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.



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 supa
 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

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


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;

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



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

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


 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

i¼0

¼

Z

a

t Dt À

Z

b

a

bDt ¼ ðb À aÞb À

b

Z

b

ðt À bÞDt

b

b



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.

aiþ1 À


 a
4 i¼0
2
i¼0
i¼0
where M ¼ supa


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

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

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

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


 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

;
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


 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

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








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


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