f tịDt ẳ a f tịDt ỵ c f tịDt, Rb Rab f tịg D tịDt ẳ fgịbị fgịaị a f D tịgrtịịDt, Raa f tịDt ẳ a Denition Let hk : T2 ! R, k N0 be defined by h0 t; sị ẳ for all s; t T and then recursively by hkỵ1 t; sị ẳ Z s t hk ðs; sÞDs for all s; t T: 756 W Liu, Q.-A Ngô / Applied Mathematics and Computation 203 (2008) 754–760 The generalized Ostrowski inequality on time scales Throughout this section, we suppose that T is a time scale and an interval means the intersection of real interval with the given time scale We are in a position to state our main result Theorem Suppose that (1) a; b T, Ik : a ¼ x0 < x1 < Á Á Á < xkÀ1 < xk ẳ b is a division of the interval ẵa; bŠ for x0 ; x1 ; ; xk T; (2) T ði ¼ 0; ; k ỵ 1ị is k ỵ points so that a0 ẳ a, ẵxi1 ; xi Š ði ¼ 1; ; kị and akỵ1 ẳ b; (3) f : ẵa; b ! R is differentiable Then we have  Z   b k kÀ1 X X   r f tịDt aiỵ1 ịf xi ị M h2 xi ; aiỵ1 ị ỵ h2 xiỵ1 ; aiỵ1 ÞÞ;    a i¼0 i¼0 ð2Þ where M ¼ sup jf D ðxÞj: a > > > > < t a2 ; Kt; Ik ị ẳ Á Á > > > > t À akÀ1 ; > > : t À ak ; b Kðt; Ik Þf D ðtÞDt; ð3Þ a t ½a; x1 Þ; t ẵx1 ; x2 ị; 4ị t ½xkÀ2 ; xkÀ1 Þ; t ½xkÀ1 ; bŠ: Proof Integrating by parts and applying Proposition 1, we have Z b Kt; Ik ịf D tịDt ẳ a k1 Z X iẳ0 ẳ k1 X xiỵ1 Kt; Ik ịf D tịDt ẳ xi k1 Z X iẳ0 xiỵ1 t aiỵ1 ịf D tịDt xi xiỵ1 aiỵ1 ịf xiỵ1 ị xi aiỵ1 ịf xi ị ẳ ! xiỵ1 r f tịDt xi iẳ0 k1 X Z aiỵ1 xi ịf xi ị ỵ xiỵ1 aiỵ1 ịf xiỵ1 ị Z ! xiỵ1 r f tịDt xi iẳ0 ẳ a1 aịf aị ỵ Z k1 k2 X X aiỵ1 xi ịf xi ị ỵ xiỵ1 aiỵ1 ịf xiỵ1 ị ỵ b ak Þf ðbÞ À i¼1 b f r ðtÞDt a i¼0 Z k1 X ẳ a1 aịf aị ỵ aiỵ1 ịf xi ị ỵ b ak ịf bị a iẳ1 b f r tịDt ẳ Z k X aiỵ1 ịf xi ị b f r tịDt; a iẳ0 i.e., (3) holds h Proof of Theorem By applying Lemma 1, we get  Z    Z   b  X  X  b k k1 Z xiỵ1 k1 Z xiỵ1 X         r D D f tịDt aiỵ1 ịf xi ị ẳ  Kt; Ik ịf tịDt  ẳ  Kðt; Ik Þf ðtÞDt jKðt; Ik Þjf D ðtÞDt      a   xi xi a iẳ0 iẳ0 iẳ0 ! Z xiỵ1 Z aiỵ1 k1 Z xiỵ1 k1 X X 6M jt aiỵ1 jDt ẳ M aiỵ1 tịDt ỵ t aiỵ1 ÞDt i¼0 ¼M kÀ1 X i¼0 xi i¼0 ðh2 ðxi ; aiỵ1 ị ỵ h2 xiỵ1 ; aiỵ1 ịị: xi aiỵ1 757 W Liu, Q.-A Ngụ / Applied Mathematics and Computation 203 (2008) 754–760 To prove the sharpness of this inequality, let f tị ẳ t; x0 ẳ a; x1 ¼ b; a0 ¼ a; a1 ¼ b; a2 ¼ b It follows that M ¼ Starting with the left-hand side of (2), we have  Z  Z  Z Z b   b   b   b k X      r f tịDt aiỵ1 ịf xi ị ẳ  rtịDt b aịa ỵ b bịbị ẳ  rtị ỵ tịDt tDt b À aÞa      a   a a a i¼0   Z  Z b Z b     b     ẳ  t ịD Dt t Dt b aịa ẳ b aịb t Dt  :     a a a Starting with the right-hand side of (2), we have M k1 X h2 xi ; aiỵ1 ị ỵ h2 xiỵ1 ; aiỵ1 ịị ẳ h2 x0 ; a1 ị ỵ h2 x1 ; a1 ị ẳ h2 a; bị ỵ h2 b; bị ẳ Z a t bịDt ỵ Z b i¼0 ¼ Z a t Dt À Z b a bDt ẳ b aịb b Z b ðt À bÞDt b b tDt: a Therefore in this particular case  Z   b k kÀ1 X X   r f ðtÞ D t À a a ịf x ị h2 aiỵ1 ; xi ị ỵ h2 aiỵ1 ; xiỵ1 ịị  iỵ1 i i  P M   a i¼0 i¼0 and by (2) also  Z   b k k1 X X   f r tịDt aiỵ1 ịf xi ị M h2 aiỵ1 ; xi ị ỵ h2 aiỵ1 ; xiỵ1 ịị:    a i¼0 i¼0 So the sharpness of the inequality (2) is shown h If we apply the inequality (2) to different time scales, we will get some well-known and some new results Corollary (Continuous case) Let T ¼ R Then our delta integral is the usual Riemann integral from calculus Hence, h2 t; sị ẳ t sị2 for all t; s R: This leads us to state the following inequality:  Z !   b k k1 k1  X X 1X xi ỵ xiỵ1 2   ; f tịdt aiỵ1 ịf xi ị M xiỵ1 xi ị ỵ aiỵ1    a iẳ0 iẳ0 iẳ0 where M ẳ supa