Some Iyengar-type inequalities on time scales for functions whose second derivatives are bounded
Introduction
Time scales essentials
Iyengar-type inequalities on time scales
Acknowledgements
References
Nội dung
Applied Mathematics and Computation 216 (2010) 3244–3251 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Some Iyengar-type inequalities on time scales for functions whose second derivatives are bounded Wenjun Liu a,*, Quô´c-Anh Ngô b,c a 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, Viet Nam c Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10 Lower Kent Ridge Road, Singapore 119076, Singapore b a r t i c l e i n f o a b s t r a c t We establish some Iyengar-type inequalities on time scales for functions whose second derivatives are bounded by using Steffensen’s inequality on time scales Ó 2010 Elsevier Inc All rights reserved Keywords: Iyengar-type inequalities Second derivative Time scales Introduction In 1938, Iyengar [18] proved the following interesting integral inequality which has received considerable attention from many researchers [2,8–11,19] Theorem Let f be a differentiable function on (a, b) and assume that there is a constant M1 > such that jf0 (x)j < M1 for x (a, b) Then Z M ðb À aÞ2 b 1 f xịdx b aịf aị ỵ f ðbÞÞ ðf ðbÞ À f ðaÞÞ2 : À a 4M ð1Þ Especially, the authors in [2,11] proved the following inequality involving bounded second-order derivatives Theorem Let f C2[a, b] and jf00 (x)j M2 Then Z 3 ! M b 1 j Dj 2 ; f ðxÞdx À ðb À aịf aị ỵ f bịị ỵ b aị f ðbÞ À f ðaÞÞ ðb À aÞ À 24 a M2 ð2Þ where D = f0 (a) À 2f0 ((a + b)/2) + f0 (b) In [12], Franjic´ et al proved the following Iyengar-type inequality and show that it is always better than (2) Theorem Let f C2[a, b] and jf00 (x)j M2 Then À M2 M2 ðb À aÞ3 þ ðk þ k3b Þ 24 a Z b 1 f ðxÞdx À ðb À aÞðf ðaÞ þ f ðbÞÞ þ ðb À aÞ2 ðf ðbÞ À f ðaÞÞ a " 3 3 # M2 M2 bÀa bÀa ; À ka þ À kb ðb À aÞ3 À 2 24 * Corresponding author E-mail addresses: wjliu@nuist.edu.cn (W Liu), bookworm_vn@yahoo.com (Q.-A Ngơ) 0096-3003/$ - see front matter Ĩ 2010 Elsevier Inc All rights reserved doi:10.1016/j.amc.2010.04.049 ð3Þ W Liu, Q.-A Ngơ / Applied Mathematics and Computation 216 (2010) 3244–3251 3245 where ka ẳ aỵb ba f0 f aị ỵ 2M2 kb ẳ aỵb ba f bị f ỵ : 2M2 and The development of the theory of time scales was initiated by Hilger [13] 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 on time scales For example, we refer the reader to [1,6,7,15–17,20–22] In the present paper we shall establish some new Iyengar-type inequalities on time scales for functions whose second derivatives are bounded by using Steffensen’s inequality on time scales Our results (see Theorems and 7) extend the results in [2,11,12] to arbitrary time scales Time scales essentials Now we briefly introduce the time scales theory and refer the reader to Hilger [13] and the books [4,5,14] for further details Definition A time scale T is an arbitrary nonempty closed subset of real numbers Definition For t T, we define the forward jump operator r : T ! T by rtị ẳ inffs T : s > tg, while the backward jump operator q : T ! T is dened by qtị ẳ supfs T : s < tg: In this definition, we put inf ; ¼ sup T (i.e r(t) = t if T has a maximum t) and sup ; ¼ inf T (i.e q(t) = t if T has a minimum t), where ; denotes the empty set If r(t) > t, then we say that t is right-scattered, while if q(t) < t then we say that t is leftscattered Points that are right-scattered and left-scattered at the same time are called isolated If r(t) = t and t < sup T, then t is called right-dense, and if q(t) = t and t > inf 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 ! ẵ0; ỵ1ị satisfying ltị :ẳ rtị t; mtị :ẳ t qtị 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 Tj :¼ T À ftg, otherwise Tj :¼ T Furthermore for a function f :¼ T ! R, we define the function f r : T ! R by fr(t) = f(r(t)) for all t T Definition Let f : T ! R be a function on time scales Then for t Tj , we define fD (t) 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 jf r ðtÞ À f ðsÞ À f D ðtÞðrðtÞ À sÞj ejrðtÞ À sj: We say that f is D-differentiable on Tj provided fD(t) exists for all t Tj We talk about the second derivative fDD provided fD 2 is differentiable on Tj ẳ Tj ịj with derivative f DD ẳ f D ịD : Tj ! R Definition A mapping f : T ! R is called rd-continuous (denoted by Crd) provided if it satisfies f is continuous at each right-dense point or maximal element of T The left-sided limit lims!t f sị ẳ f ðtÀÞ exists at each left-dense point t of T Remark It follows from Theorem 1.74 of Bohner and Peterson [4] that every rd-continuous function has an antiderivative Definition A function F : T ! R is called a D-anti-derivative of f : T ! R provided FD(t) = f(t) holds for all t Tj Then the D-integral of f is defined by Z a b f tịDt ẳ Fbị Faị: 3246 W Liu, Q.-A Ngô / Applied Mathematics and Computation 216 (2010) 3244–3251 Proposition Let f, g be rd-continuous, a; b; c T and a; b R Then (i) (ii) (iii) (iv) (v) Rb a Rb a Rb a Rb a Ra a Rb Rb ẵaf tị ỵ bgtịDt ẳ a a f tịDt ỵ b a gtịDt, Ra f tịDt ¼ À b f ðtÞDt, Rc Rb f ðtÞDt ¼ a f tịDt ỵ c f tịDt, Rb f tịg D tịDt ẳ fgịbị fgịaị a f D tịgrtịịDt, f tịDt ẳ (vi) If f(t) P for all a t < b then Rb a f ðtÞDt P Definition Let hk : T2 ! R; k N0 be defined by h0 ðt; sị ẳ for all s; t T and then recursively by hkỵ1 t; sị ẳ Z t hk ðs; sÞDs for all s; t T: s Remark It follows from Proposition 4(vi) that if s t then hk+1(t,s) P for all t; s T and all k N D Remark If we let hk ðt; sÞ denote the derivative of hk(t, s) with respect to t for each fixed s, then D hk t; sị ẳ hk1 t; sị; for k N; t Tj : The following Steffensen’s inequality on time scales was established in [3] Theorem (Steffensen’s Inequality) Let a; b Tjj and F; G : ½a; b ! R be integrable functions, with F decreasing and G Rb on [a, b] Assume k :ẳ a GtịDt such that b k; a ỵ k T Then Z b FtịDt bk Z b FtịGtịDt a Z aỵk FtịDt: 4ị a Throughout this paper, we suppose that T is a time scale, a; b T with a < b and an interval means the intersection of real interval with the given time scale Iyengar-type inequalities on time scales Our first result is embodied in the following Theorem Let a; aỵb ; b Tj and f : ½a; b ! R be a twice differentiable function and f DD : ½a; b \ Tj ! R is bounded, i.e À1 < m f DD tị M < ỵ1; t ẵa; b \ Tj : Then ! ! ba aỵb aỵb aỵb aỵb h3 a ỵ k; ỵ h3 a; Mh3 a; À mh3 b; ðM À mÞ sgn k À 2 2 Z b ba r aỵb D aỵb D ẵf aị ỵ f r bị ỵ h2 b; f bị h2 a; f ðaÞ f ðr2 ðxÞÞDx À 2 a ! ! ba aỵb aỵb aỵb aỵb M mị sgn k h3 b k; ỵ h3 b; À Mh3 a; À mh3 b; ; 2 2 5ị where kẳ ! ba aỵb ỵ f D aị 2f D ỵ f D bị Mm such that a ỵ k T; b À k T Proof To prove our result, we shall use the Steffensen’s inequality on time scales For this, let ( Fxị ẳ x a; aỵb ; h2 x; aỵb 2aỵb aỵb h2 x; x 2 ;b ; ð6Þ W Liu, Q.-A Ngơ / Applied Mathematics and Computation 216 (2010) 3244–3251 3247 and  Á DD xị < Mf x a; aỵb ; Mm Gxị ẳ aỵb : f DD xịm x 2 ;b : MÀm ð7Þ Thus it holds Gxị and kẳ Mm "Z aỵb M f DD xịịDx ỵ a Z # b aỵb f DD xị mịDx ẳ ! ba aỵb ỵ f D aị 2f D ỵ f D bị : Mm Theorem tells us that Z b FðxÞDx bÀk |{z} Z b FxịGxịDx a a Z aỵk FxịDx : a |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl} ð8Þ b Two cases are possible: À ỵ f D bị 0, and (a) f D aị 2f D aỵb (b) f D aị 2f D aỵb ỵ f D bị > ba aỵb D Case (a) f D aị 2f D aỵb ị ỵ f ðbÞ implies k and thus a þ k 6 b À k The left and right term of inequality (4) are: Z b Z bk aỵb aỵb aỵb h2 x; Dx ẳ h2 x; Dx ỵ h2 x; Dx aỵb aỵb 2 bk bk 2 aỵb aỵb ẳ h3 b; þ h3 b À k; ; 2 a¼ Z b FxịDx ẳ Z b and Z aỵk Z aỵb Z aỵb 2 aỵb aỵb aỵb FxịDx ẳ h2 x; Dx ẳ h2 x; Dx À h2 x; Dx 2 a a a aỵk aỵb aỵb þ h3 a þ k; : ¼ Àh3 a; 2 Z bẳ aỵk Case (b) Here it holds that k > ba and thus b k < aỵb < a ỵ k A similar calculation gives 2 aẳ Z b FxịDx ẳ Z aỵb aỵb aỵb aỵb aỵb h2 x; Dx ỵ h2 x; Dx ẳ h3 b; h3 b k; ; aỵb 2 2 bÀk Z bÀk b and b¼ Z aỵk FxịDx ẳ a Z a aỵb Z aỵk aỵb aỵb aỵb aỵb h2 x; Dx h2 x; Dx ẳ h3 a; h3 a ỵ k; : aỵb 2 2 Thus, (8) implies that Z b ba aỵb aỵb sgn k FtịGtịDt h3 b k; ỵ h3 b; PÀ 2 a ba aỵb aỵb h3 a ỵ k; ỵ h3 a; : P sgn k À 2 Now, we only need to calculate the middle term in Stefenssen’s inequality Obviously, Iẳ Z a b FtịGtịDt ẳ Mm "Z a aỵb # Z b aỵb aỵb DD DD h2 x; h2 x; M f xịịDx f xị mịDx : aỵb 2 ð9Þ 3248 W Liu, Q.-A Ngơ / Applied Mathematics and Computation 216 (2010) 3244–3251 Using Proposition 4(iv) and Remark 3, we have Z aỵb a Z aỵb Z aỵb 2 aỵb aỵb a ỵ b DD h2 x; h2 x; Dx À h2 x; ðM À f DD ðxÞÞDx ẳ M f xịDx 2 a a !aỵb Z a 2 aỵb aỵb D h2 x; Dx h2 x; f xị ẳ M aỵb 2 a !D Z aỵb aỵb ỵ h2 x; f D rxịịDx a Z aỵb aỵb aỵb D aỵb D h1 x; ỵ h2 a; f aị ỵ f rxịịDx: ẳ Mh3 a; 2 a By using Proposition 4(iv) and Remark again, we obtain Z aỵb a !D !aỵb Z aỵb 2 aỵb D aỵb aỵb f rxịịDx ẳ h1 x; f rxịị h1 x; h1 x; f ðrðrðxÞÞÞDx 2 a a Z aỵb aỵb aỵb ẳ Àh1 a; h0 x; f ðrðaÞÞ À f ðr2 ðxÞÞDx 2 a Z aỵb Z aỵb 2 aỵb ba f r2 xịịDx ẳ f r2 xịịDx f raịị f raịị ẳ h1 a; 2 a a which implies that Z aỵb a Z aỵb aỵb aỵb aỵb D ba h2 x; f r2 xịịDx: M f DD xịịDx ẳ Mh3 a; ỵ h2 a; f aị ỵ f raịị 2 2 a Similarly, À Z Z b aỵb aỵb aỵb D ba f DD xị mịD ẳ mh3 b; h2 b; f bị þ h2 x; f ðr2 ðxÞÞDx: f ðrðbÞÞ À aþb aỵb 2 2 2 b Therefore, Iẳ Z b FtịGtịDt a aỵb aỵb D ba r ỵ h2 a; f aị ỵ ÀMh3 a; f ðaÞ MÀm 2 # Z b Z aỵb aỵb aỵb D bÀa r 2 À h2 b; f ðbÞ þ f ðr ðxÞÞDx þ mh3 b; f ðr ðxÞÞDx f bị aỵb 2 a (Z b ba r aỵb D aỵb D f bị h2 a; f aị ẳ f r2 xịịDx ẵf aị ỵ f r bị ỵ h2 b; Mm 2 a !' aỵb aỵb mh3 b; : ỵ Mh3 a; 2 ¼ Thus, ba r aỵb D aỵb D f bị h2 a; f aị ẵf aị ỵ f r bị ỵ h2 b; 2 a ! Z b aỵb aỵb mh3 b; ẳ M mÞ FðtÞGðtÞDt À Mh3 a; 2 a Z b f ðr2 ðxÞÞDx À which implies that ! ! ba aỵb aỵb aỵb aỵb M mị sgn k h3 a þ k; þ h3 a; À Mh3 a; À mh3 b; 2 2 Z b ba r aỵb D aỵb D f r2 xịịDx ẵf aị ỵ f r bị ỵ h2 b; f ðbÞ À h2 a; f ðaÞ 2 a ! ! ba aỵb aỵb aỵb aỵb h3 b k; ỵ h3 b; Mh3 a; mh3 b; : ðM À mÞ sgn k À 2 2 This proves the theorem h ð10Þ 3249 W Liu, Q.-A Ngô / Applied Mathematics and Computation 216 (2010) 3244–3251 Remark If we apply the inequality (5) to different time scales, we can get some well-known and some new results We only give an example of the case T ¼ R here The interested reader can investigate the case T ¼ Z; T ¼ qN0 (see also [15– 17]) Corollary (Continuous case) Let T ¼ R Then it is known that hk t; sị ẳ ðt À sÞk k! for all t; s R and all k N Moreover, ba aỵb þ þ f ðbÞ7: 6f ðaÞ À 2f M À m4 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} k¼ D Thus, (5) reads C B B C 7 B B b aC aỵb aỵb C aỵb aỵb 7 B C CÀ6 M h ðM À mÞB sgn k a ỵ k; ỵ h a; a; m h b; h B C 3 3 C B @|fflfflfflfflfflffl{zfflfflfflfflffl ffl}A 2 2 C B |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}A |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}5 @ D ðaÀbÞ3 ðaÀbÞ3 ðbÀaÞ3 MÀm D ðMÀm Þ 48 48 48 Z b f ðr2 ðxÞÞ Dx À |fflfflfflfflffl{zfflfflfflfflffl} a f ðxÞ ba r aỵb D aỵb D ẵf aị ỵ f r bị ỵ h2 b; f bị À h2 a; f ðaÞ |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |{z} f aịỵf bị baị2 f bị ðbÀaÞ2 f ðaÞ C B B C 7 B B b aC aỵb aỵb C aỵb aỵb 7 B C C M h h ðM À mÞB sgn k À b k; ỵ h b; a; m h b; B C 3 3 C B @|fflfflfflfflfflffl{zfflfflfflfflffl ffl}A 2 2 C B |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}A |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}5 @ D 3 3 ðbÀaÞ ðaÀbÞ ðbÀaÞ MÀm ð DÞ 48 48 48 À MÀm ð11Þ which implies MÀm À " 3 3 # Z b bÀa jDj ðb À aÞ3 bÀa f ðbÞ À f aị ẵf aị ỵ f bị ỵ b aị2 M ỵ mị f xịdx MÀm 48 a " 3 3 # MÀm bÀa j Dj ðb À aÞ3 À M ỵ mị: Mm 48 12ị Thus we obtain Z " # M À m b À a3 jDj 3 b bÀa f ðbÞ À f ðaÞ ðb À aÞ3 f xịdx ẵf aị ỵ f bị ỵ b aị ỵ M ỵ mị a MÀm 48 which is exactly the inequality shown in Theorem of [11] We get the inequality (2) in Theorem if we set M = Àm = M2 In our next result, we shall generalize Theorem to arbitrary time scales Theorem Let a; M :ẳ aỵb ; jf DD tịj < 1: sup t2ẵa;b\Tj b Tj and f : ½a; b ! R be a twice differentiable function and f DD : ½a; b \ Tj ! R is bounded, i.e Then ! aỵb aỵb aỵb h2 x; Dx ỵ M h3 a; h3 b; aỵb 2 Àka Z b bÀa r aỵb D aỵb D f bị h2 a; f aị f r2 xịịDx ẵf aị ỵ f r bị ỵ h2 b; 2 a "Z # ! Z b aỵka aỵb aỵb aỵb aỵb h3 b; ; 2M h2 x; Dx ỵ h2 x; Dx ỵ M h3 a; 2 2 bÀkb a 2M Z aỵb ỵkb 13ị 3250 W Liu, Q.-A Ngụ / Applied Mathematics and Computation 216 (2010) 3244–3251 where ka ẳ ! ba aỵb f D aị ỵ fD 2M kb ẳ ! ba aỵb ỵ f D bị f D 2M and are such that a ỵ ka T and b À ka T À Á DD xịỵM Proof We shall apply Theorem for Fxị ẳ h2 x; aỵb We have G(x) for each x ½a; b \ Tj So, on ; Gxị ẳ f 2M j2 aỵb ẵa; \ T , Theorem tells us that Z aỵb FxịDx Z aỵb ka aỵb FxịGxịDx a Z aỵka FxịDx; a which is equivalent to 2M Z aỵb FxịDx 2M Z aỵb ka aỵb FxịGxịDx 2M a Z aỵka FxịDx; 14ị a where Z ka ẳ aỵb GxịDx ẳ a 2M Z aỵb f DD xị ỵ MịDx ẳ a ! ba aỵb f D aị : ỵ fD 2M The middle term of inequality (14) is 2M Z aỵb Z aỵb Z aỵb 2 aỵb aỵb a ỵ b DD f DD xị ỵ MịDx ẳ M f xịDx h2 x; h2 x; Dx ỵ h2 x; 2 a a a aỵb Z Z a aỵb a ỵ b DD f xịDx h2 x; Dx ỵ h2 x; ẳ M aỵb 2 a Z aỵb aỵb a ỵ b DD ỵ f xịDx: h2 x; ẳ Mh3 a; 2 a FxịGxịDx ẳ a Z aỵb With the help of Proposition 4(iv) and Remark 3, one has Z aỵb a !D !aỵb Z aỵb 2 a ỵ b DD aỵb D aỵb f xịDx ẳ h2 x; f ðxÞ À h2 x; h2 x; f D ðrðxÞÞDx 2 a a Z aỵb aỵb D aỵb D ẳ Àh2 a; h1 x; f ðaÞ À f ðrðxÞÞDx: 2 a Again by Proposition 4(iv) and Remark 3, one obtains Z aỵb a !aỵb !D Z aỵb 2 aỵb D aỵb aỵb h1 x; h1 x; f rrxịịịDx f rxịịDx ẳ h1 x; f ðrðxÞÞ À 2 a a Z aỵb Z aỵb 2 aỵb aỵb ba f raịị f r2 xịịDx ẳ h0 x; f r2 xịịDx: ẳ h1 a; f raịị 2 a a Thus, 2M Z aỵb a Z aỵb aỵb aỵb D ba r FxịGxịDx ẳ Mh3 a; f r2 xịịDx: h2 a; f aị f aị ỵ 2 a Therefore, we get 2M Z Z aỵb aỵb aỵb ba r aỵb D h2 x; Dx ỵ Mh3 a; f ðr2 ðxÞÞDx À f ðaÞ À h2 a; f aị aỵb 2 2 ka a Z aỵka aỵb aỵb : h2 x; Dx ỵ Mh3 a; 2M 2 a aỵb 15ị W Liu, Q.-A Ngụ / Applied Mathematics and Computation 216 (2010) 32443251 On aỵb 3251 ; b \ Tj , similar to (14) one has 2M Z aỵb ỵkb FxịDx 2M aỵb Z b FxịGxịDx 2M aỵb Z b FxịDx; 16ị bkb where Z kb ẳ b aỵb GxịDx ẳ 2M Z b aỵb f DD xị ỵ MịDx ẳ ! ba aỵb : þ f D ðbÞ À f D 2M The middle term of inequality (16) is 2M Z aỵb f DD xị ỵ MịDx h2 x; aỵb 2 Z b aỵb aỵb D aỵb D ỵ h2 b; f bị f rxịịDx x ẳ Mh3 b; aỵb 2 2 Z b aỵb aỵb D ba r f r2 xịịDx: ẳ Mh3 b; ỵ h2 b; f bị f bị ỵ aỵb 2 2 b aỵb FxịGxịDx ẳ Z b Therefore, we have 2M Z aỵb ỵkb aỵb Z b aỵb aỵb ba r aỵb D h2 x; Dx Mh3 b; f r2 xịịDx f bị ỵ h2 b; f bị aỵb 2 2 Z b aỵb aỵb h2 x; Dx Mh3 b; : 2M 2 bÀkb ð17Þ Addition of (15) and (17) implies (13) h Remark If we apply the inequality (13) to different time scales, we can get some well-known and some new results For example, in the special case T ¼ R, we get the inequality (3) in Theorem To be precise, we refer the reader to Corollary The interested reader can investigate the case T ¼ Z; T ¼ qN0 Acknowledgements This work was supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions (Grant No 09KJB110005) and the Science Research Foundation of Nanjing University of Information Science and Technology The author would like to express sincere gratitude to the anonymous referees for their constructive suggestions References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] R Agarwal, M Bohner, A Peterson, Inequalities on time scales: a survey, Math Inequal Appl (2001) 535–557 R.P Agarwal, V Culjak, J Pecˇaric´, Some integral inequalities involving bounded higher order derivatives, Math Comput Model 28 (1998) 51–57 D.R Anderson, Time-scale integral inequalities, JIPAM J Inequal Pure Appl Math (2005) Art ID 66 , 15pp M Bohner, A Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, 2001 M Bohner, A Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003 M Bohner, T Matthews, The Grüss inequality on time scales, Commun Math Anal (2007) 1–8 M Bohner, T Matthews, Ostrowski inequalities on time scales, JIPAM J Inequal Pure Appl Math (2008) Art 6, 8pp P Cerone, S.S Dragomir, On a weighted generalization of Iyengar type inequalities involving bounded first derivative, Math Inequal Appl (1) (2000) 35–44 P Cerone, Generalized trapezoidal rules with error involving bounds of the nth derivative, Math Inequal Appl (2002) 451–462 X.L Cheng, The Iyengar type inequality, Appl Math Lett 14 (2001) 975–978 N Elezovic´, J Pecˇaric´, Steffensen’s inequality and estimates of error in trapezoidal rule, Appl Math Lett 11 (1998) 63–69 I Franjic´, J Pecˇaric´, I Peric´, Note on an Iyengar type inequality, Appl Math Lett 19 (2006) 657–660 S Hilger, Ein Mabkettenkalkül mit Anwendung auf Zentrumsmannigfaltigkeiten, PhD thesis, Universität Würzburg, 1988 V Lakshmikantham, S Sivasundaram, B Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, 1996 W.J Liu, Q.A Ngô, A generalization of Ostrowski inequality on time scales for k points, Appl Math Comput 203 (2008) 754–760 W.J Liu, Q.A Ngô, W.B Chen, A perturbed Ostrowski type inequality on time scales for k points for functions whose second derivatives are bounded, J Inequal Appl (2008) Art ID 597241, 12pp W.J Liu, Q.A Ngô, W.B Chen, Ostrowski type inequalities on time scales for double integrals, Acta Appl Math 110 (2010) 477–497 K.S.K Iyengar, Note on an inequality, Math Student (1938) 75–76 P.M Vasic´, J Pecˇaric´, Note on the Steffensen inequality, Univ Beograd Publ Elektrotehn Fak Ser Mat Fiz 716–734 (1981) 80–82 H Roman, A time scales version of a Wirtinger-type inequality and applications, dynamic equations on time scales, J Comput Appl Math 141 (2002) 219–226 M.Z Sarikaya, On weighted Iyengar type inequalities on time scales, Appl Math Lett 22 (2009) 1340–1344 F.-H Wong, S.-L Yu, C.-C Yeh, Anderson’s inequality on time scales, Appl Math Lett 19 (2007) 931–935 ... paper we shall establish some new Iyengar-type inequalities on time scales for functions whose second derivatives are bounded by using Steffensen’s inequality on time scales Our results (see Theorems... inequality on time scales for k points for functions whose second derivatives are bounded, J Inequal Appl (2008) Art ID 597241, 12pp W.J Liu, Q.A Ngô, W.B Chen, Ostrowski type inequalities on time scales. .. 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