DSpace at VNU: A generalization of Ostrowski inequality on time scales for k points

Applied Mathematics and Computation 203 (2008) 754–760

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Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc

A generalization of Ostrowski inequality on time scales for k points
Wenjun Liu a,*, Quõc-Anh Ngô b
a
b

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, Viêt Nam

a r t i c l e

i n f o

Keywords:
Ostrowski inequality
Time scales
Simpson inequality
Trapezoid inequality
Mid-point inequality

a b s t r a c t
In this paper we first generalize the Ostrowski inequality on time scales for k points and
then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on time scales as special cases.
Ó 2008 Elsevier Inc. All rights reserved.

1. Introduction
In 1938, Ostrowski proved the following interesting integral inequality which has received considerable attention from
many researchers [10–12,14,15].
Theorem 1. Let f : ½a; bŠ ! R be continuous on ½a; bŠ and differentiable in ða; bÞ and its derivative f 0 : ða; bÞ ! R is bounded in
ða; bÞ, that is, kf 0 k1 :¼ supt2ða;bÞ jf 0 ðxÞj < 1. Then for any x 2 ½a; bŠ, we have the inequality


Z

 b


f ðtÞdt À f ðxÞðb À aÞ 6


 a


2 !
ðb À aÞ2
aþb
kf 0 k1 :
þ xÀ
2
4

The inequality is sharp in the sense that the constant

1
4

cannot be replaced by a smaller one.

The development of the theory of time scales was initiated by Hilger [8] 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 or dynamic equations on time scales. For example, we refer the reader to [1,4,5,7,13,16–18]. In [5], Bohner
and Matthews established the following so-called Ostrowski inequality on time scales.
Theorem 2 (See [5], Theorem 3.5). Let a; b; x; t 2 T, a < b and f : ½a; bŠ ! R be differentiable. Then


Z

 b


r
f ðtÞDt À f ðxÞðb À aÞ 6 M ðh2 ðx; aÞ þ h2 ðx; bÞÞ;


 a

ð1Þ

where h2 ðÁ; ÁÞ is defined by Definition 7 and M ¼ supacannot be replaced by a smaller one.

* Corresponding author.
E-mail addresses: (W. Liu), (Q.-A. Ngô).
0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2008.05.124

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

755

In the present paper we shall first generalize the above Ostrowski inequality on time scales for k points x1 ; x2 ; . . . ; xk and
then unify corresponding continuous and discrete versions. We also point out some particular Ostrowski type inequalities on
time scales as special cases.
2. Time scales essentials
Now we briefly introduce the time scales theory and refer the reader to Hilger [8] and the books [2,3,9] for further details.
Definition 1. A time scale T is an arbitrary nonempty closed subset of real numbers.
Definition 2. For t 2 T, we define the forward jump operator r : T ! T by rðtÞ ¼ inffs 2 T : s > tg, while the backward
jump operator q : T ! T is defined by qðtÞ ¼ supfs 2 T : s < tg. If rðtÞ > t, then we say that t is right-scattered, while if
qðtÞ < 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 rðtÞ ¼ t, the t is called right-dense,
and if qðtÞ ¼ t then t is called left-dense. Points that are both right-dense and left-dense are called dense.
Definition 3. Let t 2 T, then two mappings

l; m : T ! ½0; þ1Þ satisfying

lðtÞ :¼ rðtÞ À t; mðtÞ :¼ t À qðtÞ
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
T :¼ T À ftg, otherwise Tj :¼ T. Furthermore for a function f : T ! R, we define the function f r : T ! R by f r ðtÞ ¼ f ðrðtÞÞ
for all t 2 T.
j

Definition 4. Let f : T ! R be a function on time scales. Then for t 2 Tj , we define f D ðtÞ to be the number, if one exists, such
that for all e > 0 there is a neighborhood U of t such that for all s 2 U

 r

f ðtÞ À f ðsÞ À f D ðtÞðrðtÞ À sÞ 6 ejrðtÞ À sj:
We say that f is D-differentiable on Tj provided f D ðtÞ exists for all t 2 Tj .
Definition 5. A mapping f : T ! R is called rd-continuous (denoted by C rd ) provided if it satisfies
(1) f is continuous at each right-dense point or maximal element of T.
(2) The left-sided limit lims!tÀ f ðsÞ ¼ f ðtÀÞ exists at each left-dense point t of T.

Remark 1. It follows from Theorem 1.74 of Bohner and Peterson [2] that every rd-continuous function has an antiderivative.
Definition 6. A function F : T ! R is called a D-antiderivative of f : T ! R provided F D ðtÞ ¼ f ðtÞ holds for all t 2 Tj . Then the
D-integral of f is defined by

Z

b

f ðtÞDt ¼ FðbÞ À FðaÞ:

a

Proposition 1. Let f ; g be rd-continuous, a; b; c 2 T and a; b 2 R. Then
(1)
(2)
(3)
(4)
(5)

Rb
Rb
Rb

ðaf ðtÞ þ bgðtÞÞDt ¼ a a f ðtÞDt þ b a gðtÞDt,
Ra
Rab
f ðtÞDt ¼ À b f ðtÞDt,
Rab
Rc
Rb
f ðtÞDt ¼ a f ðtÞDt þ c f ðtÞDt,
Rb
Rab
f ðtÞg D ðtÞDt ¼ ðfgÞðbÞ À ðfgÞðaÞ À a f D ðtÞgðrðtÞÞDt,
Raa
f ðtÞDt ¼ 0.
a

Definition 7. Let hk : T2 ! R, k 2 N0 be defined by

h0 ðt; sÞ ¼ 1 for all s; t 2 T
and then recursively by

hkþ1 ðt; sÞ ¼

Z
s

t

hk ðs; sÞDs for all s; t 2 T:

756

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

3. The generalized Ostrowski inequality on time scales
Throughout this section, we suppose that T is a time scale and an interval means the intersection of real interval with the
given time scale. We are in a position to state our main result.
Theorem 3. Suppose that
(1) a; b 2 T, Ik : a ¼ x0 < x1 < Á Á Á < xkÀ1 < xk ¼ b is a division of the interval ½a; bŠ for x0 ; x1 ; . . . ; xk 2 T;
(2) ai 2 T ði ¼ 0; . . . ; k þ 1Þ is ‘‘k þ 2” points so that a0 ¼ a, ai 2 ½xiÀ1 ; xi Š ði ¼ 1; . . . ; kÞ and akþ1 ¼ b;
(3) f : ½a; bŠ ! R is differentiable.
Then we have


Z

 b
k
kÀ1
X
X


r
f ðtÞDt À
ðaiþ1 À ai Þf ðxi Þ 6 M
ðh2 ðxi ; aiþ1 Þ þ h2 ðxiþ1 ; aiþ1 ÞÞ;


 a

i¼0
i¼0

ð2Þ

where

M ¼ sup jf D ðxÞj:
a
This inequality is sharp in the sense that the right-hand side of (2) cannot be replaced by a smaller one.
To prove Theorem 3, we need the following generalized Montgomery identity.
Lemma 1 (Generalized Montgomery identity). Under the assumptions of Theorem 3, we have

Z
k
X
ðaiþ1 À ai Þf ðxi Þ ¼

f r ðtÞDt þ

Z

a

i¼0

where

b

8
t À a1 ;
>
>
>
>
>
>
< t À a2 ;
Kðt; Ik Þ ¼ Á Á Á
>
>
>
> t À akÀ1 ;
>
>
:
t À ak ;

b

Kðt; Ik Þf D ðtÞDt;

ð3Þ

a

t 2 ½a; x1 Þ;
t 2 ½x1 ; x2 Þ;

ÁÁÁ

ð4Þ

t 2 ½xkÀ2 ; xkÀ1 Þ;
t 2 ½xkÀ1 ; bŠ:

Proof. Integrating by parts and applying Proposition 1, we have

Z

b

Kðt; Ik Þf D ðtÞDt ¼

a

kÀ1 Z
X
i¼0

¼

kÀ1
X

xiþ1

Kðt; Ik Þf D ðtÞDt ¼

xi

kÀ1 Z
X
i¼0

xiþ1

ðt À aiþ1 Þf D ðtÞDt

xi

ðxiþ1 À aiþ1 Þf ðxiþ1 Þ À ðxi À aiþ1 Þf ðxi Þ À

¼

!

xiþ1

r

f ðtÞDt

xi

i¼0
kÀ1
X

Z

ðaiþ1 À xi Þf ðxi Þ þ ðxiþ1 À aiþ1 Þf ðxiþ1 Þ À

Z

!

xiþ1

r

f ðtÞDt

xi

i¼0

¼ ða1 À aÞf ðaÞ þ

Z
kÀ1
kÀ2
X
X
ðaiþ1 À xi Þf ðxi Þ þ
ðxiþ1 À aiþ1 Þf ðxiþ1 Þ þ ðb À ak Þf ðbÞ À
i¼1

b

f r ðtÞDt

a

i¼0

Z
kÀ1
X
¼ ða1 À aÞf ðaÞ þ
ðaiþ1 À ai Þf ðxi Þ þ ðb À ak Þf ðbÞ À
a

i¼1

b

f r ðtÞDt ¼

Z
k
X
ðaiþ1 À ai Þf ðxi Þ À

b

f r ðtÞDt;

a

i¼0

i.e., (3) holds. h
Proof of Theorem 3. By applying Lemma 1, we get

 Z
 

Z
  b
 X
 X
 b
k
kÀ1 Z xiþ1
kÀ1 Z xiþ1
X


 
 


r
D
D
f ðtÞDt À
ðaiþ1 À ai Þf ðxi Þ ¼ 
Kðt; Ik Þf ðtÞDt  ¼ 

Kðt; Ik Þf ðtÞDt 6
jKðt; Ik Þjf D ðtÞDt




 a


xi
xi
a
i¼0
i¼0
i¼0
!
Z xiþ1
Z aiþ1
kÀ1 Z xiþ1
kÀ1
X
X
6M
jt À aiþ1 jDt ¼ M
ðaiþ1 À tÞDt þ
ðt À aiþ1 ÞDt
i¼0

¼M

kÀ1
X
i¼0

xi

i¼0

ðh2 ðxi ; aiþ1 Þ þ h2 ðxiþ1 ; aiþ1 ÞÞ:

xi

aiþ1


757

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

To prove the sharpness of this inequality, let f ðtÞ ¼ t; x0 ¼ a; x1 ¼ b; a0 ¼ a; a1 ¼ b; a2 ¼ b. It follows that M ¼ 1. Starting with
the left-hand side of (2), we have

 Z
 Z

Z
Z b
  b
  b


 b
k
X
 
 

r
f ðtÞDt À
ðaiþ1 À ai Þf ðxi Þ ¼ 
rðtÞDt À ððb À aÞa þ ðb À bÞbÞ ¼  ðrðtÞ þ tÞDt À tDt À ðb À aÞa




 a


a
a
a
i¼0


Z

Z b
Z b
 

 b

 


¼  ðt 2 ÞD Dt À
t Dt À ðb À aÞa ¼ ðb À aÞb À
t Dt  :
 

 a
a
a
Starting with the right-hand side of (2), we have

M

kÀ1
X

ðh2 ðxi ; aiþ1 Þ þ h2 ðxiþ1 ; aiþ1 ÞÞ ¼ h2 ðx0 ; a1 Þ þ h2 ðx1 ; a1 Þ ¼ h2 ða; bÞ þ h2 ðb; bÞ ¼

Z

a

ðt À bÞDt þ

Z

b

i¼0

¼

Z

a

t Dt À

Z

b

a

bDt ¼ ðb À aÞb À

b

Z

b

ðt À bÞDt

b

b

tDt:

a

Therefore in this particular case


Z

 b
k
kÀ1
X
X


r
f
ðtÞ
D
t
À
ð
a
À
a
Þf
ðx
Þ
ðh2 ðaiþ1 ; xi Þ þ h2 ðaiþ1 ; xiþ1 ÞÞ


iþ1
i
i  P M

 a
i¼0
i¼0
and by (2) also


Z

 b
k
kÀ1
X
X


f r ðtÞDt À
ðaiþ1 À ai Þf ðxi Þ 6 M
ðh2 ðaiþ1 ; xi Þ þ h2 ðaiþ1 ; xiþ1 ÞÞ:


 a
i¼0
i¼0
So the sharpness of the inequality (2) is shown. h
If we apply the inequality (2) to different time scales, we will get some well-known and some new results.

Corollary 1 (Continuous case). Let T ¼ R. Then our delta integral is the usual Riemann integral from calculus. Hence,

h2 ðt; sÞ ¼

ðt À sÞ2
2

for all t; s 2 R:

This leads us to state the following inequality:


Z
!

 b
k
kÀ1
kÀ1 
X
X
1X
xi þ xiþ1 2


2
;
f ðtÞdt À
ðaiþ1 À ai Þf ðxi Þ 6 M
ðxiþ1 À xi Þ þ

aiþ1 À


 a
4 i¼0
2
i¼0
i¼0
where M ¼ supa
1
4

in the right-hand side is the best possible.

Remark 2. The inequality (5) is exactly the generalized Ostrowski inequality shown in [6].
Corollary 2 (Discrete case). Let T ¼ Z, a ¼ 0, b ¼ n. Suppose that
(1) Ik : 0 ¼ j0 < j1 < Á Á Á < jkÀ1 < jk ¼ n is a division of ½0; nŠ \ Z for j0 ; k1 ; . . . ; jk 2 Z;
(2) pi 2 Z ði ¼ 0; . . . ; k þ 1Þ is ‘‘k þ 2” points so that p0 ¼ 0, pi 2 ½jiÀ1 ; ji Š \ Z ði ¼ 1; . . . ; kÞ and pkþ1 ¼ n;
(3) f ðkÞ ¼ xk .
Then, we have



2 X
!

X
n
k

kÀ1
kÀ1 
kÀ1 
X
X
1X
ji þ jiþ1
ji þ jiþ1


2
x À
ðp À pi Þxji  6 M
ðj À ji Þ þ
piþ1 À
þ
piþ1 À


 j¼1 j i¼0 iþ1
4 i¼0 iþ1
2
2
i¼0
i¼0
for all i ¼ 1; n, where M ¼ supi¼1;...;nÀ1 jDxi j and the constant
Proof. It is known that

hk ðt; sÞ ¼



tÀs



k

for all t; s 2 Z:

Therefore,


h2 ðji ; piþ1 Þ ¼

ji À piþ1
2


¼

ðji À piþ1 Þðji À piþ1 À 1Þ
2

1
4

in the right-hand side is the best possible.

ð5Þ

758

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

and

h2 ðjiþ1 ; piþ1 Þ ¼



jiþ1 À piþ1
2


¼

ðjiþ1 À piþ1 Þðjiþ1 À piþ1 À 1Þ
:
2

The conclusion is obtained by some easy calculation. h
Corollary 3 (Quantum calculus case). Let T ¼ qN0 , q > 1, a ¼ qm ; b ¼ qn with m < n. Suppose that
(1) Ik : qm ¼ qj0 < qj1 < Á Á Á < qjkÀ1 < qjk ¼ qn is a division of ½qm ; qn Š \ qN0 for j0 ; k1 ; . . . ; jk 2 N0 ;
(2) qpi 2 qN0 ði ¼ 0; . . . ; k þ 1Þ is ‘‘k þ 2” points so that qp0 ¼ qm , qpi 2 ½qjiÀ1 ; qji Š \ qN0 ði ¼ 1; . . . ; kÞ and qpkþ1 ¼ qm ;
(3) f : ½qm ; qn Š ! R is differentiable.
Then, we have



Z n

 q
k
X

r
piþ1
pi
ji 
f ðtÞDt À
ðq
À q Þf ðq Þ


 qm
i¼0
0
!2
!
À Á2 p
1þq
kÀ1
ðqpi þ qpiþ1 Þ
2ðq2pi þ q2piþ1 Þ À 1þq
ðq i þ qpiþ1 Þ2
2M X
ji
2ji
2

2
@
6
þ q ðq À 1Þ ;
q À
þ
2
4
1 þ q i¼0
where



f ðqtÞ À f ðtÞ

M ¼ sup 

qm and the constant

1
4

in the right-hand side is the best possible.

Proof. In this situation, one has

hk ðt; sÞ ¼

kÀ1

Y
t À qm s
Pm
l
l¼0 q
m¼0

for all t; s 2 qN0 :

Therefore,

À j
ÁÀ
Á
À
Á
q i À qpiþ1 qji À qpiþ1 þ1
h2 qji ; qpiþ1 ¼
1þq
and

À j
ÁÀ
Á
À
Á
q iþ1 À qpiþ1 qjiþ1 À qpiþ1 þ1
:
h2 qjiþ1 ; qpiþ1 ¼
1þq

The conclusion is easy obtained by some simple calculation. h

4. Some particular Ostrowski type inequalities on time scales
In this section we point out some particular Ostrowski type inequalities on time scales as special cases, such as: 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 2 T with a < b; f : ½a; bŠ ! R is differentiable. We
denote

M ¼ sup jf D ðxÞj:
a
Proposition 2. Suppose that a 2 ½a; bŠ \ T. Then we have the sharp rectangle inequality on time scales


Z

 b


r
f ðtÞDt À ðða À aÞf ðaÞ þ ðb À aÞf ðbÞÞ 6 Mðh2 ða; aÞ þ h2 ðb; aÞÞ:


 a
Proof. We choose k ¼ 1; x0 ¼ a; x1 ¼ b; a0 ¼ a; a1 ¼ a and a2 ¼ b in Theorem 3 to get the result. h

ð6Þ

759

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

Remark 3
(a) If we choose a ¼ b in (6), we get the sharp left rectangle inequality on time scales


Z

 b


r
f ðtÞDt À ðb À aÞf ðaÞ 6 Mh2 ða; bÞ:


 a

ð7Þ

(b) If we choose a ¼ a in (6), we get the sharp right rectangle inequality on time scales


Z

 b


r

f ðtÞDt À ðb À aÞf ðbÞ 6 Mh2 ða; bÞ:


 a

ð8Þ

(c) If we choose a ¼ aþb
in (6), we get the sharp trapezoid inequality on time scales
2


Z
 




 b
f ðaÞ þ f ðbÞ
aþb
aþb


r
f ðtÞDt À
ðb À aÞ 6 M h2 a;
þ h2 b;
:



 a
2
2
2

ð9Þ

Proposition 3. Suppose that x 2 ½a; bŠ \ T, a1 2 ½a; xŠ \ T, a2 2 ½x; bŠ \ T. Then we have the sharp inequality on time scales


Z

 b


r
f ðtÞDt À ðða1 À aÞf ðaÞ þ ða2 À a1 Þf ðxÞ þ ðb À a2 Þf ðbÞÞ 6 M ðh2 ða; a1 Þ þ h2 ðx; a1 Þ þ h2 ðx; a2 Þ þ h2 ðb; a2 ÞÞ:


 a

ð10Þ

Proof. We choose k ¼ 2, x0 ¼ a, x1 ¼ x, x2 ¼ b and ai ði ¼ 0; 3Þ is as in Theorem 3 to get the result. h
Remark 4
(a) If we choose a1 ¼ a and a2 ¼ b in Proposition 3, we get exactly Theorem 2. Therefore, Theorem 3 is a generalization of
Theorem 3.5 in [5].
(b) If we choose x ¼ aþb
in (1), we get the sharp mid-point inequality on time scales

2


Z


 




 b
aþb
aþb
aþb


r
6
M
h
ðb
À
aÞ
f
ðtÞ
D
t
À
f

;
a
þ
h
;
b
:


2
2

 a
2
2
2

Corollary 4. Suppose that a1 ¼ 5aþb
2 T, a2 ¼ aþ5b
2 T, and x 2
6
6

Â5aþb
6

ð11Þ

Ã
; aþ5b

\ T. Then we have the sharp inequality on time scales
6

Z


 







 b
b À a f ðaÞ þ f ðbÞ
5a þ b
5a þ b
a þ 5b
a þ 5b


r
þ 2f ðxÞ  6 M h2 a;
þ h2 x;
þ h2 x;
þ h2 b;
:
f ðtÞDt À



 a
3
2
6
6
6
6

ð12Þ
Remark 5. If we choose x ¼ aþb
in (12), we get the sharp Simpson inequality on time scales
2

Z



 b
b À a f ðaÞ þ f ðbÞ
a þ b 

r
þ 2f
f ðtÞDt À



 a
3

2
2
 







5a þ b
a þ b 5a þ b
a þ b a þ 5b
a þ 5b
6 M h2 a;
þ h2
þ h2
þ h2 b;
:
;
;
6
2
6
2
6
6

Â
Ã

Â
Ã
Corollary 5. Suppose that a1 2 a; aþb
; b \ T. Then we have the sharp inequality on time scales
\ T and a2 2 aþb
2
2

Z




 b
aþb


r
þ ðb À a2 Þf ðbÞ 
f ðtÞDt À ða1 À aÞf ðaÞ þ ða2 À a1 Þf


 a
2







aþb
aþb
6 M h2 ða; a1 Þ þ h2
; a 1 þ h2
; a2 þ h2 ðb; a2 Þ :
2
2

ð13Þ

Remark 6. If we choose a1 ¼ 3aþb
and a2 ¼ aþ3b
in (13), we get the sharp averaged mid-point-trapezoid inequality on time scales
4
4

Z



 b
b À a f ðaÞ þ f ðbÞ
a þ b 

r
þf
f ðtÞDt À


 a


2
2
2
 







3a þ b
a þ b 3a þ b
a þ b a þ 3b
a þ 3b
6 M h2 a;
þ h2
;
þ h2
;
þ h2 b;
:
4
2
4
2
4
4

760

W. Liu, Q.-A. Ngô / Applied Mathematics and Computation 203 (2008) 754–760

Acknowledgements
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 and the Natural Science Foundation of Jiangsu Province Education Department under Grant No.07KJD510133.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]

R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: a survey, Math. Inequal. Appl. 4 (4) (2001) 535–557.
M. Bohner, A. Peterson, Dynamic Equations on Time Series, Birkhäuser, Boston, 2001.
M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Series, Birkhäuser, Boston, 2003.

M. Bohner, T. Matthews, The Grüss inequality on time scales, Commun. Math. Anal. 3 (1) (2007) 1–8.
M. Bohner, T. Matthewa, Ostrowski inequalities on time scales, J. Inequal. Pure Appl. Math. 9 (1) (2008). Art. 6, 8 pp.
S.S. Dragomir, A generalization of Ostrowski integral inequality for mappings whose derivatives belong to L1 and applications in numerical integration,
J. KSIAM 5 (2) (2001) 117–136.
F. Geng, D. Zhu, Multiple results of p-Laplacian dynamic equations on time scales, Appl. Math. Comput. 193 (2007) 311–320.
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrmsmannigfaltingkeiten, Ph.D. Thesis, Univarsi. Würzburg, 1988.
V. Lakshmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, 1996.
W.J. Liu, Q.L. Xue, S.F. Wang, Several new perturbed Ostrowski-like type inequalities, J. Inequal. Pure Appl. Math. 8 (4) (2007). Art. 110, 6 pp.
W.J. Liu, C.C. Li, Y.M. Hao, Further generalization of some double integral inequalities and applications, Acta Math. Univ. Comenianae 77 (1) (2008)
147–154.
W.J. Liu, Several error inequalities for a quadrature formula with a parameter and applications, Comput. Math. Appl., doi:10.1016/j.camwa.2008.
04.016.
W.J. Liu, Q.A. Ngô, An Ostrowski-Grüss type inequality on time scales. Availabel from: <arXiv:0804.3231>.
D.S. Mitrinovic´, J. Pecaric´, A.M. Fink, Inequalities for Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, 1994.
D.S. Mitrinovic´, J. Pecaric´, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
H. Roman, A time scales version of a Wirtinger-type inequality and applications, dynamic equations on time scales, J. Comput. Appl. Math. 141 (1/2)
(2002) 219–226.
H. Su, M. Zhang, Solutions for higher-order dynamic equations on time scales, Appl. Math. Comput. 200 (1) (2008) 413–428.
F.-H. Wong, S.-L. Yu, C.-C. Yeh, Andersons inequality on time scales, Appl. Math. Lett. 19 (2007) 931–935.