Volume 2008, Article ID 597241,12pages doi:10.1155/2008/597241
Research Article
A Perturbed Ostrowski-Type Inequality on
Time Scales for
k
Points for Functions Whose
Second Derivatives Are Bounded
Wenjun Liu,1Quoc Anh Ng ˆo,´ˆ 2, 3 and Wenbing Chen1
1College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2Department of Mathematics, College of Science, Viet Nam National University, Hanoi, Vietnam
3Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Correspondence should be addressed to Wenjun Liu,[email protected]
Received 6 May 2008; Accepted 13 August 2008
Recommended by Kunquan Lan
We first derive a perturbed Ostrowski-type inequality on time scales forkpoints for functions whose second derivatives are bounded and then unify corresponding continuous and discrete versions. We also point out some particular perturbed integral inequalities on time scales for functions whose second derivatives are bounded as special cases.
Copyrightq2008 Wenjun Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The following integral inequality which was first proved by Ostrowski in 1938 has received considerable attention from many researchers1–9.
Theorem 1.1. Let f : a, b → R be continuous on a, b and differentiable in a, b and its
derivativef : a, b → Ris bounded on a, b, that is, f∞ : supt∈a,b|fx| < ∞. Then
for anyx∈a, b, the following inequality holds:
fx− 1
b−a
b
a
ftdt≤
1 4
x−ab/22
b−a2
b−af∞. 1.1
The inequality is sharp in the sense that the constant1/4cannot be replaced by a smaller one.
studied the theory of certain integral inequalities on time scales. For example, we refer the reader to 11–18. In 15, Bohner and Matthews established the following so-called Ostrowski inequality on time scales.
Theorem 1.2see15, Theorem 3.5. Leta, b, x, t∈T,a < b, andf:a, b → Rbe differentiable.
Then
b
a
fσtΔt−fxb−a≤Mh2x, a h2x, b, 1.2
wherehk·,·is defined byDefinition 2.9below andM supa<x<b|fΔx|.This inequality is sharp in the sense that the right-hand side of1.1cannot be replaced by a smaller one.
Liu and Ng ˆo then generalize the above Ostrowski inequality on time scales forkpoints
x1, x2, . . . , xk in19. They also extended the result by considering functions whose second
derivatives are bounded in20. They obtained the following theorem.
Theorem 1.3. Leta, b, x, t ∈ T,a < b, andf : a, b → Rbe a twice differentiable function on
a, bandfΔΔ:a, b → Rbounded, that is,M:supa<t<b|fΔΔx|<∞. Then
b
a
fσtΔt−fσxb−a h2x, a−h2x, bfΔx
≤Mh3x, a−h3x, b. 1.3
Theorem 1.3may be thought of as a perturbed version ofTheorem 1.2. In the present paper we will first derive a perturbed Ostrowski-type inequality on time scales for k
points x1, x2, . . . , xk for functions whose second derivatives are bounded and then unify corresponding continuous and discrete versions. We also point out some particular perturbed integral inequalities on time scales for functions whose second derivatives are bounded as special cases.
2. Time scales essentials
In this section, we briefly introduce the time scales theory and refer the reader to Hilger10 and the books21–23for further detailssee also19,20.
Definition 2.1. A time scaleTis an arbitrary nonempty closed subset of the real numbers.
Definition 2.2. Fort∈T, one defines theforward jump operatorσ:T → Tbyσt inf{s∈T:
s > t},while thebackward jump operatorρ:T → Tis defined byρt sup{s∈T:s < t}.
In this definition, we put inf∅ supT i.e.,σt t if T has a maximum tand sup∅infTi.e.,ρt tifThas a minimumt, where∅denotes the empty set. Ifσt> t, then we say thattisright-scattered, while ifρt< t, then we say thattisleft-scattered. Points that are right-scattered and left-scattered at the same time are called isolated. Ifσt tand
Definition 2.3. Lett∈T, then two mappingsμ, ν:T → 0,∞satisfying
μt:σt−t, νt:t−ρt 2.1
are called thegraininess functions.
We now introduce the setTκwhich is derived from the time scalesTas follows. IfT has a left-scattered maximumt, thenTκ : T− {t}, otherwiseTκ : T. Furthermore, for a
functionf:T → R, we define the functionfσ :T → Rbyfσt fσtfor allt∈T.
Definition 2.4. Letf :T → Rbe a function on time scales. Then fort∈Tκ, one defines fΔt
to be the number, if one exists, such that for allε >0 there is a neighborhoodUoftsuch that for alls∈U
fσt−fs−fΔtσt−s≤ε|σt−s|. 2.2
We say thatfisΔ-differentiable onTκprovidedfΔtexists for allt∈Tκ. We talk about the
second derivative fΔΔ providedfΔ is differentiable onTκ2
Tκκ with derivativefΔΔ fΔΔ:Tκ2
→ R.
Definition 2.5. A mappingf :T → Ris calledrd-continuousdenoted byf ∈Crdprovided
that it satisfies
1fis continuous at each right-dense point ofT;
2the left-sided limit lims→t−fs ft−exists at each left-dense pointtofT.
Remark 2.6. It follows from Theorem 1.74 of Bohner and Peterson 21 that every
rd-continuous function has an antiderivative.
Definition 2.7. A functionF : T → Ris called aΔ-antiderivative off : T → Rprovided
FΔt ftholds for allt∈Tκ. Then theΔ-integral offis defined by
b
a
ftΔtFb−Fa. 2.3
Proposition 2.8. Letf, gbe rd-continuous,a, b, c∈T, andα, β∈R. Then
1abαft βgtΔtαabftΔtβabgtΔt,
2abftΔt−baftΔt,
3abftΔtacftΔtcbftΔt,
4abftgΔtΔt fgb−fga−abfΔtgσtΔt, 5aaftΔt0,
Definition 2.9. Lethk:T2 → R,k∈N
0be defined by
h0t, s 1 ∀s, t∈T 2.4
and then recursively by
hk1t, s t
s
hkτ, sΔτ ∀s, t∈T. 2.5
Remark 2.10. It follows fromProposition 2.86that ifs ≤t, thenhk1t, s≥0 for allt, s ∈T and allk∈N.
Remark 2.11. If we lethΔkt, sdenote for each fixedsthe derivative ofhkt, swith respect to
t, then
hΔkt, s hk−1t, s, fork∈N, t∈Tκ. 2.6
3. The perturbed Ostrowski inequality on time scales
Our main result reads as follows .
Theorem 3.1. Suppose that
1a, b ∈ T,Ik :a x0 < x1 < · · · < xk−1 < xk bis a division of the intervala, bfor
x0, x1, . . . , xk∈T;
2αi∈Ti0, . . . , k1is “k2” points so thatα0 a,αi ∈xi−1, xi i1, . . . , kand
αk1b;
3f : a, b → Ris a twice differentiable function ona, band fΔΔ : a, b → R is
bounded, that is,M:supa<t<b|fΔΔt|<∞.
Then
b
a
fσtΔt− k
i0
αi1−αifσxi
k−1
i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi
≤M k−1
i0
h3xi1, αi1−h3xi, αi1.
3.1
Lemma 3.2generalized montgomery identity. Under the assumptions ofTheorem 3.1,
k
i0
αi1−αifσxi b
a
fσtΔt−
b
a
Kt, IkfΔΔΔt
k−1 i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi ,
3.2
where
Kt, Ik
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
h2t, α1, t∈a, x1,
h2t, α2, t∈x1, x2,
.. .
h2t, αk−1, t∈xk−2, xk−1,
h2t, αk, t∈xk−1, b.
3.3
Proof. Integrating by parts and applyingProposition 2.84, we have
b
a
Kt, IkfΔΔtΔt
k−1 i0
xi1
xi
Kt, IkfΔΔtΔt
k−1 i0
xi1
xi
h2t, αi1fΔΔtΔt
k−1 i0
αi1
xi
h2t, αi1fΔΔtΔt xi1
αi1
h2t, αi1fΔΔtΔt
k−1 i0
h2αi1, αi1fΔαi1−h2xi, αi1fΔxi− αi1
xi
fΔσthΔ2t, αi1Δt
h2xi1, αi1fΔxi1−h2αi1, αi1fΔαi1− xi1
αi1
fΔσthΔ2t, αi1Δt
k−1 i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi
− αi1
xi
fΔσtt−αi1Δt− xi1
αi1
fΔσtt−αi1Δt
k−1 i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi fσxixi−αi1
αi1
x1
fσtΔt−fσxi1xi1−αi1 xi1
αi1
fσtΔt
b
a
fσtΔt k−1
i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi fσaa−α1
k−1 i1
fσxix
i−αi1−fσbb−αk−
k−2
i0
fσx
i1xi1−αi1
b
a
fσtΔt− k
i0
fσxiαi1−αi
k−1
i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi ,
3.4
that is,3.2holds.
Proof ofTheorem 3.1. By applyingLemma 3.2, we get
b
a
fσtΔt− k
i0
αi1−αifσxi
k−1
i0
h2xi1, αi1fΔxi1−h2xi, αi1fΔxi
b
a
Kt, IkfΔΔtΔt
k−1
i0 xi1
xi
Kt, IkfΔΔtΔt
≤k−1
i0 xi1
xi
|Kt, Ik||fΔΔt|Δt≤M k−1
i0 xi1
xi
|h2t, αi1|Δt
M k−1
i0
αi1
xi
αi1
t
αi1−τΔτ
Δt
xi1
αi1
h2t, αi1Δt
M k−1
i0
αi1
xi
h2t, αi1Δt xi1
αi1
h2t, αi1Δt
M k−1
i0
h3xi1, αi1−h3xi, αi1.
3.5
The proof is complete.
If we apply the the inequality3.1to different time scales, we will get some well-known and some new results.
Corollary 3.3continuous case. LetTR. Then our delta integral is the usual Riemann integral
from calculus. Hence,
h2t, s t−s 2
This leads us to state the following inequality:
b
a
ftΔt−k
i0
αi1−αifxi 1 2
k−1
i0
xi1−αi12fxi1−xi−αi12fxi
≤ M 6
k−1
i0
xi1−αi13−xi−αi13 ,
3.7
whereMsupa<x<b|fx|.
Remark 3.4. The inequality 3.7is exactly the generalized Ostrowski inequality shown in
24.
Corollary 3.5discrete case. LetTZ,a0,bn. Suppose that
1Ik: 0j0< j1<· · ·< jk−1< jknis a division of0, n∩Zforj0, k1, . . . , jk∈Z; 2pi ∈Zi0, . . . , k1is “k2” points so thatp00,pi ∈ji−1, ji∩Zi1, . . . , k
andpk1n;
3fk xk.
Then,
n
j1
xj− k
i0
pi1−pixji1
k−1
i0
h2ji1, pi1Δxji1−h2ji, pi1Δxji
≤M k−1
i0
h3ji1, pi1−h3ji, pi1
3.8
for alli1, n, where
M sup 1≤i≤n−1
Δ2x
i, hkt, s
t−s k
3.9
for allt, s∈Z.
Corollary 3.6quantum calculus case. LetTqN0,q >1,aqm, bqnwithm < n. Suppose
that
1Ik:qmqj0 < qj1 <· · ·< qjk−1 < qjk qnis a division ofqm, qn∩qN0forj0, j1, . . . , jk∈
N0;
2qpi ∈qN0 i0, . . . , k1is “k2” points so thatqp0 qm,qpi ∈qji−1, qji∩qN0 i
1, . . . , kandqpk1qm;
Then,
qn
qm
fqtΔt−k
i0
qpi1−qpifqji1
k−1 i0
h2qji1, qpi1
fqji11−fqji1 q−1qji1 −h2q
ji, qpifq
ji1−fqji
q−1qji
≤M k−1
i0
h3qji1, qpi1−h3qji, qpi1,
3.10
where
M sup
qm<t<qn
fq2t−qqq−1f12qt t2 qft
, hkt, s k−1
ν0
t−qνs
ν μ0qμ
, ∀t, s∈qN0.
3.11
4. Some particular perturbed integral inequalities on time scales
In this section, we point out some particular perturbed integral inequalities on time scales for functions whose second derivatives are bounded as special cases, such as perturbed rectangle inequality on time scales, perturbed trapezoid inequality on time scales, perturbed mid-point inequality on time scales, perturbed Simpson inequality on time scales, perturbed averaged mid-point-trapezoid inequality on time scales, and others. Throughout this section, we always assume thata, b∈Twitha > bandf :a, b → Ris differentiable. We denote
M sup
a<x<b
fΔΔx<∞. 4.1
Proposition 4.1. Suppose thatα ∈ a, b∩T. Then one has the perturbed rectangle inequality on
time scales
b
a
fσtΔt−α−afσa b−αfσb h2b, αfΔb−h2a, αfΔa
≤Mh3b, α−h3a, α.
4.2
Proof. We choosex0 a,x1 b,α0 a,α1 α∈a, bandα2 binTheorem 3.1to get the
result.
Remark 4.2. aIf we chooseαbin4.2, we get the perturbed left rectangle inequality on
time scales
b
a
fσtΔt−b−afσa−h
2a, bfΔa
bIf we chooseαain4.2, we get the perturbed right rectangle inequality on time scales
b
a
fσtΔt−b−afσb h2b, afΔb
≤Mh3b, a. 4.4
cIf we chooseα ab/2 in4.2, we get the perturbed trapezoid inequality on time scales
b
a
fσtΔt−f
σa fσb
2 b−a
h2
b,ab
2
fΔb−h2
a,ab
2
fΔa
≤M
h3
b,ab
2
−h3
a,ab
2
.
4.5
Proposition 4.3. Suppose thatx∈a, b∩T,α1∈a, x∩Tandα2∈x, b∩T. Then one has the
perturbed inequality on time scales
b
a
fσtΔt−α1−afσa α2−α1fσx b−α2fσb
h2x, α1fΔx−h2a, α1fΔa h2b, α2fΔb−h2x, α2fΔx
≤Mh3x, α1−h3a, α1 h3b, α2−h3x, α2.
4.6
Remark 4.4. If we choose α1 aandα2 binProposition 4.3, we get exactlyTheorem 1.3.
Therefore,Theorem 3.1is a generalization of Theorem 4 in20. If we choosex ab/2 in
3.1, we get the perturbed mid-point inequality on time scales
b
a
fσtΔt−fσ
ab
2
b−a
h2
ab
2 , a
−h2
ab
2 , b
fΔ
ab
2 ≤M h3
ab
2 , a
−h3
ab
2 , b
.
4.7
Corollary 4.5. Suppose thatx∈5ab/6,a5b/6∩T,α1 5ab/6andα2 a5b/6.
Then one has the perturbed inequality on time scales
b
a
fσtΔt−b−a
3
fσa fσb
2 2f
σx h2
x,5ab
6
fΔx−h2
a,5ab
6
fΔah2
b,a5b
6
fΔb−h2
x,a5b
6
fΔx
≤M
h3
x,5ab
6
−h3
a,5ab
6
h3
b,a5b
6
−h3
x,a5b
6
.
Remark 4.6. If we choosex ab/2 in4.8, we get the perturbed Simpson inequality on time scales
b
a
fσtΔt−b−a
3
fσa fσb
2 2f
σ
ab
2 h2
ab
2 , 5ab
6
fΔ
ab
2
−h2
a,5ab
6
fΔa
h2
b,a5b
6
fΔb−h2
ab
2 ,
a5b
6
fΔ
ab
2 ≤M h3
ab
2 , 5ab
6
−h3
a,5ab
6
h3
b,a5b
6
−h3
ab
2 ,
a5b
6
.
4.9
Corollary 4.7. Suppose thatab/2∈T,α1 ∈a,ab/2∩Tandα2 ∈ab/2, b∩T.
Then one has the perturbed inequality on time scales
b
a
fσtΔt−
α1−afσa α2−α1fσ
ab
2
b−α2fσb
h2
ab
2 , α1
fΔ
ab
2
−h2a, α1fΔa h2b, α2fΔb−h2
ab
2 , α2
fΔ
ab
2 ≤M h3
ab
2 , α1
−h3a, α1 h3b, α2−h3
ab
2 , α2
.
4.10
Remark 4.8. If we chooseα1 3ab/4 andα2 a3b/4 in4.10, we get the perturbed
averaged mid-point-trapezoid inequality on time scales
b
a
fσtΔt−b−a
2
fσa fσb
2 f
σ
ab
2 h2
ab
2 , 3ab
4
fΔ
ab
2
−h2
a,3ab
4
fΔa
h2
b,a3b
4
fΔb−h2
ab
2 ,
a3b
4
fΔ
ab
2 ≤M h3
ab
2 , 3ab
4
−h3
a,3ab
4
h3
b,a3b
4
−h3
ab
2 ,
a3b
4
.
Acknowledgments
The authors wish to express their gratitude to the anonymous referees for a number of valuable comments and suggestions. This work was supported by the Science Research Foundation of Nanjing University of Information Science and Technology.
References
1 P. Cerone, S. S. Dragomir, and J. Roumeliotis, “An inequality of Ostrowski-Gr ¨uss type for twice differentiable mappings and applications in numerical integration,”Kyungpook Mathematical Journal, vol. 39, no. 2, pp. 333–341, 1999.
2 S. S. Dragomir, R. P. Agarwal, and P. Cerone, “On Simpson’s inequality and applications,”Journal of Inequalities and Applications, vol. 5, no. 6, pp. 533–579, 2000.
3 S. S. Dragomir, P. Cerone, and J. Roumeliotis, “A new generalization of Ostrowski’s integral inequality for mappings whose derivatives are bounded and applications in numerical integration and for special means,”Applied Mathematics Letters, vol. 13, no. 1, pp. 19–25, 2000.
4 W. Liu, Q.-L. Xue, and S.-F. Wang, “Several new perturbed Ostrowski-like type inequalities,”Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 4, article 110, pp. 1–6, 2007.
5 W. Liu, C. Li, and Y. Hao, “Further generalization of some double integral inequalities and applications,”Acta Mathematica Universitatis Comenianae, vol. 77, no. 1, pp. 147–154, 2008.
6 W. Liu, “Several error inequalities for a quadrature formula with a parameter and applications,” Computers & Mathematics with Applications, vol. 56, no. 7, pp. 1766–1772, 2008.
7 D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.
8 D. S. Mitrinovi´c, J. E. Peˇcari´c, and A. M. Fink, Classical and New Inequalities in Analysis, vol. 61 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
9 N. Ujevi´c, “A generalization of Ostrowski’s inequality and applications in numerical integration,” Applied Mathematics Letters, vol. 17, no. 2, pp. 133–137, 2004.
10 S. Hilger,Ein maßkettenkalk ¨ul mit anwendung auf zentrumsmannigfaltigkeiten, Ph.D. thesis, Universitaat W ¨urzburg, W ¨urzburg, Germany, 1988.
11 R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001.
12 M. R. S. Ammi, R. A. C. Ferreira, and D. F. M. Torres, “Diamon-αJensen’s inequality on time scales,” Journal of Inequalities and Applications, vol. 2008, Article ID 576876, 13 pages, 2008.
13 M. Bohner, S. Clark, and J. Ridenhour, “Lyapunov inequalities for time scales,”Journal of Inequalities and Applications, vol. 7, no. 1, pp. 61–77, 2002.
14 M. Bohner and T. Matthews, “The Gr ¨uss inequality on time scales,”Communications in Mathematical Analysis, vol. 3, no. 1, pp. 1–8, 2007.
15 M. Bohner and T. Matthews, “Ostrowski inequalities on time scales,”Journal of Inequalities in Pure and Applied Mathematics, vol. 9, no. 1, article 6, pp. 1–8, 2008.
16 W. N. Li and W. Sheng, “Some nonlinear integral inequalities on time scales,”Journal of Inequalities and Applications, vol. 2007, Article ID 70465, 15 pages, 2007.
17 F.-H. Wong, S.-L. Yu, and C.-C. Yeh, “Anderson’s inequality on time scales,”Applied Mathematics Letters, vol. 19, no. 9, pp. 931–935, 2006.
18 C.-C. Yeh, H.-L. Hong, H.-J. Li, and S.-L. Yu, “Some complements of Cauchy’s inequality on time scales,”Journal of Inequalities and Applications, vol. 2006, Article ID 97430, 19 pages, 2006.
19 W. Liu and Q.-A. Ng ˆo, “A generalization of Ostrowski inequality on time scales forkpoints,”Applied Mathematics and Computation, vol. 203, no. 2, pp. 754–760, 2008.
20 W. Liu and Q.-A. Ng ˆo, “An Ostrowski type inequality on time scales for functions whose second derivatives are bounded,” to appear in Inequality Theory and Applications, vol. 6, 2009.
22 M. Bohner and A. Peterso, Eds.,Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston, Mass, USA, 2003.
23 V. Lakshmikantham, S. Sivasundaram, and B. Kaymakcalan,Dynamic Systems on Measure Chains, vol. 370 ofMathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.
