An. Şt. Univ. Ovidius Constanţa
Vol. 17(2), 2009, 101–114
A NEW GENERALIZATION OF
OSTROWSKI TYPE INEQUALITY ON
TIME SCALES
´ c-Anh Ngô and Wenbing Chen
Wenjun Liu, Quô
Abstract
In this paper, by introducing a parameter, we first extend a generalization of Ostrowski type inequality on time scales for functions whose
derivatives are bounded and then unify corresponding continuous and
discrete versions. We also point out some particular integral inequalities
on time scales as special cases.
1
Introduction
The following integral inequality was first established by Ostrowski in 1938.
Theorem 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 in (a, b), that is, kf ′ k∞ :=
sup |f ′ (x)| < ∞. Then for any x ∈ [a, b], we have the inequality:
t∈(a,b)
Z b
1
f (t)dt 6
f (x) −
b−a a
2 !
x − a+b
1
2
(b − a)kf ′ k∞ .
+
4
(b − a)2
The inequality is sharp in the sense that the constant
a smaller one.
1
4
(1)
cannot be replaced by
Key Words: Ostrowski’s inequality; generalization; time scales; Simpson inequality;
trapezoid inequality; mid-point inequality.
Mathematics Subject Classification: 26D15, 39A10, 39A12, 39A13.
Received: May, 2009
Accepted: September, 2009
101
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
102
For some extensions, generalizations and similar results, please see [6, 8, 9,
14, 15, 19, 20, 21] and references therein.
The development of the theory of time scales was initiated by Hilger [10]
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, 4, 5, 7, 11, 13]. In [5], Bohner and Matthews established the following
so-called Ostrowski’s inequality on time scales.
Theorem 2 (See [5], Theorem 3.5) Let a, b, s, t ∈ T, a < b and f : [a, b] → R
be differentiable. Then
Z b
1
M1 h2 (t, a) + h2 (t, b) ,
(2)
f σ (s)∆s 6
f (t) −
b−a
b−a a
where M1 = sup |f ∆ (t)|. This inequality is sharp in the sense that the righta<t<b
hand side of (2) cannot be replaced by a smaller one.
Liu and Ngô [16] then generalized the above Ostrowski inequality on time
scales for k points x1 , x2 , · · · , xk for functions whose derivatives are bounded.
They also extended the result by considering functions whose second derivatives are bounded in [17]. They obtained:
Theorem 3 Let a, b, x, t ∈ T, a < b and f : [a, b] → R be a twice differentiable function on (a, b) and f ∆∆ : (a, b) → R is bounded, i.e. M2 :=
sup |f ∆∆ (x)| < ∞. Then we have
a<t<b
Z
b
σ
σ
∆
f (t)∆t − f (x)(b − a) + (h2 (x, a) − h2 (x, b))f (x) 6 M2 h3 (x, a)−h3 (x, b) .
a
The Theorem 3 may be thought of as a perturbed version of the Theorem
2. In [18], the authors derive a perturbed Ostrowski type inequality on time
scales for k points x1 , x2 , · · · , xk for functions whose second derivatives are
bounded.
In the present paper, by introducing a parameter, we shall first extend
another generalization of Ostrowski type inequality on time scales for functions
whose derivatives are bounded and then unify corresponding continuous and
discrete versions. We also point out some particular integral inequalities on
time scales as special cases.
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
2
103
The generalized Ostrowski type inequality on time
scales
A time scale T is an arbitrary nonempty closed subset of the real numbers.
For a general introduction to the theory of time scale we refer the reader to
Hilger [10] (see also [16, 17, 18]) and the books [2, 3, 12] .
Definition 1 Let hk : T2 → R, k ∈ N0 be defined by
h0 (t, s) = 1
for all
s, t ∈ T
and then recursively by
hk+1 (t, s) =
Z
t
hk (τ, s) ∆τ
for all
s, t ∈ T .
(3)
s
Our main result reads as follow.
Theorem 4 Let a, b, s, t ∈ T, a < b and f : [a, b] → R be differentiable. Then
Z b
f (a) + f (b)
1
σ
−
f (s)∆s
(1 − λ)f (t) + λ
2
b−a a
M
b−a
b−a
6
(4)
h2 a, a + λ
+ h2 t, a + λ
b−a
2
2
!
b−a
b−a
+ h2 b, b − λ
+h2 t, b − λ
2
2
for all λ ∈ [0, 1] such that a + λ(b − a)/2 and b − λ(b − a)/2 are in T and
b−a
∆
t ∈ [a + λ b−a
2 , b − λ 2 ] ∩ T, where M := sup |f (t)| < ∞. This inequality
a<t<b
is sharp provided
λ2
λ
2
a(b − a) +
(b − a) 6
2
4
Z
a+λ b−a
2
s∆s.
(5)
a
Remark 1 We note that the condition (5) is trivial if λ = 0.
To prove the Theorem 4, we need the following Generalized Montgomery
Identity. This is motivated by the ideas of Dragomir et al. in [9], where the
continuous version of a generalized Ostrowski integral inequality for mappings
whose derivatives are bounded was proved.
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
104
Lemma 1 (Generalized Montgomery Identity) Under the assumptions
of Theorem 4, we have
(1 − λ)f (t) + λ
where
1
f (a) + f (b)
=
2
b−a
Z
b
f σ (s)∆s +
a

b−a


,
s
−
a
+
λ


2
K(t, s) =

b−a


,
 s− b−λ
2
1
b−a
Z
b
K(t, s)f ∆ (s)∆s,
a
s ∈ [a, t),
(6)
s ∈ [t, b].
Proof. Integrating by parts, we have
Z
b
K(t, s)f ∆ (s)∆s
a
Z t
Z b
b−a
b−a
s− a+λ
s− b−λ
f ∆ (s)∆s +
f ∆ (s)∆s
2
2
a
t
Z t
b−a
λ
= t− a+λ
f σ (s)∆s
f (t) + (b − a)f (a) −
2
2
a
Z b
λ
b−a
f (t) + (b − a)f (b) −
f σ (s)∆s
− t− b−λ
2
2
t
Z b
f (a) + f (b)
f σ (s)∆s,
−
=(b − a) (1 − λ)f (t) + λ
2
a
=
from which we get the desired identity.
Corollary 1 (Continuous case) Let T = R. Then
f (a) + f (b)
1
(1−λ)f (t)+λ
=
2
b−a
Z
b
a
1
f (s) ds+
b−a
Z
b
K(t, s)f ′ (s) ds. (7)
a
This is the Montgomery identity in the continuous case, which can be found
in [9].
Corollary 2 (Discrete case) Let T = Z, a = 0, b = n, s = j, t = i and
f (k) = xk . Then
(1 − λ)xi + λ
n
n−1
1X
1X
x0 + xn
=
xj +
K (i, j) ∆xj ,
2
n j=1
n j=0
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
105
where
K(i, 0) = −
nλ
,
2
nλ
for 1 6 j 6 n − 1,
K(1, j) = j − n −
2
nλ
K(n, j) = j −
for 0 6 j 6 n − 1,
( 2
j ∈ [0, i),
j − nλ
2 ,
K(i, j) =
j − n − nλ
j ∈ [i, n − 1],
2 ,
as we just need 1 6 i 6 n and 1 6 j 6 n − 1.
Corollary 3 (Quantum calculus case) Let T = q N0 , q > 1, a = q m , b =
q n with m < n. Then
(1 − λ)f (t) + λ
n−1
P
q k f (q k+1 )
= k=m n−1
P
k=m
where
f (q m ) + f (q n )
2
+
qk
n−1
X
1
f (q k+1 ) − f (q k ) K(t, q k ),
n
m
q −q
k=m

qm − qn

k
m

,
q
−
q
+
λ


2
k
K(t, q ) =

qm − qn

k
n

,
 q − q −λ
2
q k ∈ [q m , t),
q k ∈ [t, q n ].
Proof. (Proof of the Theorem 4) By applying Lemma 1, we get
Z b
f (a) + f (b)
1
σ
f (s)∆s
−
(1 − λ)f (t) + λ
2
b−a a
Z
1 b
6
K(t, s)f ∆ (s)∆s
b−a a
!
Z b
Z t
M
6
|K (t, s)| ∆s
|K (t, s)| ∆s +
b−a
t
a
!
Z b
Z t
b
−
a
b
−
a
M
s − b − λ
s − a + λ
∆s +
∆s
=
b−a
2
2
t
a
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
106
Z t
Z a+λ b−a
2
b − a M
s − a + λ b − a ∆s
s
−
a
+
λ
∆s
+
=
b−a
2
2
a+λ b−a
a
2
!
Z b−λ b−a
Z b
2
b − a b − a +
s − b − λ 2
s − b − λ 2
∆s +
∆s
t
b−λ b−a
2
Z t
Z a
b−a
b−a
M
s− a+λ
s− a+λ
∆s +
∆s
=
b−a
2
2
a+λ b−a
a+λ b−a
2
2
!
Z b
Z t
b−a
b−a
s− b−λ
s− b−λ
∆s +
∆s
+
2
2
b−λ b−a
b−λ b−a
2
2
M
b−a
b−a
=
h2 a, a + λ
+ h2 t, a + λ
b−a
2
2
b−a
b−a
+ h2 b, b − λ
,
+h2 t, b − λ
2
2
which completes the first part of our proof.
To prove the sharpness of this inequality, let f (t) = t, t = b − λ b−a
2 . It
follows that M = 1. Starting with the righ-hand side of (4), we have
b−a
b−a
M
h2 a, a + λ
+ h2 t, a + λ
b−a
2
2
!
b−a
b−a
+ h2 t, b − λ
+ h2 b, b − λ
2
2
!
b−a
b−a
b−a
b−a
1
h2 a, a + λ
+ h2 b − λ
+ h2 b, b − λ
.
,a + λ
=
b−a
2
2
2
2
Moreover,
h2
b−a
a, a + λ
2
=
Z
=
Z
=
Z
a
a+λ b−a
2
a
a+λ b−a
2
a
a+λ b−a
2
b−a
s− a+λ
∆s
2
b−a
b−a
s∆s − a + λ
a− a+λ
2
2
b−a
b−a
s∆s + a + λ
λ
.
2
2
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
107
Z b−λ b−a 2
b−a
b−a
b−a
s− a+λ
h2 b − λ
=
∆s
,a + λ
2
2
2
a+λ b−a
2
Z b−λ b−a
2
b−a
b−a
b−a
b−λ
− a+λ
s∆s − a + λ
=
2
2
2
a+λ b−a
2
Z b−λ b−a
2
b−a
=
(b − a) (1 − λ) .
s∆s − a + λ
b−a
2
a+λ 2
Z b
b−a
b−a
h2 b, b − λ
s− b−λ
=
∆s
2
2
b−λ b−a
2
Z b
b−a
b−a
b− b−λ
=
s∆s − b − λ
2
2
b−λ b−a
2
Z b
b−a
b−a
=
s∆s − b − λ
λ
.
b−a
2
2
b−λ 2
Thus, in this situation, the right-hand side of (4) equals to
!
Z b
Z a+λ b−a
Z b−λ b−a
2
2
1
s∆s +
s∆s
−
s∆s +
b−a
b−λ b−a
a
a+λ b−a
2
2
b−a λ
b−a λ
b−a
+ a+λ
(1 − λ) − b − λ
− a+λ
2
2
2
2
2
!
Z a+λ b−a
Z
b
2
1
−2
s∆s − (a + λ(b − a)) (1 − λ) .
s∆s +
=
b−a
a
a
Starting with the left-hand side of (4), we have
Z b
1
f (a) + f (b)
σ
f (s) ∆s
−
(1 − λ) f (t) + λ
2
b−a a
Z b
a+b
1
b−a
+λ
−
σ (s) ∆s
= (1 − λ) b − λ
2
2
b−a a
Z b
a+b
1
b−a
s∆s − b − a
+λ
+
= (1 − λ) b − λ
2
2
b−a a
Z
b
λ
1
= −λ 1 −
(b − a) − a +
s∆s ,
2
b−a a
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
108
where we have used
Z
b
σ (s) ∆s =
a
So, if
Z
b
a
σ(s)+s ∆s−
Z
b
s∆s =
a
Z
b
(s2 )∆ ∆s−
a
λ
λ2
2
a(b − a) +
(b − a) 6
2
4
Z
Z
b
s∆s = b2 −a2 −
a
Z
b
s∆s.
a
a+λ b−a
2
s∆s
a
holds true, then
Z b
λ
1
(b − a) − a +
s∆s
−λ 1−
2
b−a a
Z b
λ
1
> −λ 1 −
(b − a) − a +
s∆s
2
b−a a
!
Z a+λ b−a
Z b
2
1
>
−2
s∆s − a + λ(b − a) (1 − λ) ,
s∆s +
b−a
a
a
which helps us to complete our proof.
If we apply the the inequality (4) to different time scales, we will get some
well-known and some new results.
Corollary 4 (Continuous case) Let T = R. Then our delta integral is the
usual Riemann integral from calculus. Hence,
2
h2 (t, s) =
(t − s)
,
2
for all
t, s ∈ R.
This leads us to state the following inequality
Z b
f (a) + f (b)
1
−
f (s) ds
(1 − λ)f (t) + λ
2
b−a a
6M
1
1
(b − a) (1 − λ)2 + λ2 +
4
b−a
a+b
x−
2
2 !
b−a
′
for all λ ∈ [0, 1] and a + λ b−a
2 6 t 6 b − λ 2 , where M = sup |f (x)| < ∞,
x∈(a,b)
which is exactly the generalized Ostrowski type inequality shown in Theorem 2
of [9].
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
Corollary 5 (Discrete case) Let T =
f (k) = xk . Thus, we have
n
X
M
(1 − λ)xi + λ x0 + xn − 1
x
j 6
2
n j=1 n
for all λ ∈ [0, 1] such that λn
2 and n −
where M = max |∆xi | < ∞.
109
Z, a = 0, b = n, s = j, t = i and
!
2
2
2
n
+
1
(2λ
−
2λ
+
1)n
−
1
i −
+
2 4
λn
2
are in Z and i ∈
16i6n−1
Proof. In this situation, it is known that
!
t−s
,
for all
hk (t, s) =
k
λn
2
,n −
λn
2
∩ T,
t, s ∈ Z.
Therefore,
b−a
=
h2 a, a + λ
2
b−a
h2 t, a + λ
=
2
b−a
=
h2 t, b − λ
2
− nλ
2
2
i−
!
nλ
2
=
!
2
i−n+
and
h2
b−a
t, b − λ
2
=
+1
2
i−
=
nλ
2
2
nλ
2
nλ
2
!
=
nλ
2
2
nλ
2
,
i−
2
i−n+
!
=
nλ
2
nλ
2
nλ
2
nλ
2
2
−1
,
i−n+
2
−1
nλ
2
−1
,
,
Thus, we get the desired result.
Corollary 6 (Quantum calculus case). Let T = q N0 , q > 1, a = q m , b = q n
with m < n. Then
Z qn
1
f (q m ) + f (q n )
σ
f
(s)∆s
− n
(1 − λ)f (t) + λ
2
q − q m qm
6
M
2t2 − (1 + q)(q m + q n )t
(1 + q)(q n − q m )
!
3
λ m
2
2m+1
2n+1
m+n+1
n 2
+
2λ − λ + 1 (q
+q
) − λ(3 − 2λ)q
+ (q − q )
2
2
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
110
n
m
n
m
for all λ ∈ [0, 1] such that q m + λ q −q
and q n − λ q −q
are in T and
2
2
qn − qm
qn − qm n
∩T
,q − λ
t ∈ qm + λ
2
2
where
M=
f (qt) − f (t) (q − 1)t .
t∈(q m ,q n )
sup
Proof. In this situation, one has
hk (t, s) =
and
k−1
Y
t − qν s
,
ν
P
ν=0
qµ
for all
t, s ∈ T
µ=0
f ∆ (t) =
f (qt) − f (t)
.
(q − 1)t
Therefore,
m
m+1 λ n+1 λ m
λ
n
− 2q
qn − qm
m m
2 (q − q ) q − 1 − 2 q
=
,
h2 q , q + λ
2
1+q
t − 1 − λ2 q m − λ2 q n t − 1 − λ2 q m+1 − λ2 q n+1
qn − qm
m
h2 t, q + λ
=
,
2
1+q
t − 1 − λ2 q n − λ2 q m t − 1 − λ2 q n+1 − λ2 q m+1
qn − qm
=
,
h2 t, q n − λ
2
1+q
and
h2
qn − qm
q ,q − λ
2
m
n
=
λ n
2 (q
− q m ) q n − 1 − λ2 q n+1 − λ2 q m+1
.
1+q
Thus, we get the result.
3
Some particular integral inequalities on time scales
In this section we point out some particular integral inequalities on time scales
as special cases, such as: rectangle inequality on time scales, trapezoid inequality on time scales, mid-point inequality on time scales, Simpson inequality on
time scales, averaged mid-point-trapezoid inequality on time scales and others.
Throughout this section, we always assume T is a time scale; a, b ∈ T with
a < b; f : [a, b] → R is differentiable. We denote
M = sup |f ∆ (x)|.
a<x<b
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
Corollary 7 Under the assumptions of Theorem 4 with λ = 1 and t =
T. Then we have the trapezoid inequality on time scales
111
a+b
2
∈
Z b
f (a) + f (b)
a+b
a+b
1
M
σ
h2 a,
+ h2 b,
.
−
f (s)∆s 6
b−a
2
b−a a
2
2
(8)
Remark 2 If we take λ = 0 in Theorem 4, then Theorem 2 is recaptured.
Therefore, Theorem 4 may be regarded as a generalization of Theorem 2.
Corollary 8 Under the assumptions of Theorem 4 with λ = 13 . Then we have
the following integral inequality on time scales
Z b
1
1
f σ (s)∆s
(f (a) + f (b) + 4f (t)) −
6
b−a a
M
5a + b
5a + b
6
h2 a,
+ h2 t,
b−a
6
6
!
a + 5b
a + 5b
+h2 t,
+ h2 b,
6
6
for all t ∈
5a+b
6
(9)
, a+5b
∩ T.
6
Remark 3 If we choose t =
time scales
a+b
2
in (9), we get the Simpson inequality on
Z b
1
1
a+b
f (a) + 4f
+ f (b) −
f σ (s)∆s
6
2
b−a a
5a + b
a + b 5a + b
M
h2 a,
+ h2
,
6
b−a
6
2
6
!
a + b a + 5b
a + 5b
+h2
+ h2 b,
.
,
2
6
6
Corollary 9 Under the assumptions of Theorem 4 with λ = 12 . Then we have
112
Wenjun J. Liu, Qu´ôc-Anh Ngô and Wenbing B. Chen
the following integral inequality on time scales
Z b
1 f (a) + f (b)
1
f σ (s)∆s
+ f (t) −
2
2
b−a a
M
3a + b
3a + b
6
h2 a,
+ h2 t,
b−a
4
4
!
a + 3b
a + 3b
+ h2 b,
+h2 t,
4
4
a+3b
for all t ∈ 3a+b
∩ T.
4 , 4
(10)
Remark 4 If we choose t = a+b
in (10), we get the averaged mid-point2
trapezoid inequality on time scales
Z b
1 f (a) + f (b)
1
a+b
σ
−
+f
f (s)∆s
2
2
2
b−a a
M
3a + b
a + b 3a + b
6
h2 a,
+ h2
,
b−a
4
2
4
!
a + 3b
a + b a + 3b
+ h2 b,
.
,
+h2
2
4
4
Corollary 10 Under the assumptions of Theorem 4 with t = a+b
2 ∈ T. Then
we have the following integral inequality on time scales
Z b
a+b
f (a) + f (b)
1
+λ
−
f σ (s)∆s
(1 − λ)f
2
2
b−a a
M
b−a
a+b
b−a
6
h2 a, a + λ
+ h2
,a + λ
(11)
b−a
2
2
2
!
b−a
b−a
a+b
+ h2 b, b − λ
,b − λ
+h2
2
2
2
for all λ ∈ [0, 1] such that a + λ(b − a)/2 and b − λ(b − a)/2 are in T.
Remark 5 If we choose λ = 0 in (11), we get the mid-point inequality on
time scales
!
Z b
a+b
M
a+b
1
a+b
σ
−
h2
, a + h2
,b
.
f (t)∆t 6
f
b−a
2
b−a a
2
2
A NEW GENERALIZATION OF OSTROWSKI TYPE INEQUALITY ON TIME
SCALES
113
Acknowledgements
The work was supported by the Science Research Foundation of Nanjing University of Information Science and Technology (Grant No. 20080295) and
the Natural Science Foundation of the Jiangsu Higher Education Institutions
(Grant No. 09KJB110005).
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(Wenjun J. Liu) College of Mathematics and Physics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China,
E-mail: wjliu@nuist.edu.cn
´ c-Anh Ngô) Department of Mathematics,
(Quô
College of Science,
Viê.t Nam National University,
Hà Nô.i, Viê.t Nam,
E-mail: bookworm vn@yahoo.com
(Wenbing B. Chen) College of Mathematics and Physics,
Nanjing University of Information Science and Technology,
Nanjing 210044, China,
E-mail: chenwb@nuist.edu.cn