Volume 9 (2008), Issue 4, Article 112, 7 pp.
GENERALIZED OSTROWSKI’S INEQUALITY ON TIME SCALES
BAŞAK KARPUZ AND UMUT MUTLU ÖZKAN
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE AND A RTS , ANS C AMPUS
A FYON KOCATEPE U NIVERSITY
03200 A FYONKARAHISAR , T URKEY.
bkarpuz@gmail.com
D EPARTMENT OF M ATHEMATICS
FACULTY OF S CIENCE AND A RTS , ANS C AMPUS
A FYON KOCATEPE U NIVERSITY
03200 A FYONKARAHISAR , T URKEY.
umut_ozkan@aku.edu.tr
Received 31 July, 2008; accepted 08 October, 2008
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Montgomery’s identity, Ostrowski’s inequality, time scales.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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)
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
f (t) =
f σ (η)∆η +
Ψ(t, η)f ∆ (η)∆η
b−a
a
a
215-08
2
BA ŞAK K ARPUZ AND U MUT M UTLU Ö ZKAN
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
for s, t ∈ [a, b]T .
1
Theorem A (Ostrowski’s inequality). Let a, b ∈ T with a < b and f ∈ Crd
([a, b]T , R). Then
(
)
Z b
h2 (t, a) + h2 (t, b)
σ
∆
f (t) − 1
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.
2. T IME S CALES E SSENTIALS
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 left-dense points of T. The set
1
of rd-continuous functions is denoted by Crd (T, R), and Crd
(T, R) denotes the set of functions
for which the delta derivative belongs to Crd (T, R).
Theorem 2.1 (Existence of antiderivatives). Let f be a rd-continuous function. Then f has an
antiderivative F such that F ∆ = f holds.
Definition 2.5. 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 2.2. 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,
Rb
Ra
(2) a 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.
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 112, 7 pp.
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O STROWSKI ’ S I NEQUALITY ON T IME S CALES
Definition 2.6. Let hk : T2 → R be defined as follows:

1,
(2.1)
hk (t, s) := R
 t h (η, s)∆η,
s k−1
3
k=0
k∈N
for all s, t ∈ T and k ∈ N0 .
Note that the function hk satisfies
t
h∆
k (t, s) =
(2.2)

0,
k=0
h (t, s),
k−1
k∈N
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.
3. G ENERALIZATION
BY
G ENERALIZED P OLYNOMIALS
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
a
Now, we give a generalization for Montgomery’s identity as follows:
1
1
Lemma 3.2. Assume that a, b ∈ T and f ∈ Crd
([a, b]T , R). Define Ψ, Φ ∈ Crd
([a, b]T , R) by


hk (s, a), s ∈ [a, t)T
hk−1 (s, a), s ∈ [a, t)T
Ψ(t, s) :=
and Φ(t, s) :=
h (s, b), s ∈ [t, b]
h (s, b), s ∈ [t, b]
k
T
k−1
T
for s, t ∈ [a, b]T and k ∈ N. Then
(3.1)
1
f (t) =
hk (t, a) − hk (t, b)
b
Z
Z
σ
Φ(t, η)f (η)∆η +
a
b
∆
Ψ(t, η)f (η)∆η
a
is true for all t ∈ [a, b]T and all k ∈ N.
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
a
Z tZ
=
a
(3.2)
a
σ(η)
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
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 112, 7 pp.
a
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4
BA ŞAK K ARPUZ AND U MUT M UTLU Ö ZKAN
Applying Lemma 3.1 and considering (2.1), the right-hand side of (3.2) takes the form
Z tZ
t
Z tZ
∆
η
hk−1 (ξ, a)f ∆ (η)∆ξ∆η
hk−1 (η, a)f (ξ)∆η∆ξ + f (a)hk (t, a) +
a
ξ
a
Z tZ
hk−1 (η, a)f ∆ (ξ)∆η∆ξ + f (a)hk (t, a)
=
a
a
t
a
= f (t)hk (t, a),
(3.3)
and very similarly, from Lemma 3.1, (3.1), (2.1) and (2.2), we obtain
Z
b
σ
Z
b
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 bZ
σ(η)
Z bZ
∆
hk−1 (η, b)f (ξ)∆ξ∆η − f (t)hk (t, b) −
=
t
t
Z bZ
=
t
t
b
hk−1 (η, b)f ∆ (ξ)∆η∆ξ − f (t)hk (t, b) −
ξ
∆
hk (ξ, b)f ∆ (η) ξ ∆ξ∆η,
η
Z bZ
t
b
b
hk−1 (ξ, b)f ∆ (η)∆ξ∆η
η
(3.4) = −f (t)hk (t, b).
By summing (3.3) and (3.4), we get the desired result.
Now, we give the following generalization of Ostrowski’s inequality.
1
Theorem 3.3. Assume that a, b ∈ T and f ∈ Crd
([a, b]T , R). 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
(3.5)
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
Z b
Z t
M
,
≤
h
(η,
a)∆η
+
h
(η,
b)∆η
k
k
hk (t, a) − hk (t, b) a
t
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 112, 7 pp.
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O STROWSKI ’ S I NEQUALITY ON T IME S CALES
5
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)
which completes the proof.
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.
4. A PPLICATIONS FOR G ENERALIZED P OLYNOMIALS
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
(t − a)k+1 + (b − t)k+1
M
,
≤
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 .
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 112, 7 pp.
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6
BA ŞAK K ARPUZ AND U MUT M UTLU Ö ZKAN
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 ,
j=1
(t − s)kq :=
k−1
Y
(t − q j s)
for s, t ∈ q N0 and k ∈ N0 .
j=0
It is shown in [1, Example 1.104] that
(t − s)kq
hk (t, s) :=
[k]!
for s, t ∈ q N0 and k ∈ N0
holds.
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)
k
k
(t − a)q − (t − b)q
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].
5. G ENERALIZATION BY A RBITRARY F UNCTIONS
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
for s, t ∈ [a, b]T . Then
1
f (t) =
ψ(t) − φ(t)
1
=
ψ(t) − φ(t)
Z
b
∆η
Ψ(t, η)f (η) ∆η
a
Z
b
σ
Z
Φ(t, η)f (η)∆η +
a
b
∆
Ψ(t, η)f (η)∆η
a
is true for all t ∈ [a, b]T .
J. Inequal. Pure and Appl. Math., 9(4) (2008), Art. 112, 7 pp.
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O STROWSKI ’ S I NEQUALITY ON T IME S CALES
7
1
1
Theorem 5.2. 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 . Then
Z b
Z b
M
1
σ
f (t) −
Φ(t, η)f (η)∆η ≤
|Ψ(t, η)|∆η
ψ(t) − φ(t)
|ψ(t) − φ(t)|
a
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)|.
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.
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