GENERALIZED OSTROWSKI’S INEQUALITY ON
TIME SCALES
Ostrowski’s Inequality on
Time Scales
BAŞAK KARPUZ AND UMUT MUTLU ÖZKAN
Department of Mathematics
Faculty of Science and Arts, ANS Campus
Afyon Kocatepe University
03200 Afyonkarahisar, Turkey.
EMail: bkarpuz@gmail.com
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
Title Page
umut_ozkan@aku.edu.tr
Received:
31 July, 2008
Accepted:
08 October, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15.
Key words:
Montgomery’s identity, Ostrowski’s inequality, time scales.
Abstract:
In this paper, we generalize Ostrowski’s inequality and Montgomery’s identity
on arbitrary time scales which were given in a recent paper [J. Inequal. Pure.
Appl. Math., 9(1) (2008), Art. 6] by Bohner and Matthews. Some examples for
the continuous, discrete and the quantum calculus cases are given as well.
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Contents
1
Introduction
3
2
Time Scales Essentials
5
3
Generalization by Generalized Polynomials
7
4
Applications for Generalized Polynomials
11
5
Generalization by Arbitrary Functions
13
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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1.
Introduction
In 1937, Ostrowski gave a very useful formula to estimate the absolute value of
derivation of a differentiable function by its integral mean. In [9], the so-called
Ostrowski’s inequality
)
(
Z b
2
2
1
(t
−
a)
+
(b
−
t)
0
f (t) −
f (η)dη ≤ sup |f (η)|
b−a a
2(b − a)
η∈(a,b)
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
is shown by the means of the Montgomery’s identity (see [6, pp. 565]).
In a very recent paper [2], the Montgomery identity and the Ostrowski inequality
were generalized respectively as follows:
1
Lemma A (Montgomery’s identity). Let a, b ∈ T with a < b and f ∈ Crd
([a, b]T , R).
Then
Z b
Z b
1
σ
∆
f (t) =
f (η)∆η +
Ψ(t, η)f (η)∆η
b−a
a
a
holds for all t ∈ T, where Ψ : [a, b]2T → R is defined as follows:
(
s − a, s ∈ [a, t)T ;
Ψ(t, s) :=
s − b, s ∈ [t, b]T
vol. 9, iss. 4, art. 112, 2008
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for s, t ∈ [a, b]T .
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1
Crd
([a, b]T , R).
Theorem A (Ostrowski’s inequality). Let a, b ∈ T with a < b and f ∈
Then
)
(
Z b
1
h2 (t, a) + h2 (t, b)
σ
∆
f (t) −
f (η)∆η ≤ sup |f (η)|
b−a a
b−a
η∈(a,b)
holds for all t ∈ T. Here, h2 (t, s) is the second-order generalized polynomial on
time scales.
In this paper, we shall apply a new method to generalize Lemma A, Theorem A,
which is completely different to the method employed in [2], however following the
routine steps in [2], our results may also be proved.
The paper is arranged as follows: in §2, we quote some preliminaries on time
scales from [1]; §3 includes our main results which generalize Lemma A and Theorem A by the means of generalized polynomials on time scales; in §4, as applications, we consider particular time scales R, Z and q N0 ; finally, in §5, we give
extensions of the results stated in §3.
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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2.
Time Scales Essentials
Definition 2.1. A time scale is a nonempty closed subset of reals.
Definition 2.2. On an arbitrary time scale T the following are defined: the forward
jump operator σ : T → T is defined by σ(t) := inf(t, ∞)T for t ∈ T, the backward
jump operator ρ : T → T is defined by ρ(t) := sup(−∞, t)T for t ∈ T, and the
graininess function µ : T → R+
0 is defined by µ(t) := σ(t) − t for t ∈ T. For
convenience, we set inf ∅ := sup T and sup ∅ := inf T.
Definition 2.3. Let t be a point in T. If σ(t) = t holds, then t is called right-dense,
otherwise it is called right-scattered. Similarly, if ρ(t) = t holds, then t is called
left-dense, a point which is not left-dense is called left-scattered.
Definition 2.4. A function f : T → R is called rd-continuous provided that it is
continuous at right-dense points of T and its left-sided limits exist (finite) at leftdense points of T. The set of rd-continuous functions is denoted by Crd (T, R),
1
and Crd
(T, R) denotes the set of functions for which the delta derivative belongs
to Crd (T, R).
Theorem 2.5 (Existence of antiderivatives). Let f be a rd-continuous function.
Then f has an antiderivative F such that F ∆ = f holds.
Definition 2.6. If f ∈ Crd (T, R) and s ∈ T, then we define the integral
Z t
F (t) :=
f (η)∆η for t ∈ T.
s
Theorem 2.7. Let f, g be rd-continuous functions, a, b, c ∈ T and α, β ∈ R. Then,
the following are true:
Rb
Rb
Rb
1. a αf (η) + βg(η) ∆η = α a f (η)∆η + β a g(η)∆η,
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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Ra
f (η)∆η = − b f (η)∆η,
Rc
Rb
Rc
3. a f (η)∆η = a f (η)∆η + b f (η)∆η,
Rb
Rb
4. a f (η)g ∆ (η)∆η = f (b)g(b) − f (a)g(a) − a f ∆ (η)g(σ(η))∆η.
2.
Rb
a
Definition 2.8. Let hk : T2 → R be defined as follows:

1,
k=0
(2.1)
hk (t, s) := R
 t h (η, s)∆η, k ∈ N
s k−1
for all s, t ∈ T and k ∈ N0 .
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
Title Page
Note that the function hk satisfies
(2.2)
t
h∆
k (t, s) =
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
0,
k=0
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h (t, s),
k−1
k∈N
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I
for all s, t ∈ T and k ∈ N0 .
Property 1. Using induction it is easy to see that hk (t, s) ≥ 0 holds for all k ∈ N
and s, t ∈ T with t ≥ s and (−1)k hk (t, s) ≥ 0 holds for all k ∈ N and s, t ∈ T with
t ≤ s.
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3.
Generalization by Generalized Polynomials
We start this section by quoting the following useful change of order formula for
double(iterated) integrals which is employed in our proofs.
Lemma 3.1 ([8, Lemma 1]). Assume that a, b ∈ T and f ∈ Crd (T2 , R). Then
Z bZ b
Z b Z σ(η)
f (η, ξ)∆η∆ξ =
f (η, ξ)∆ξ∆η.
a
ξ
a
Ostrowski’s Inequality on
Time Scales
a
B. Karpuz and U.M. Özkan
Now, we give a generalization for Montgomery’s identity as follows:
Lemma 3.2. Assume that a, b ∈ T and f ∈
by

hk (s, a), s ∈ [a, t)T
Ψ(t, s) :=
and
h (s, b), s ∈ [t, b]
k
T
1
Crd
([a, b]T , R).
vol. 9, iss. 4, art. 112, 2008
Define Ψ, Φ ∈
1
Crd
([a, b]T , R)
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Φ(t, s) :=

hk−1 (s, a), s ∈ [a, t)T
h (s, b),
k−1
s ∈ [t, b]T
for s, t ∈ [a, b]T and k ∈ N. Then
(3.1)
f (t) =
1
hk (t, a) − hk (t, b)
Z
b
Φ(t, η)f σ (η)∆η +
a
Z
b
Ψ(t, η)f ∆ (η)∆η
a
is true for all t ∈ [a, b]T and all k ∈ N.
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Proof. Note that we have Ψ∆s = Φ. Clearly, for all t ∈ [a, b]T and all k ∈ N, from
(3.1), (2.1) and (2.2) we have
Z t
Z t
σ
Φ(t, η)f (η)∆η +
Ψ(t, η)f ∆ (η)∆η
a
a
Z t
Z t
σ
=
hk−1 (η, a)f (η)∆η +
hk (η, a)f ∆ (η)∆η
a
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a
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Z tZ
σ(η)
=
(3.2)
a
a
hk−1 (η, a)f ∆ (ξ)∆ξ∆η + f (a)hk (t, a)
Z tZ η
∆
+
hk (ξ, a)f ∆ (η) ξ ∆ξ∆η.
a
a
Applying Lemma 3.1 and considering (2.1), the right-hand side of (3.2) takes the
form
Z tZ t
hk−1 (η, a)f ∆ (ξ)∆η∆ξ + f (a)hk (t, a)
a
ξ
Z tZ
+
a
Z tZ
=
a
(3.3)
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
η
hk−1 (ξ, a)f ∆ (η)∆ξ∆η
a
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t
hk−1 (η, a)f ∆ (ξ)∆η∆ξ + f (a)hk (t, a)
a
= f (t)hk (t, a),
and very similarly, from Lemma 3.1, (3.1), (2.1) and (2.2), we obtain
Z b
Z b
σ
Φ(t, η)f (η)∆η +
Ψ(t, η)f ∆ (η)∆η
t
t
Z b
Z b
σ
=
hk−1 (η, b)f (η)∆η +
hk (η, b)f ∆ (η)∆η
t
t
Z b Z σ(η)
=
hk−1 (η, b)f ∆ (ξ)∆ξ∆η − f (t)hk (t, b)
t
t
Z bZ b
∆
−
hk (ξ, b)f ∆ (η) ξ ∆ξ∆η,
t
η
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Z bZ
=
t
b
hk−1 (η, b)f ∆ (ξ)∆η∆ξ − f (t)hk (t, b)
ξ
Z bZ
−
t
(3.4)
b
hk−1 (ξ, b)f ∆ (η)∆ξ∆η
η
= −f (t)hk (t, b).
Ostrowski’s Inequality on
Time Scales
By summing (3.3) and (3.4), we get the desired result.
Now, we give the following generalization of Ostrowski’s inequality.
Theorem 3.3. Assume that a, b ∈ T and f ∈
1
Crd
([a, b]T , R).
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
Then
Z b
1
σ
f (t) −
Φ(t,
η)f
(η)∆η
hk (t, a) − hk (t, b) a
hk+1 (t, a) + (−1)k+1 hk+1 (t, b)
≤M
hk (t, a) − hk (t, b)
is true for all t ∈ [a, b]T and all k ∈ N, where Φ is as introduced in (3.1) and
M := supη∈(a,b) |f ∆ (η)|.
Proof. From Lemma 3.2 and (3.1), for all k ∈ N and t ∈ [a, b]T , we get
Z b
1
σ
f (t) −
Φ(t,
η)f
(η)∆η
hk (t, a) − hk (t, b) a
Z b
1
= Ψ(t, η)f ∆ (η)∆η hk (t, a) − hk (t, b) a
Z t
Z b
1
∆
∆
=
hk (η, a)f (η)∆η +
hk (η, b)f (η)∆η hk (t, a) − hk (t, b)
a
t
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(3.5)
M
≤
hk (t, a) − hk (t, b)
Z b
Z t
hk (η, a)∆η + ,
h
(η,
b)∆η
k
a
t
and considering Property 1 and (2.1) on the right-hand side of (3.5), we have
Z t
Z b
M
k
hk (η, a)∆η +
(−1) hk (η, b)∆η
hk (t, a) − hk (t, b)
a
t
Z t
Z t
M
k+1
hk (η, a)∆η + (−1)
hk (η, b)∆η
=
hk (t, a) − hk (t, b)
a
b
hk+1 (t, a) + (−1)k+1 hk+1 (t, b)
,
=M
hk (t, a) − hk (t, b)
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
Title Page
which completes the proof.
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Remark 1. It is clear that Lemma 3.2 and Theorem 3.3 reduce to Lemma A and
Theorem A respectively by letting k = 1.
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4.
Applications for Generalized Polynomials
In this section, we give examples on particular time scales for Theorem 3.3. First,
we consider the continuous case.
Example 4.1. Let T = R. Then, we have hk (t, s) = (t − s)k /k! = (−1)k (s − t)k /k!
for all s, t ∈ R and k ∈ N. In this case, Ostrowski’s inequality reads as follows:
Z b
k!
f (t) −
Φ(t,
η)f
(η)dη
(t − a)k + (−1)k+1 (b − t)k a
M
(t − a)k+1 + (b − t)k+1
,
≤
k + 1 (t − a)k + (−1)k+1 (b − t)k
where M is the maximum value of the absolute value of the derivative f 0 over [a, b]R ,
and Φ(t, s) = (s − a)k /k! for s ∈ [a, t)R and Φ(t, s) = (s − b)k /k! for s ∈ [t, b]R .
Next, we consider the discrete calculus case.
Example 4.2. Let T = Z. Then, we have hk (t, s) = (t − s)(k) /k! = (−1)k (s − t +
k)(k) /k! for all s, t ∈ Z and k ∈ N, where the usual factorial function (k) is defined
by n(k) := n!/k! for k ∈ N and n(0) := 1 for n ∈ Z. In this case, Ostrowski’s
inequality reduces to the following inequality:
b−1
X
k!
Φ(t, η)f (η + 1)
f (t) −
(k)
k+1
(k)
(t − a) + (−1) (b − t + k) η=a
M
(t − a)(k+1) + (b − t + k)(k+1)
,
≤
k + 1 (t − a)(k) + (−1)k+1 (b − t + k)(k)
where M is the maximum value of the absolute value of the difference ∆f over
[a, b − 1]Z , and Φ(t, s) = (s − a)(k) /k! for s ∈ [a, t − 1]Z and Φ(t, s) = (s − b)(k) /k!
for s ∈ [t, b]Z .
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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Before giving the quantum calculus case, we need to introduce the following
notations from [7]:
qk − 1
[k]q :=
for q ∈ R/{1} and k ∈ N0 ,
q−1
k
Y
[k]! :=
[j]q
for k ∈ N0 ,
Ostrowski’s Inequality on
Time Scales
j=1
(t − s)kq :=
k−1
Y
(t − q j s)
for s, t ∈ q N0 and k ∈ N0 .
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
j=0
It is shown in [1, Example 1.104] that the following holds:
(t − s)kq
hk (t, s) :=
for s, t ∈ q N0 and k ∈ N0 .
[k]!
And finally, we consider the quantum calculus case.
Example 4.3. Let T = q N0 with q > 1. Therefore, for the quantum calculus case,
Ostrowski’s inequality takes the following form:
logq (b/(qa))
X
[k]!(q
−
1)a
η
η
η+1
f (t) −
q
Φ(t,
q
a)f
(q
a)
(t − a)kq − (t − b)kq
η=0
!
k+1
k+1
(t − a)k+1
+
(−1)
(t
−
b)
M
q
q
≤
,
[k + 1]q
(t − a)kq − (t − b)kq
where M is the maximum value of the absolute value of the q-difference Dq f over
[a, b/q]qN0 , and Φ(t, s) = (s − a)kq /[k]! for s ∈ [a, t/q]qN0 and Φ(t, s) = (s − b)k /[k]!
for s ∈ [t, b]qN0 . Here, the q-difference operator Dq is defined by Dq f (t) := [f (qt) −
f (t)]/[(q − 1)t].
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5.
Generalization by Arbitrary Functions
In this section, we replace the generalized polynomials hk (t, s) appearing in the
definitions of Φ(t, s) and Ψ(t, s) by arbitrary functions.
Since the proof of the following results can be done easily, we just give the statements of the results without proofs.
1
1
Lemma 5.1. Assume that a, b ∈ T, f ∈ Crd
([a, b]T , R), and that ψ, φ ∈ Crd
([a, b]T , R)
with ψ(b) = φ(a) = 0 and ψ(t) − φ(t) 6= 0 for all t ∈ [a, b]T . Set Ψ, Φ ∈
Crd ([a, b]T , R) by
(
φ(s), s ∈ [a, t)T
(5.1)
Ψ(t, s) :=
and Φ(t, s) := Ψ∆s (t, s)
ψ(s), s ∈ [t, b]T
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
Title Page
Contents
for s, t ∈ [a, b]T . Then
Z b
∆η
1
f (t) =
Ψ(t, η)f (η) ∆η
ψ(t) − φ(t) a
Z b
Z b
1
σ
∆
=
Φ(t, η)f (η)∆η +
Ψ(t, η)f (η)∆η
ψ(t) − φ(t)
a
a
is true for all t ∈ [a, b]T .
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1
Crd
([a, b]T , R),
1
Crd
([a, b]T , R)
Theorem 5.2. Assume that a, b ∈ T, f ∈
and that ψ, φ ∈
with ψ(b) = φ(a) = 0 and ψ(t) − φ(t) 6= 0 for all t ∈ [a, b]T . Then
Z b
Z b
1
M
σ
f (t) −
Φ(t, η)f (η)∆η ≤
|Ψ(t, η)|∆η
ψ(t) − φ(t) a
|ψ(t) − φ(t)|
a
is true for all t ∈ [a, b]T , where Ψ, Φ are as introduced in (5.1) and M := supη∈(a,b) |f ∆ (η)|.
Close
Remark 2. Letting φ(t) = hk (t, a) and ψ(t) = hk (t, b) for some k ∈ N, we obtain
the results of §3, which reduce to the results in [2, § 3] by letting k = 1. This is for
Ostrowski-polynomial type inequalities.
Remark 3. For instance, we may let φ(t) = eλ (t, a) − 1 and ψ(t) = eλ (t, b) − 1 for
some λ ∈ R+ ([a, b]T , R+ ) to obtain new Ostrowski-exponential type inequalities.
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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References
[1] M. BOHNER AND A. PETERSON, Dynamic Equations on Time Scales: An
Introduction with Applications, Boston, MA, Birkhäuser Boston Inc., 2001.
[2] M. BOHNER AND T. MATTHEWS, Ostrowski inequalities on time scales, J.
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vu.edu.au/article.php?sid=940].
[3] S.S. DRAGOMIR, The discrete version of Ostrowski’s inequality in normed
linear spaces, J. Inequal. Pure Appl. Math., 3(1) (2002), Art. 2. [ONLINE:
http://jipam.vu.edu.au/article.php?sid=155].
[4] S.S. DRAGOMIR, Ostrowski type inequalities for isotonic linear functionals, J.
Inequal. Pure Appl. Math., 3(5) (2002), Art. 68. [ONLINE: http://jipam.
vu.edu.au/article.php?sid=220].
[5] B. GAVREA AND I. GAVREA, Ostrowski type inequalities from a linear functional point of view, J. Inequal. Pure Appl. Math., 1(2) (2000), Art. 11. [ONLINE: http://jipam.vu.edu.au/article.php?sid=104].
Ostrowski’s Inequality on
Time Scales
B. Karpuz and U.M. Özkan
vol. 9, iss. 4, art. 112, 2008
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[6] D.S. MITRINOVIĆ, J. E. PEČARIĆ AND A.M. FINK, Inequalities Involving
Functions and their Integrals and Derivatives, Mathematics and its Applications, Dordrecht, Kluwer Academic Publishers, vol. 53, 1991.
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[7] V. KAC AND P. CHEUNG, Quantum Calculus, Universitext, Springer, New
York, NY, USA, 2002.
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[8] B. KARPUZ, Unbounded oscillation of higher-order nonlinear delay dynamic
equations of neutral type with oscillating coefficients, (submitted).
[9] A. OSTROWSKI, Über die absolutabweichung einer differenzierbaren funktion
von ihrem integralmittelwert, Comment. Math. Helv., 10(1) (1937), 226–227.
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