OSTROWSKI INEQUALITIES ON TIME SCALES
MARTIN BOHNER AND THOMAS MATTHEWS
Missouri University of Science and Technology
Department of Mathematics and Statistics
Rolla, MO 65409-0020, USA
EMail: bohner@mst.edu
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
tmnqb@mst.edu
Title Page
Received:
06 July, 2007
Accepted:
15 February, 2008
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
26D15.
Key words:
Ostrowski inequality, Montgomery identity, Time scales.
Abstract:
We prove Ostrowski inequalities (regular and weighted cases) on time scales and
thus unify and extend corresponding continuous and discrete versions from the
literature. We also apply our results to the quantum calculus case.
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Acknowledgements:
The authors thank the referees for their careful reading of the manuscript and
insightful comments.
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Contents
1
Introduction
3
2
Time Scales Essentials
4
3
The Ostrowski Inequality on Time Scales
7
4
The Weighted Case
13
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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1.
Introduction
In 1938, Ostrowski derived a formula to estimate the absolute deviation of a differentiable function from its integral mean. As shown in [6], the so-called Ostrowski
inequality
"
#
Z b
a+b 2
t
−
1
1
2
f (t) −
(1.1)
f (s)ds ≤ sup |f 0 (t)| (b − a)
+
b−a a
(b − a)2
4
a<t<b
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
holds and can be shown by using the Montgomery identity [5]. These two properties
will be proved for general time scales, which unify discrete, continuous and many
other cases. The setup of this paper is as follows. In Section 2 we first give some
preliminary results on time scales that are needed in the remainder of this paper.
Next, in Section 3 we prove time scales versions of the Montgomery identity and of
the Ostrowski inequality (1.1) (the question of sharpness is also addressed), while in
Section 4 we offer several weighted time scales versions of the Ostrowski inequality.
Throughout, we apply our results to the special cases of continuous, discrete, and
quantum time scales.
vol. 9, iss. 1, art. 6, 2008
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2.
Time Scales Essentials
Definition 2.1. A time scale is an arbitrary nonempty closed subset of the real numbers.
The most important examples of time scales are R, Z and q N0 := {q k | k ∈ N0 }.
Definition 2.2. If T is a time scale, then we define the forward jump operator σ :
T → T by σ(t) := inf {s ∈ T| s > t} for all t ∈ T, the backward jump operator
ρ : T → T by ρ(t) := sup {s ∈ T| s < t} for all t ∈ T, and the graininess function
µ : T → [0, ∞) by µ(t) := σ(t) − t for all t ∈ T. Furthermore for a function
f : T → R, we define f σ (t) = f (σ(t)) for all t ∈ T and f ρ (t) = f (ρ(t)) for all
t ∈ T. In this definition we use inf ∅ = sup T (i.e., σ(t) = t if t is the maximum of
T) and sup ∅ = inf T (i.e., ρ(t) = t if t is the minimum of T).
These definitions allow us to characterize every point in a time scale as displayed
in Table 1.
t right-scattered
t right-dense
t left-scattered
t left-dense
t isolated
t dense
t < σ(t)
t = σ(t)
ρ(t) < t
ρ(t) = t
ρ(t) < t < σ(t)
ρ(t) = t = σ(t)
Table 1: Classification of Points
Definition 2.3. A function f : T → R is called rd-continuous (denoted by Crd ) if
it is continuous at right-dense points of T and its left-sided limits exist (finite) at
left-dense points of T.
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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Theorem 2.4 (Existence of Antiderivatives). Let f be rd-continuous. Then f has
an antiderivative F satisfying F ∆ = f .
Proof. See [1, Theorem 1.74].
Definition 2.5. If f is rd-continuous and t0 ∈ T, then we define the integral
Z t
(2.1)
F (t) =
f (τ )∆τ for t ∈ T.
t0
Therefore for f ∈ Crd we have
Rb
a
Thomas Matthews
f (τ )∆τ = F (b) − F (a), where F
∆
Theorem 2.6. Let f, g be rd-continuous, a, b, c ∈ T and α, β ∈ R. Then
Rb
Rb
Rb
1. a [αf (t) + βg(t)]∆t = α a f (t)∆t + β a g(t)∆t,
2.
Rb
3.
Rb
4.
Rb
5.
Ra
a
a
a
a
f (t)∆t = −
f (t)∆t =
Ra
Rc
a
b
f (t)∆t,
f (t)∆t +
Rb
c
Ostrowski Inequalities on
Time Scales
Martin Bohner and
f (t)∆t,
f (t)g ∆ (t)∆t = (f g)(b) − (f g)(a) −
Rb
a
f ∆ (t)g(σ(t))∆t,
= f.
vol. 9, iss. 1, art. 6, 2008
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f (t)∆t = 0.
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Proof. See [1, Theorem 1.77].
Close
Definition 2.7. Let gk , hk : T2 → R, k ∈ N0 be defined by
g0 (t, s) = h0 (t, s) = 1 for all
s, t ∈ T
and then recursively by
t
Z
gk+1 (t, s) =
gk (σ(τ ), s)∆τ
for all
s, t ∈ T
s
and
Z
hk+1 (t, s) =
t
hk (τ, s)∆τ
s, t ∈ T.
for all
s
Theorem 2.8 (Hölder’s Inequality). Let a, b ∈ T and f, g : [a, b] → R be rdcontinuous. Then
Z b
p1 Z b
1q
Z b
p
q
(2.2)
|f (t)g(t)| ∆t ≤
|f (t)| ∆t
|g(t)| ∆t ,
a
where p > 1 and
a
1
p
+
1
q
a
Ostrowski Inequalities on
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Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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= 1.
Proof. See [1, Theorem 6.13].
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3.
The Ostrowski Inequality on Time Scales
Lemma 3.1 (Montgomery Identity). Let a, b, s, t ∈ T, a < b and f : [a, b] → R be
differentiable. Then
Z b
Z b
1
1
σ
f (s)∆s +
p(t, s)f ∆ (s)∆s,
(3.1)
f (t) =
b−a a
b−a a
where
(
s − a, a ≤ s < t,
Thomas Matthews
p(t, s) =
(3.2)
Ostrowski Inequalities on
Time Scales
Martin Bohner and
vol. 9, iss. 1, art. 6, 2008
s − b, t ≤ s ≤ b.
Proof. Using Theorem 2.6 (4), we have
Z t
Z t
∆
(s − a)f (s)∆s = (t − a)f (t) −
f σ (s)∆s
a
a
and similarily
Z
t
b
(s − b)f ∆ (s)∆s = (b − t)f (t) −
Z
b
f σ (s)∆s.
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t
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Therefore
1
b−a
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Z
b
a
i.e., (3.1) holds.
Z b
1
f (s)∆s +
p(t, s)f ∆ (s)∆s
b−a a
Z b
Z b
1
1
σ
σ
=
f (s)∆s +
(b − a)f (t) −
f (s)∆s
b−a a
b−a
a
= f (t),
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σ
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If we apply Lemma 3.1 to the discrete and continuous cases, we have the following results.
Corollary 3.2 (discrete case). We let T = Z. Let a = 0, b = n, s = j, t = i and
f (k) = xk . Then
n
n−1
1X
1X
xi =
xj +
p(i, j)∆xj ,
n j=1
n j=0
where
p(i, 0) = 0,
p(1, j) = j − n
p(n, j) = j
(
p(i, j) =
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
for 1 ≤ j ≤ n − 1,
for
j,
Ostrowski Inequalities on
Time Scales
Martin Bohner and
0 ≤ j ≤ n − 1,
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0 ≤ j < i,
j − n, i ≤ j ≤ n − 1
as we just need 1 ≤ i ≤ n and 0 ≤ j ≤ n − 1. This result is the same as in [2,
Theorem 2.1].
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Corollary 3.3 (continuous case). We let T = R. Then
Z b
Z b
1
1
f (t) =
f (s)ds +
p(t, s)f 0 (s)ds.
b−a a
b−a a
This is the Montgomery identity in the continuous case, which can be found in [5, p.
565].
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Corollary 3.4 (quantum calculus case). We let T = q N0 , q > 1, a = q m and b = q n
with m < n. Then
n−1
P
f (t) =
q k f (q k+1 )
k=m
n−1
P
qk
n−1
X
k+1
1
+ n
f (q ) − f (q k ) p(t, q k ),
m
q − q k=m
Ostrowski Inequalities on
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Martin Bohner and
k=m
where
(
k
p(t, q ) =
q k − q m , q m ≤ q k < t,
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
qk − qn, t ≤ qk ≤ qn.
Theorem 3.5 (Ostrowski Inequality). Let a, b, s, t ∈ T, a < b and f : [a, b] → R
be differentiable. Then
Z b
1
M
σ
f (t) −
(3.3)
f (s)∆s ≤
(h2 (t, a) + h2 (t, b)) ,
b−a a
b−a
where
M = sup |f ∆ (t)|.
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a<t<b
This inequality is sharp in the sense that the right-hand side of (3.3) cannot be
replaced by a smaller one.
Proof. With Lemma 3.1 and p(t, s) defined as in (3.2), we have
Z b
Z b
1
1
σ
∆
f (t) −
f (s)∆s = p(t, s)f (s)∆s
b−a a
b−a a
Z t
Z b
M
≤
|s − a| ∆s +
|s − b| ∆s
b−a
a
t
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Z t
Z b
M
=
(s − a) ∆s +
(b − s) ∆s
b−a
a
t
M
=
(h2 (t, a) + h2 (t, b)) .
b−a
Note that, since p(t, a) = 0, the smallest value attaining the supremum in M is
greater than a. To prove the sharpness of this inequality, let f (t) = t, a = t1 , b = t2
and t = t2 . It follows that f ∆ (t) = 1 and M = 1. Starting with the left-hand side of
(3.3), we have
Z b
Z t2
1
1
σ
f (t) −
f (s)∆s = t2 −
σ(s)∆s
b−a a
t2 − t1 t1
Z t2
Z t2
1
= t2 −
(σ(s) + s)∆s −
s∆s t2 − t1
t
t
Z 1t2
Z t2 1 1
= t2 −
(s2 )∆ ∆s −
s∆s t2 − t1
t
t1
Z 1t2
1
= −t1 +
s∆s .
t2 − t1 t1
Starting with the right-hand side of (3.3), we have
Z t2
Z t2
M
1
(h2 (t, a) + h2 (t, b)) =
(s − t1 )∆s +
(s − t2 )∆s
b−a
t2 − t1
t1
t2
Z t2
1
2
−t1 t2 + t1 +
s∆s
=
t2 − t1
t1
Z t2
1
s∆s.
= −t1 +
t2 − t1 t1
Ostrowski Inequalities on
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Martin Bohner and
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vol. 9, iss. 1, art. 6, 2008
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Therefore in this particular case
Z b
1
σ
f (t) −
≥ M (h2 (t, a) + h2 (t, b))
f
(s)∆s
b−a
b−a a
and by (3.3) also
f (t) −
1
b−a
Z
a
b
M
f (s)∆s ≤
(h2 (t, a) + h2 (t, b)) .
b−a
σ
So the sharpness of the Ostrowski inequality is shown.
Ostrowski Inequalities on
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Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
Corollary 3.6 (discrete case). Let T = Z. Let a = 0, b = n, s = j, t = i and
f (k) = xk . Then
"
#
2
n
M 2
X
n
+
1
n
−
1
1
i −
+
(3.4)
xj ≤
,
xi −
n j=1 n 2 4
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M = max |∆xi | .
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This is the discrete Ostrowski inequality from [2, Theorem 3.1], where the constant
1
in the right-hand side of (3.4) is the best possible in the sense that it cannot be
4
replaced by a smaller one.
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1≤i≤n−1
Corollary 3.7 (continuous case). If T = R, then
"
#
Z b
a+b 2
t
−
1
1
2
f (t) −
f (s)ds ≤ M (b − a)
+
,
b−a a
(b − a)2
4
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where
M = sup |f 0 (t)| .
a<t<b
This is the Ostrowski inequality in the continuous case [6, p. 226–227], where again
the constant 41 in the right-hand side is the best possible.
Corollary 3.8 (quantum calculus case). Let T = q N0 , q > 1, a = q m and b = q n
with m < n. Then


!2
Z qn
1+q m
n
(q
+
q
)
1
M
2
f (t) −

 t− 2
f σ (s)∆s ≤ n
q n − q m qm
q − qm 1 + q
2
!#
2 m
n 2
2m
2n
− 1+q
(q
+
q
)
+
(2(1
+
q)
−
2)
(q
+
q
)
2
,
+
4
where
and the constant
f (qt) − f (t) ,
M = sup m
n
(q
−
1)t
q <t<q
1
4
Ostrowski Inequalities on
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Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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in the right-hand side is the best possible.
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4.
The Weighted Case
The following weighted Ostrowski inequality on time scales holds.
Theorem 4.1. Let a, b, s, t, τ ∈ T, a < b and f : [a, b] → R be differentiable,
q ∈ Crd . Then
Z b
σ
σ
f
(t)
−
q
(s)f
(s)∆s
a
Z b
(4.1)
≤
q σ (s) |σ(s) − t| M ∆s
a
 1q
p1 R
Rb

b
q
p
σ

(q (s)) ∆s , p1 + 1q = 1, p > 1;
|σ(s) − t| ∆s


a
a


sup q σ (s) [g2 (a, t) + g2 (b, t)] ;
(4.2)
≤M

a≤s<b




 b−σ(a) + t − b+σ(a) ,
2
2
where
M=
sup
∆ f (τ )
Ostrowski Inequalities on
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Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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σ(a)≤τ <b
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and
Z
b
q σ (s)∆s = 1,
q(s) ≥ 0.
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a
Proof. We have
Z b
σ
σ
f (t) −
q (s)f (s)∆s
a
Z b
= q σ (s) (f (t) − f σ (s)) ∆s
a
Close
Z
t
σ
Z
b
q (s) |f (t) − f (s)| ∆s +
q σ (s) |f (t) − f σ (s)| ∆s
a
t
Z t
Z t
Z b
Z σ(s)
∆ σ
∆
σ
f (τ ) ∆τ ∆s +
f (τ ) ∆τ ∆s
≤
q (s)
q (s)
≤
a
σ
σ(s)
Z
≤M
t
t
b
q σ (s) |σ(s) − t| ∆s,
a
and therefore (4.1) is shown.
The first part of (4.2) can be done easily by applying (2.2). By factoring sup q σ (s),
a≤s<b
we have
Z t
Z b
Z b
σ
σ
q (s) |σ(s) − t| ∆s ≤ sup q (s)
(t − σ(s))∆s +
(σ(s) − t)∆s
a
a≤s<b
a
= sup q σ (s)
a≤s<b
Z
Ostrowski Inequalities on
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Martin Bohner and
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vol. 9, iss. 1, art. 6, 2008
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t
a
Z
(σ(s) − t)∆s +
t
b
(σ(s) − t)∆s
t
σ
= sup q (s) [g2 (a, t) + g2 (b, t)] ,
a≤s<b
and therefore the second part of (4.2) holds. Finally for proving the third inequality,
we use the fact that
b − σ(a) b + σ(a) sup {|σ(s) − t|} = max {b − t, t − σ(a)} =
+ t −
.
2
2
a≤s<b
Thus (4.2) is shown.
Remark 1. Theorem 4.1 states a similar result as shown
R b in [3, Theorem 3.1], if we
consider the normalized isotonic functional A(f ) = a q σ (s)f σ (s)∆s. Moreover
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the second inequality of (4.2) is comparable to the achievement in [4, Theorem 3.1]
for the continuous case (see [4, Corollary 3.3]).
Corollary 4.2 (discrete
Pncase). Let T = Z. Let a = 0, b = n, s = j, t = i, τ = k
and f (k) = xk . Then i=1 qi = 1, qi ≤ 0 and
n
n
X
X
qj x j ≤
qj |j − i| M
xi −
j=1
j=1
 1
1 Pn
p p Pn

q q

, p1 + 1q = 1, p > 1;

j=1 qj
j=1 |j − i|


h 2
i
≤M
n+1 2
n −1
max q(j);
+ i− 2

4

j=1..n


 n−1 ,
+ i − n+1
2
2
where
M = max |∆xk | .
k=1..n−1
This is the result given in [2, Theorem 4.1].
Rb
Corollary 4.3 (continuous case). If T = R, then a q(s)ds = 1, q(s) ≥ 0 and
Z b
Z b
f (t) −
≤
q(s)f
(s)ds
q(s) |s − t| M ds
a
a
 p1 R
1q
Rb

b
p
q


|s
−
t|
ds
(q(s))
ds
, p1 + 1q = 1, p > 1;

a
a


a+b 2
≤M
1
2 (t− 2 )
sup q(s)(b − a)
+4 ;


(b−a)2

a≤s<b


 b−a ,
+ t − b+a
2
2
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vol. 9, iss. 1, art. 6, 2008
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where
M = sup |f 0 (τ )| .
a≤τ <b
Another interesting conclusion of Theorem 4.1 is the following corollary.
Corollary 4.4. Let a, b, s, t ∈ T and f differentiable. Then
Z b
1
σ
≤ M (h2 (t, a) + h2 (t, b)) ,
(4.3)
f
(t)
−
f
(s)∆s
b−a
b−a a
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
where
vol. 9, iss. 1, art. 6, 2008
M=
∆
sup |f (t)|.
σ(a)≤t<b
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Note that this was shown in a different manner in Theorem 3.5. In (4.3) we use
the fact that the functions g2 and h2 satisfy g2 (s, t) = (−1)2 h2 (t, s) for all t ∈ T and
all s ∈ Tκ (see [1, Theorem 1.112]).
Remark 2. Moreover note that there is a small difference of (4.2) in comparison to
Theorem 3.5, as we have sup instead of sup . This is just important if a is
σ(a)≤t<b
a<t<b
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right-dense, i.e., σ(a) = a. But in those cases the inequality does not change and is
still sharp. Furthermore in the proof of Theorem 3.5 we could have picked sup as
a≤t<b
explained before.
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References
[1] M. BOHNER AND A. PETERSON, Dynamic equations on time scales, Boston,
MA: Birkhäuser Boston Inc., 2001.
[2] 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]
[3] 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]
[4] 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]
[5] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities involving
functions and their integrals and derivatives, vol. 53 of Mathematics and its
Applications (East European Series). Dordrecht: Kluwer Academic Publisher
Group, p. 565, 1991.
[6] A. OSTROWSKI, Über die Absolutabweichung einer differenzierbaren Funktion von ihrem Integralmittelwert, Comment. Math. Helv., 10(1) (1937), 226–
227.
Ostrowski Inequalities on
Time Scales
Martin Bohner and
Thomas Matthews
vol. 9, iss. 1, art. 6, 2008
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Contents
JJ
II
J
I
Page 17 of 17
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