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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 597241, 12 pages
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,
1
Qu
´
ˆoc Anh Ng
ˆ
o,
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,
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.
Copyright q 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 researchers 1–9.
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
∞
: sup
t∈a,b
|f
x| < ∞.Then
for any x ∈ a, b, the following inequality holds:
fx −
1
b − a
b
a
ftdt
≤
1
4
x − a b/2
2
b − a
2
b − af
∞
. 1.1
The inequality is sharp in the sense that the constant 1/4 cannot be replaced by a smaller one.
In 1988, Hilger 10 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
2 Journal of Inequalities and Applications
studied the theory of certain integral inequalities on time scales. For example, we refer
the reader to 11–18.In15, 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
≤ Mh
2
x, ah
2
x, b, 1.2
where h
k
·, · is defined by Definition 2.9 below and M sup
a<x<b
|f
Δ
x|. This inequality is sharp
in the sense that the right-hand side of 1.1 cannot be replaced by a smaller one.
Liu and Ng
ˆ
o then generalize the above Ostrowski inequality on time scales for k points
x
1
,x
2
, ,x
k
in 19. They also extended the result by considering functions whose second
derivatives are bounded in 20. 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 f
ΔΔ
: a, b → R bounded, that is, M : sup
a<t<b
|f
ΔΔ
x| < ∞.Then
b
a
f
σ
tΔt − f
σ
xb − ah
2
x, a − h
2
x, bf
Δ
x
≤ Mh
3
x, a − h
3
x, b. 1.3
Theorem 1.3 may be thought of as a perturbed version of Theorem 1.2. In the present
paper we will first derive a perturbed Ostrowski-type i nequality on time scales for k
points x
1
,x
2
, ,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 Hilger 10
and the books 21–23 for further details see also 19, 20.
Definition 2.1. A time scale T 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 σtinf{s ∈ T :
s>t}, while the backward jump operator ρ : T → T is defined by ρtsup{s ∈ T : s<t}.
In this definition, we put inf ∅ sup T i.e., σtt 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 σtt and
t<sup T, then t is called right dense,andifρtt and t>inf T, then t is called left dense.
Points that are both right dense and left dense are called dense.
Wenjun Liu et al. 3
Definition 2.3. Let t ∈ T, then two mappings μ, ν : T → 0, ∞ satisfying
μt : σt − t, νt : t − ρt2.1
are called the graininess functions.
We now introduce the set T
κ
which is derived from the time scales T as follows. If T
has a left-scattered maximum t, then T
κ
: T −{t}, otherwise T
κ
: T. Furthermore, for a
function f : T → R, we define the function f
σ
: T → R by f
σ
tfσt for all t ∈ T.
Definition 2.4. Let f : T → R be a function on time scales. Then for t ∈ T
κ
, one defines f
Δ
t
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
f
σ
t − fs − f
Δ
tσt − s
≤ ε|σt − s|. 2.2
We say that f is Δ-differentiable on T
κ
provided f
Δ
t exists for all t ∈ T
κ
. We talk about the
second derivative f
ΔΔ
provided f
Δ
is differentiable on T
κ
2
T
κ
κ
with derivative f
ΔΔ
f
Δ
Δ
: T
κ
2
→ R.
Definition 2.5. A mapping f : T → R is called rd-continuous denoted by f ∈ C
rd
provided
that it satisfies
1 f is continuous at each right-dense point of T;
2 the left-sided limit lim
s → t−
fsft− 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
F
Δ
tft holds for all t ∈ T
κ
. Then the Δ-integral of f is defined by
b
a
ftΔt Fb − Fa. 2.3
Proposition 2.8. Let f, g be rd-continuous, a, b, c ∈ T, and α, β ∈ R.Then
1
b
a
αftβgtΔt α
b
a
ftΔt β
b
a
gtΔt,
2
b
a
ftΔt −
a
b
ftΔt,
3
b
a
ftΔt
c
a
ftΔt
b
c
ftΔt,
4
b
a
ftg
Δ
tΔt fgb − fga −
b
a
f
Δ
tgσtΔt,
5
a
a
ftΔt 0,
6 If ft ≥ 0 for all a ≤ t<bthen
b
a
ftΔt ≥ 0.
4 Journal of Inequalities and Applications
Definition 2.9. Let h
k
: T
2
→ R, k ∈ N
0
be defined by
h
0
t, s1 ∀s, t ∈ T 2.4
and then recursively by
h
k1
t, s
t
s
h
k
τ,sΔτ ∀s, t ∈ T. 2.5
Remark 2.10. It follows from Proposition 2.86 that if s ≤ t, then h
k1
t, s ≥ 0 for all t, s ∈ T
and all k ∈ N.
Remark 2.11. If we let h
Δ
k
t, s denote for each fixed s the derivative of h
k
t, s with respect to
t, then
h
Δ
k
t, sh
k−1
t, s, for k ∈ N,t∈ T
κ
. 2.6
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 x
0
<x
1
< ··· <x
k−1
<x
k
b is a division of the interval a, b for
x
0
,x
1
, ,x
k
∈ T;
2 α
i
∈ T i 0, ,k 1 is “k 2” points so that α
0
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 f
ΔΔ
: a, b → R is
bounded, that is, M : sup
a<t<b
|f
ΔΔ
t| < ∞.
Then
b
a
f
σ
tΔt −
k
i0
α
i1
− α
i
f
σ
x
i
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
≤ M
k−1
i0
h
3
x
i1
,α
i1
− h
3
x
i
,α
i1
.
3.1
To prove Theorem 3.1, we need the following generalized montgomery identity for
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.
Wenjun Liu et al. 5
Lemma 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
Kt, I
k
f
ΔΔ
Δt
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
,
3.2
where
Kt, I
k
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
h
2
t, α
1
,t∈ a, x
1
,
h
2
t, α
2
,t∈ x
1
,x
2
,
.
.
.
h
2
t, α
k−1
,t∈ x
k−2
,x
k−1
,
h
2
t, α
k
,t∈ x
k−1
,b.
3.3
Proof. Integrating by parts and applying Proposition 2.84 , we have
b
a
Kt, I
k
f
ΔΔ
tΔt
k−1
i0
x
i1
x
i
Kt, I
k
f
ΔΔ
tΔt
k−1
i0
x
i1
x
i
h
2
t, α
i1
f
ΔΔ
tΔt
k−1
i0
α
i1
x
i
h
2
t, α
i1
f
ΔΔ
tΔt
x
i1
α
i1
h
2
t, α
i1
f
ΔΔ
tΔt
k−1
i0
h
2
α
i1
,α
i1
f
Δ
α
i1
− h
2
x
i
,α
i1
f
Δ
x
i
−
α
i1
x
i
f
Δ
σth
Δ
2
t, α
i1
Δt
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
α
i1
,α
i1
f
Δ
α
i1
−
x
i1
α
i1
f
Δ
σth
Δ
2
t, α
i1
Δt
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
−
α
i1
x
i
f
Δ
σtt − α
i1
Δt −
x
i1
α
i1
f
Δ
σtt − α
i1
Δt
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
f
σ
x
i
x
i
− α
i1
α
i1
x
1
f
σ
tΔt − f
σ
x
i1
x
i1
− α
i1
x
i1
α
i1
f
σ
tΔt
6 Journal of Inequalities and Applications
b
a
f
σ
tΔt
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
f
σ
aa − α
1
k−1
i1
f
σ
x
i
x
i
− α
i1
− f
σ
bb − α
k
−
k−2
i0
f
σ
x
i1
x
i1
− α
i1
b
a
f
σ
tΔt −
k
i0
f
σ
x
i
α
i1
− α
i
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
,
3.4
that is, 3.2 holds.
Proof of Theorem 3.1. By applying Lemma 3.2,weget
b
a
f
σ
tΔt −
k
i0
α
i1
− α
i
f
σ
x
i
k−1
i0
h
2
x
i1
,α
i1
f
Δ
x
i1
− h
2
x
i
,α
i1
f
Δ
x
i
b
a
Kt, I
k
f
ΔΔ
tΔt
k−1
i0
x
i1
x
i
Kt, I
k
f
ΔΔ
tΔt
≤
k−1
i0
x
i1
x
i
|Kt, I
k
||f
ΔΔ
t|Δt ≤ M
k−1
i0
x
i1
x
i
|h
2
t, α
i1
|Δt
M
k−1
i0
α
i1
x
i
α
i1
t
α
i1
− τΔτ
Δt
x
i1
α
i1
h
2
t, α
i1
Δt
M
k−1
i0
α
i1
x
i
h
2
t, α
i1
Δt
x
i1
α
i1
h
2
t, α
i1
Δt
M
k−1
i0
h
3
x
i1
,α
i1
− h
3
x
i
,α
i1
.
3.5
The proof is complete.
If we apply the the inequality 3.1 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,
h
2
t, s
t − s
2
2
, ∀t, s ∈ R. 3.6
Wenjun Liu et al. 7
This leads us to state the following inequality:
b
a
ftΔt −
k
i0
α
i1
− α
i
fx
i
1
2
k−1
i0
x
i1
− α
i1
2
f
x
i1
− x
i
− α
i1
2
f
x
i
≤
M
6
k−1
i0
x
i1
− α
i1
3
− x
i
− α
i1
3
,
3.7
where M sup
a<x<b
|f
x|.
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 j
0
<j
1
< ··· <j
k−1
<j
k
n is a division of 0,n ∩ Z for j
0
,k
1
, ,j
k
∈ Z;
2 p
i
∈ Z i 0, ,k 1 is “k 2” points so that p
0
0, p
i
∈ j
i−1
,j
i
∩ Z i 1, ,k
and p
k1
n;
3 fkx
k
.
Then,
n
j1
x
j
−
k
i0
p
i1
− p
i
x
j
i
1
k−1
i0
h
2
j
i1
,p
i1
Δx
j
i
1
− h
2
j
i
,p
i1
Δx
j
i
≤ M
k−1
i0
h
3
j
i1
,p
i1
− h
3
j
i
,p
i1
3.8
for all i
1,n,where
M sup
1≤i≤n−1
Δ
2
x
i
,h
k
t, s
t − s
k
3.9
for all t, s ∈ Z.
Corollary 3.6 quantum calculus case. Let T q
N
0
, q>1, a q
m
,b q
n
with m<n. Suppose
that
1 I
k
: q
m
q
j
0
<q
j
1
< ···<q
j
k−1
<q
j
k
q
n
is a division of q
m
,q
n
∩ q
N
0
for j
0
,j
1
, ,j
k
∈
N
0
;
2 q
p
i
∈ q
N
0
i 0, ,k 1 is “k 2” points so that q
p
0
q
m
, q
p
i
∈ q
j
i−1
,q
j
i
∩ q
N
0
i
1, ,k and q
p
k1
q
m
;
3 f : q
m
,q
n
→ R is differentiable.
8 Journal of Inequalities and Applications
Then,
q
n
q
m
fqtΔt −
k
i0
q
p
i1
− q
p
i
fq
j
i
1
k−1
i0
h
2
q
j
i1
,q
p
i1
fq
j
i1
1
− fq
j
i1
q − 1q
j
i1
− h
2
q
j
i
,q
p
i
fq
j
i
1
− fq
j
i
q − 1q
j
i
≤ M
k−1
i0
h
3
q
j
i1
,q
p
i1
− h
3
q
j
i
,q
p
i1
,
3.10
where
M sup
q
m
<t<q
n
fq
2
t − q 1fqtqft
qq − 1
2
t
2
,h
k
t, s
k−1
ν0
t − q
ν
s
ν
μ0
q
μ
, ∀t, s ∈ q
N
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>band f : a, b → R is differentiable. We denote
M sup
a<x<b
f
ΔΔ
x
< ∞. 4.1
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
ab − αf
σ
b
h
2
b, αf
Δ
b − h
2
a, αf
Δ
a
≤ Mh
3
b, α − h
3
a, α.
4.2
Proof. We choose x
0
a, x
1
b, α
0
a, α
1
α ∈ a, b and α
2
b in Theorem 3.1 to 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 − h
2
a, bf
Δ
a
≤−Mh
3
a, b. 4.3
Wenjun Liu et al. 9
b 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
bh
2
b, af
Δ
b
≤ Mh
3
b, a. 4.4
c If we choose α a b/2in4.2, we get the perturbed trapezoid inequality on
time scales
b
a
f
σ
tΔt −
f
σ
af
σ
b
2
b − a
h
2
b,
a b
2
f
Δ
b − h
2
a,
a b
2
f
Δ
a
≤ M
h
3
b,
a b
2
− h
3
a,
a b
2
.
4.5
Proposition 4.3. Suppose that x ∈ a, b ∩ T, α
1
∈ a, x ∩ T and α
2
∈ x, b ∩ T. Then one has the
perturbed inequality on time scales
b
a
f
σ
tΔt −
α
1
− af
σ
aα
2
− α
1
f
σ
xb − α
2
f
σ
b
h
2
x, α
1
f
Δ
x − h
2
a, α
1
f
Δ
ah
2
b, α
2
f
Δ
b − h
2
x, α
2
f
Δ
x
≤ Mh
3
x, α
1
− h
3
a, α
1
h
3
b, α
2
− h
3
x, α
2
.
4.6
Remark 4.4. If we choose α
1
a and α
2
b in Proposition 4.3, we get exactly Theorem 1.3.
Therefore, Theorem 3.1 is a generalization of Theorem 4 in 20. If we choose x a b/2in
3.1, we get the perturbed mid-point inequality on time scales
b
a
f
σ
tΔt − f
σ
a b
2
b − a
h
2
a b
2
,a
− h
2
a b
2
,b
f
Δ
a b
2
≤ M
h
3
a b
2
,a
− h
3
a b
2
,b
.
4.7
Corollary 4.5. Suppose that x ∈ 5a b/6, a 5b/6 ∩ T, α
1
5a b/6 and α
2
a5b/6.
Then one has the perturbed inequality on time scales
b
a
f
σ
tΔt −
b − a
3
f
σ
af
σ
b
2
2f
σ
x
h
2
x,
5ab
6
f
Δ
x−h
2
a,
5a b
6
f
Δ
ah
2
b,
a 5b
6
f
Δ
b−h
2
x,
a 5b
6
f
Δ
x
≤ M
h
3
x,
5a b
6
− h
3
a,
5a b
6
h
3
b,
a 5b
6
− h
3
x,
a 5b
6
.
4.8
10 Journal of Inequalities and Applications
Remark 4.6. If we choose x a b/2in4.8, we get the perturbed Simpson inequality on
time scales
b
a
f
σ
tΔt −
b − a
3
f
σ
af
σ
b
2
2f
σ
a b
2
h
2
a b
2
,
5a b
6
f
Δ
a b
2
− h
2
a,
5a b
6
f
Δ
a
h
2
b,
a 5b
6
f
Δ
b − h
2
a b
2
,
a 5b
6
f
Δ
a b
2
≤ M
h
3
a b
2
,
5a b
6
− h
3
a,
5a b
6
h
3
b,
a 5b
6
− h
3
a b
2
,
a 5b
6
.
4.9
Corollary 4.7. Suppose that a b/2 ∈ T, α
1
∈ a, a b/2 ∩ T and α
2
∈ a b/2,b ∩ T.
Then one has the perturbed inequality on time scales
b
a
f
σ
tΔt −
α
1
− af
σ
aα
2
− α
1
f
σ
a b
2
b − α
2
f
σ
b
h
2
a b
2
,α
1
f
Δ
a b
2
− h
2
a, α
1
f
Δ
ah
2
b, α
2
f
Δ
b − h
2
a b
2
,α
2
f
Δ
a b
2
≤ M
h
3
a b
2
,α
1
− h
3
a, α
1
h
3
b, α
2
− h
3
a b
2
,α
2
.
4.10
Remark 4.8. If we choose α
1
3a b/4andα
2
a 3b/4in4.10, we get the perturbed
averaged mid-point-trapezoid inequality on time scales
b
a
f
σ
tΔt −
b − a
2
f
σ
af
σ
b
2
f
σ
a b
2
h
2
a b
2
,
3a b
4
f
Δ
a b
2
− h
2
a,
3a b
4
f
Δ
a
h
2
b,
a 3b
4
f
Δ
b − h
2
a b
2
,
a 3b
4
f
Δ
a b
2
≤ M
h
3
a b
2
,
3a b
4
− h
3
a,
3a b
4
h
3
b,
a 3b
4
− h
3
a b
2
,
a 3b
4
.
4.11
Wenjun Liu et al. 11
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.
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