Volume 9 (2008), Issue 1, Article 6, 8 pp.
OSTROWSKI INEQUALITIES ON TIME SCALES
MARTIN BOHNER AND THOMAS MATTHEWS
M ISSOURI U NIVERSITY OF S CIENCE AND T ECHNOLOGY
D EPARTMENT OF M ATHEMATICS AND S TATISTICS
ROLLA , MO 65409-0020, USA
bohner@mst.edu
tmnqb@mst.edu
Received 06 July, 2007; accepted 15 February, 2008
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Ostrowski inequality, Montgomery identity, Time scales.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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
2
f (t) − 1
(1.1)
f (s)ds ≤ sup |f 0 (t)| (b − a)
+
2
b−a a
(b − a)
4
a<t<b
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.
The authors thank the referees for their careful reading of the manuscript and insightful comments.
226-07
2
M ARTIN B OHNER AND T HOMAS M ATTHEWS
2. T IME S CALES E SSENTIALS
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 2.1.
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 2.1: 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.
Theorem 2.1 (Existence of Antiderivatives). Let f be rd-continuous. Then f has an antiderivative F satisfying F ∆ = f .
Proof. See [1, Theorem 1.74].
Definition 2.4. 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
f (τ )∆τ = F (b) − F (a), where F ∆ = f .
Theorem 2.2. 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,
Rb
Ra
(2) a f (t)∆t = − b f (t)∆t,
Rb
Rc
Rb
(3) a f (t)∆t = a f (t)∆t + c f (t)∆t,
Rb
Rb
(4) a f (t)g ∆ (t)∆t = (f g)(b) − (f g)(a) − a f ∆ (t)g(σ(t))∆t,
Ra
(5) a f (t)∆t = 0.
Proof. See [1, Theorem 1.77].
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 6, 8 pp.
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O STROWSKI I NEQUALITIES ON T IME S CALES
3
Definition 2.5. 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
t
hk+1 (t, s) =
hk (τ, s)∆τ
for all s, t ∈ T.
s
Theorem 2.3 (Hölder’s Inequality). Let a, b ∈ T and f, g : [a, b] → R be rd-continuous. Then
1q
Z b
p1 Z b
Z b
(2.2)
|f (t)g(t)| ∆t ≤
|f (t)|p ∆t
|g(t)|q ∆t ,
a
where p > 1 and
1
p
a
+
1
q
a
= 1.
Proof. See [1, Theorem 6.13].
3. T HE O STROWSKI I NEQUALITY ON T IME S CALES
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
(3.1)
f (t) =
f (s)∆s +
p(t, s)f ∆ (s)∆s,
b−a a
b−a a
where
(
s − a, a ≤ s < t,
p(t, s) =
(3.2)
s − b, t ≤ s ≤ b.
Proof. Using Theorem 2.2 (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
b
∆
Z
(s − b)f (s)∆s = (b − t)f (t) −
t
b
f σ (s)∆s.
t
Therefore
1
b−a
Z
a
b
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),
σ
i.e., (3.1) holds.
If we apply Lemma 3.1 to the discrete and continuous cases, we have the following results.
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 6, 8 pp.
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4
M ARTIN B OHNER AND T HOMAS M ATTHEWS
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 for 1 ≤ j ≤ n − 1,
p(n, j) = j for 0 ≤ j ≤ n − 1,
(
j,
0 ≤ j < i,
p(i, j) =
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].
Corollary 3.3 (continuous case). We let T = R. Then
Z b
Z b
1
1
f (s)ds +
p(t, s)f 0 (s)ds.
f (t) =
b−a a
b−a a
This is the Montgomery identity in the continuous case, which can be found in [5, p. 565].
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 k k+1
q f (q )
n−1
X
k+1
1
k=m
k
f (t) =
+
f
(q
)
−
f
(q
)
p(t, q k ),
n−1
P k
q n − q m k=m
q
k=m
where
(
k
p(t, q ) =
q k − q m , q m ≤ q k < t,
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)|.
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
σ
∆
f (t) − 1
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
Z t
Z b
M
=
(s − a) ∆s +
(b − s) ∆s
b−a
a
t
M
=
(h2 (t, a) + h2 (t, b)) .
b−a
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O STROWSKI I NEQUALITIES ON T IME S CALES
5
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
σ
f (t) − 1
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
t1
Z 1t2
Z t2
1
2
∆
= t2 −
(s ) ∆s −
s∆s t2 − t1
t1
t1
Z t2
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
Therefore in this particular case
Z b
1
M
σ
f (t) −
f (s)∆s ≥
(h2 (t, a) + h2 (t, b))
b−a a
b−a
and by (3.3) also
Z b
1
σ
≤ M (h2 (t, a) + h2 (t, b)) .
f (t) −
f
(s)∆s
b−a
b−a
a
So the sharpness of the Ostrowski inequality is shown.
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
1
n
+
1
n
−
1
i −
+
(3.4)
xj ≤
,
xi −
n j=1 n 2 4
where
M = max |∆xi | .
1≤i≤n−1
This is the discrete Ostrowski inequality from [2, Theorem 3.1], where the constant 41 in the
right-hand side of (3.4) is the best possible in the sense that it cannot be replaced by a smaller
one.
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
where
M = sup |f 0 (t)| .
a<t<b
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 6, 8 pp.
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6
M ARTIN B OHNER AND T HOMAS M ATTHEWS
This is the Ostrowski inequality in the continuous case [6, p. 226–227], where again the constant 14 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
2
M
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
1
4
f (qt) − f (t) ,
M = sup (q − 1)t q m <t<q n
in the right-hand side is the best possible.
4. T HE W EIGHTED C ASE
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
q σ (s)f σ (s)∆s
f (t) −
a
Z b
(4.1)
≤
q σ (s) |σ(s) − t| M ∆s
a
 p1 R
1q
Rb

b
p
q
σ

|σ(s) − t| ∆s
(q (s)) ∆s , p1 + 1q = 1, p > 1;


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 )
σ(a)≤τ <b
and
b
Z
q σ (s)∆s = 1,
q(s) ≥ 0.
a
Proof. We have
Z b
Z b
σ
σ
σ
σ
f (t) −
=
q
(s)f
(s)∆s
q
(s)
(f
(t)
−
f
(s))
∆s
a
a
Z t
Z b
σ
σ
≤
q (s) |f (t) − f (s)| ∆s +
q σ (s) |f (t) − f σ (s)| ∆s
a
Z
≤
t
t
Z
σ
t
q (s)
a
∆ f (τ ) ∆τ ∆s +
σ(s)
Z
≤M
Z
b
σ
Z
q (s)
t
σ(s)
∆ f (τ ) ∆τ ∆s
t
b
q σ (s) |σ(s) − t| ∆s,
a
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O STROWSKI I NEQUALITIES ON T IME S CALES
7
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), we have
a≤s<b
Z
b
σ
Z
σ
t
q (s) |σ(s) − t| ∆s ≤ sup q (s)
a
(t − σ(s))∆s +
a≤s<b
a
Z
σ
b
Z
(σ(s) − t)∆s
t
a
Z
(σ(s) − t)∆s +
= sup q (s)
a≤s<b
t
b
(σ(s) − t)∆s
t
= sup q σ (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 4.2. Theorem 4.1 states a similar result
in [3, Theorem 3.1], if we consider
R b σ as shown
σ
the normalized isotonic functional A(f ) = a q (s)f (s)∆s. Moreover 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
Pn 4.3 (discrete case). 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

! p1
! 1q


n
n
P
P

1


|j − i|p
qjq
,
+ 1q = 1, p > 1;

p

 j=1
j=1
h 2
i
≤M
n+1 2
n −1

+
i
−
max q(j);

4
2

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 4.4 (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

q
p

|s
−
t|
ds
(q(s))
ds
, p1 + 1q = 1,

a
a




a+b 2
≤M
1
2 (t− 2 )
+4 ;
sup q(s)(b − a)


(b−a)2

a≤s<b



 b−a + t − b+a ,
2
J. Inequal. Pure and Appl. Math., 9(1) (2008), Art. 6, 8 pp.
p > 1;
2
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8
M ARTIN B OHNER AND T HOMAS M ATTHEWS
where
M = sup |f 0 (τ )| .
a≤τ <b
Another interesting conclusion of Theorem 4.1 is the following corollary.
Corollary 4.5. Let a, b, s, t ∈ T and f differentiable. Then
Z b
1
σ
f (t) −
≤ M (h2 (t, a) + h2 (t, b)) ,
(4.3)
f
(s)∆s
b−a
b−a
a
where
M=
sup |f ∆ (t)|.
σ(a)≤t<b
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 4.6. 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 right-dense, i.e.,
σ(a)≤t<b
a<t<b
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 explained before.
a≤t<b
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220]
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http://jipam.vu.edu.au/