Volume 2008, Article ID 597241, 12 pagesdoi:10.1155/2008/597241 Research Article A Perturbed Ostrowski-Type Inequality on Second Derivatives Are Bounded Wenjun Liu, 1 Qu ´ ˆoc Anh Ng ˆo,
Trang 1Volume 2008, Article ID 597241, 12 pages
doi:10.1155/2008/597241
Research Article
A Perturbed Ostrowski-Type Inequality on
Second Derivatives Are Bounded
Wenjun Liu, 1 Qu ´ ˆoc Anh Ng ˆo, 2, 3 and Wenbing Chen 1
1 College of Mathematics and Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
2 Department of Mathematics, College of Science, Viet Nam National University, Hanoi, Vietnam
3 Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Correspondence should be addressed to Wenjun Liu,wjliu@nuist.edu.cn
Received 6 May 2008; Accepted 13 August 2008
Recommended by Kunquan Lan
We first derive a perturbed Ostrowski-type inequality on time scales for k points 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
Copyrightq 2008 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
derivative f : a, b → R is bounded on a, b, that is, f∞ : supt ∈a,b |fx| < ∞ Then
for any x ∈ a, b, the following inequality holds:
fx − b − a1 b
a
f tdt
≤
1
4 x − a b/22
b − a2
The inequality is sharp in the sense that the constant 1/4 cannot be replaced by a smaller one.
In 1988, Hilger10 developed the theory of time scales as a theory capable to contain both difference and differential calculus in a consistent way Since then, many authors have
Trang 2studied 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 1.2 see 15, Theorem 3.5 Let a, b, x, t ∈ T, a < b, and f : a, b → R be differentiable
Then
b
a
f σ tΔt − fxb − a
where h k ·, · is defined by Definition 2.9 below and M supa<x<b |fΔx| This inequality is sharp
in the sense that the right-hand side of1.1 cannot be replaced by a smaller one.
Liu and Ng ˆo then generalize the above Ostrowski inequality on time scales for k points
x1, x2, , x k in19 They also extended the result by considering functions whose second derivatives are bounded in20 They obtained the following theorem
Theorem 1.3 Let a, b, x, t ∈ T, a < b, and f : a, b → R be a twice differentiable function on
a, b and fΔΔ:a, b → R bounded, that is, M : sup a<t<b |fΔΔx| < ∞ Then
b
a
f σ tΔt − f σ xb − a h2x, a − h2x, bfΔx
≤ Mh3x, a − h3x, b. 1.3
paper we will first derive a perturbed Ostrowski-type inequality on time scales for k points x1, x2, , x k 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 books21–23 for further details see also 19,20
Definition 2.1 A time scaleT is an arbitrary nonempty closed subset of the real numbers
Definition 2.2 For t ∈ T, one defines the forward jump operator σ : T → T by σt inf{s ∈ T :
s > t }, while the backward jump operator ρ : T → T is defined by ρt sup{s ∈ T : s < t}.
In this definition, we put inf∅ sup T i.e., σt t if T has a maximum t and
sup∅ inf T i.e., ρt t if T has a minimum t, where ∅ denotes the empty set If σt > t, then we say that t is right-scattered, while if ρt < 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 σt t and
t < sup T, then t is called right dense, and if ρt t and t > inf T, then t is called left dense.
Points that are both right dense and left dense are called dense
Trang 3Definition 2.3 Let t ∈ T, then two mappings μ, ν : T → 0, ∞ satisfying
are called the graininess functions.
We now introduce the setTκwhich is derived from the time scalesT as follows If T
has a left-scattered maximum t, thenTκ : T − {t}, otherwise Tκ : T Furthermore, for a
function f : T → R, we define the function f σ :T → R by f σ t fσt for all t ∈ T.
Definition 2.4 Let f : T → R be a function on time scales Then for t ∈ T κ , one defines fΔt
to be the number, if one exists, such that for all ε > 0 there is a neighborhood U of t such that for all s ∈ U
We say that f isΔ-differentiable on Tκ provided fΔt exists for all t ∈ T κ We talk about the
second derivative fΔΔ provided fΔ is differentiable on Tκ2
Tκκ with derivative fΔΔ
fΔΔ:Tκ2
→ R
Definition 2.5 A mapping f : T → R is called rd-continuous denoted by f ∈ Crd provided that it satisfies
1 f is continuous at each right-dense point of T;
2 the left-sided limit lims → t− f s ft− exists at each left-dense point t of T.
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 function F : T → R is called a Δ-antiderivative of f : T → R provided
FΔt ft holds for all t ∈ T κ Then theΔ-integral of f is defined by
b
a
Proposition 2.8 Let f, g be rd-continuous, a, b, c ∈ T, and α, β ∈ R Then
1b
a αft βgtΔt αb
a f tΔt βb
a g tΔt,
2b
a f tΔt −a
b f tΔt,
3b
a f tΔt c
a f tΔt b
c f tΔt,
4b
a f tgΔtΔt fgb − fga −b
a fΔtgσtΔt,
5a
a f tΔt 0,
6 If ft ≥ 0 for all a ≤ t < b thenb
a f tΔt ≥ 0.
Trang 4Definition 2.9 Let h k:T2 → R, k ∈ N0be defined by
and then recursively by
h k1t, s
t
s
Remark 2.10 It follows fromProposition 2.86 that if s ≤ t, then h k1t, s ≥ 0 for all t, s ∈ T and all k∈ N
Remark 2.11 If we let hΔk t, s denote for each fixed s the derivative of h k t, s with respect to
t, then
3 The perturbed Ostrowski inequality on time scales
Our main result reads as follows
Theorem 3.1 Suppose that
1 a, b ∈ T, I k : a x0 < x1 < · · · < x k−1 < x k b is a division of the interval a, b for
x0, x1, , x k∈ T;
2 α i ∈ T i 0, , k 1 is “k 2” points so that α0 a, α i ∈ x i−1, x i i 1, , k and
α k1 b;
3 f : a, b → R is a twice differentiable function on a, b and fΔΔ : a, b → R is
bounded, that is, M : supa<t<b |fΔΔt| < ∞.
Then
b
a
f σ tΔt −k
i0
α i1− α i f σ x i k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i
≤ Mk−1
i0
h3x i1, α i1 − h3x i , α i1.
3.1
twice differentiable function This is motivated by the ideas of Sofo and Dragomir in 24, where the continuous version of a perturbed Ostrowski inequality for twice differentiable mappings was proved
Trang 5Lemma 3.2 generalized montgomery identity Under the assumptions of Theorem 3.1 ,
k
i0
α i1− α i f σ x i
b
a
f σ tΔt −
b
a
K t, I k fΔΔΔt
k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i ,
3.2
where
K t, I k
⎧
⎪
⎪
⎪
⎪
⎪
⎪
h2t, α1, t ∈ a, x1,
h2t, α2, t ∈ x1, x2,
.
h2t, α k−1, t ∈ x k−2, x k−1,
h2t, α k , t ∈ x k−1, b .
3.3
Proof Integrating by parts and applyingProposition 2.84 , we have
b
a
K t, I k fΔΔtΔt
k−1
i0
x i1
x i
K t, I k fΔΔtΔt
k−1
i0
x i1
x i
h2t, α i1fΔΔtΔt
k−1
i0
α i1
x i
h2t, α i1fΔΔtΔt
x i1
α i1
h2t, α i1fΔΔtΔt
k−1
i0
h2α i1, α i1fΔα i1 − h2x i , α i1fΔx i −
α i1
x i
fΔσthΔ
2t, α i1Δt
h2x i1, α i1fΔx i1 − h2α i1, α i1fΔα i1 −
x i1
α i1
fΔσthΔ
2t, α i1Δt
k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i
−
α i1
x i
fΔσtt − α i1Δt −
x i1
α i1
fΔσtt − α i1Δt
k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i f σ x i x i − α i1
α i1
x
f σ tΔt − f σ x i1x i1− α i1
x i1
α
f σ tΔt
Trang 6
b
a
f σ tΔt k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i f σ aa − α1
k−1
i1
f σ x i x i − α i1 − f σ bb − α k −k−2
i0
f σ x i1x i1− α i1
b
a
f σ tΔt −k
i0
f σ x i α i1− α i k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i ,
3.4 that is,3.2 holds
Proof of Theorem 3.1 By applyingLemma 3.2, we get
b
a
f σ tΔt −k
i0
α i1− α i f σ x i k−1
i0
h2x i1, α i1fΔx i1 − h2x i , α i1fΔx i
b
a
K t, I k fΔΔtΔt
k−1
i0
x i1
x i
K t, I k fΔΔtΔt
≤k−1
i0
x i1
x i
|Kt, I k ||fΔΔt|Δt ≤ Mk−1
i0
x i1
x i
|h2t, α i1|Δt
M k−1
i0
α i1
x i
α i1
t
α i1− τΔτ
Δt
x i1
α i1
h2t, α i1Δt
M k−1
i0
α i1
x i
h2t, α i1Δt
x i1
α i1
h2t, α i1Δt
M k−1
i0
h3x i1, α i1 − h3x i , α i1.
3.5
The proof is complete
If we apply the the inequality3.1 to different time scales, we will get some well-known and some new results
Corollary 3.3 continuous case Let T R Then our delta integral is the usual Riemann integral
from calculus Hence,
h2t, s t − s2
Trang 7This leads us to state the following inequality:
b
a
f tΔt −k
i0
α i1− α i fx i 1
2
k−1
i0
x i1− α i12fx i1 − x i − α i12fx i
6
k−1
i0
x i1− α i13− x i − α i13
,
3.7
where M supa<x<b |fx|.
Remark 3.4 The inequality 3.7 is exactly the generalized Ostrowski inequality shown in
24
Corollary 3.5 discrete case Let T Z, a 0, b n Suppose that
1 I k: 0 j0< j1< · · · < j k−1< j k n is a division of 0, n ∩ Z for j0, k1, , j k∈ Z;
2 p i ∈ Z i 0, , k 1 is “k 2” points so that p0 0, p i ∈ j i−1, j i ∩ Z i 1, , k
and p k1 n;
3 fk x k
Then,
n
j1
x j−k
i0
p i1− p i x j i1k−1
i0
h2j i1, p i1Δx j i1− h2j i , p i1Δx j i
≤ M k−1
i0
h3j i1, p i1 − h3j i , p i1
3.8
for all i 1, n, where
1≤i≤n−1Δ2x i, h k t, s t − s
k
3.9
for all t, s ∈ Z.
Corollary 3.6 quantum calculus case Let T qN 0, q > 1, a q m , b q n with m < n Suppose that
1 I k : q m q j0 < q j1 < · · · < q j k−1 < q j k q n is a division of q m , q n ∩ qN 0for j0, j1, , j k∈
N0;
2 q p i ∈ qN 0 i 0, , k 1 is “k 2” points so that q p0 q m , q p i ∈ q j i−1, q j i ∩ qN 0 i
1, , k and q p k1 q m ;
3 f : q m , q n → R is differentiable.
Trang 8
q n
q m
f qtΔt −k
i0
q p i1− q p i fq j i1
k−1
i0
h2q j i1, q p i1f q j i11 − fq j i1
q − 1q j i1 − h2q j i , q p if q j i1 − fq j i
q − 1q j i
≤ Mk−1
i0
h3q j i1, q p i1 − h3q j i , q p i1,
3.10
where
q m <t<q n
f q2t − q 1fqt qft q q − 12t2
, h k t, s k−1
ν0
t − q ν s
ν
μ0q μ , ∀t, s ∈ qN 0.
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 that a, b ∈ T with a > b and f : a, b → R is differentiable We denote
a<x<b
Proposition 4.1 Suppose that α ∈ a, b ∩ T Then one has the perturbed rectangle inequality on
time scales
b
a
f σ tΔt − α − af σ a b − αf σ b h2b, αfΔb − h2a, αfΔa
≤ Mh3b, α − h3a, α.
4.2
Proof We choose x0 a, x1 b, α0 a, α1 α ∈ a, b and α2 b inTheorem 3.1to get the result
Remark 4.2 a If we choose α b in 4.2, we get the perturbed left rectangle inequality on time scales
b
a
f σ tΔt − b − af σ a − h2a, bfΔa
Trang 9b If we choose α a in 4.2, we get the perturbed right rectangle inequality on time scales
b
a
f σ tΔt − b − af σ b h2b, afΔb
c If we choose α a b/2 in 4.2, we get the perturbed trapezoid inequality on time scales
b
a
f σ tΔt − f σ a f σ b
h2
b, a b
2
fΔb − h2
a, a b
2
fΔa
≤ M
h3
b, a b
2
− h3
a, a b
2
.
4.5
Proposition 4.3 Suppose that x ∈ a, b ∩ T, α1 ∈ a, x ∩ T and α2∈ x, b ∩ T Then one has the
perturbed inequality on time scales
b
a
f σ tΔt −α1− af σ a α2− α1f σ x b − α2f σ b
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 a and α2 b inProposition 4.3, we get exactlyTheorem 1.3 Therefore,Theorem 3.1is a generalization of Theorem 4 in20 If we choose x a b/2 in
3.1, we get the perturbed mid-point inequality on time scales
b
a
f σ tΔt − f σ
a b
2
b − a
h2
a b
− h2
a b
2 , b
fΔ
a b
2
≤ M
h3
a b
− h3
a b
2 , b
.
4.7
Corollary 4.5 Suppose that x ∈ 5ab/6, a5b/6∩T, α1 5ab/6 and α2 a5b/6.
Then one has the perturbed inequality on time scales
b
a
f σ tΔt − b − a
3
f σ a f σ b
h2
x, 5ab
6
fΔx−h2
a, 5a b
6
fΔah2
b, a 5b
6
fΔb−h2
x, a 5b
6
fΔx
≤ M
h3
x, 5a b
6
− h3
a, 5a b
6
h3
b, a 5b
6
− h3
x, a 5b
6
.
4.8
Trang 10Remark 4.6 If we choose x a b/2 in 4.8, we get the perturbed Simpson inequality on time scales
b
a
f σ tΔt − b − a
3
f σ a f σ b
a b
2
h2
a b
5a b
6
fΔ
a b
2
− h2
a, 5a b
6
fΔa
h2
b, a 5b
6
fΔb − h2
a b
a 5b
6
fΔ
a b
2
≤ M
h3
a b
5a b
6
− h3
a, 5a b
6
h3
b, a 5b
6
− h3
a b
a 5b
6
.
4.9
Corollary 4.7 Suppose that a b/2 ∈ T, α1 ∈ a, a b/2 ∩ T and α2 ∈ a b/2, b ∩ T.
Then one has the perturbed inequality on time scales
b
a
f σ tΔt −
α1− af σ a α2− α1f σ
a b
2
b − α2f σ b
h2
a b
2 , α1
fΔ
a b
2
− h2a, α1fΔa h2b, α2fΔb − h2
a b
2 , α2
fΔ
a b
2
≤ M
h3
a b
2 , α1
− h3a, α1 h3b, α2 − h3
a b
2 , α2
.
4.10
Remark 4.8 If we choose α1 3a b/4 and α2 a 3b/4 in 4.10, we get the perturbed averaged mid-point-trapezoid inequality on time scales
b
a
f σ tΔt − b − a
2
f σ a f σ b
a b
2
h2
a b
3a b
4
fΔ
a b
2
− h2
a, 3a b
4
fΔa
h2
b, a 3b
4
fΔb − h2
a b
a 3b
4
fΔ
a b
2
≤ M
h3
a b
3a b
4
− h3
a, 3a b
4
h3
b, a 3b
4
− h3
a b
a 3b
4
.
4.11