ON AN INTEGRATION-BY-PARTS FORMULA FOR
MEASURES
Integration-by-parts Formula
A. ČIVLJAK
Lj. DEDIĆ AND M. MATIĆ
American College of Management & Technology
Rochester Institute of Technology
1Don Frana Bulica 6,
20000 Dubrovnik, Croatia
EMail: acivljak@acmt.hr
Department of Mathematics
Fac. of Natural Sciences, Mathematics & Education
1060 University of Split
Teslina 12, 21000 Split, Croatia
EMail: {ljuban,mmatic}@pmfst.hr
Received:
21 June, 2007
Accepted:
15 November, 2007
Communicated by:
W.S. Cheung
2000 AMS Sub. Class.:
26D15, 26D20, 26D99.
Key words:
Integration-by-parts formula, Harmonic sequences, Inequalities.
Abstract:
An integration-by-parts formula, involving finite Borel measures supported by
intervals on real line, is proved. Some applications to Ostrowski-type and Grüsstype inequalities are presented.
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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Contents
1
Introduction
3
2
Integration-by-parts Formula for Measures
5
3
Some Ostrowski-type Inequalities
13
Integration-by-parts Formula
4
Some Grüss-type Inequalities
22
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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1.
Introduction
In the paper [4], S.S. Dragomir introduced the notion of a w0 -Appell type sequence
of functions as a sequence w0 , w1 , . . . , wn , for n ≥ 1, of real absolutely continuous
functions defined on [a, b], such that
wk0 = wk−1 , a.e. on [a, b],
k = 1, . . . , n.
Integration-by-parts Formula
For such a sequence the author proved a generalisation of Mitrinović-Pečarić integrationby-parts formula
Z b
(1.1)
w0 (t)g(t)dt = An + Bn ,
a
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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where
An =
n
X
(−1)k−1 wk (b)g (k−1) (b) − wk (a)g (k−1) (a)
k=1
and
n
Z
Bn = (−1)
b
for every g : [a, b]→R such that g
is absolutely continuous on [a, b] and wn g
L1 [a, b]. Using identity (1.1) the author proved the following inequality
Z b
≤ kwn k kg (n) kq ,
(1.2)
w
(t)g(t)dt
−
A
0
n
p
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wn (t)g (n) (t)dt,
a
(n−1)
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(n)
∈
a
for wn ∈ Lp [a, b], g (n) ∈ Lp [a, b], where p, q ∈ [1, ∞] and 1/p + 1/q = 1, giving
explicitly some interesting special cases. For some similar inequalities, see also [5],
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[6] and [7]. The aim of this paper is to give a generalization of the integration-byparts formula (1.1), by replacing the w0 -Appell type sequence of functions by a more
general sequence of functions, and to generalize inequality (1.2), as well as to prove
some related inequalities.
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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2.
Integration-by-parts Formula for Measures
For a, b ∈ R, a < b, let C[a, b] be the Banach space of all continuous functions
f : [a, b]→R with the max norm, and M [a, b] the Banach space of all real Borel
measures on [a, b] with the total variation norm. For µ ∈ M [a, b] define the function
µ̌n : [a, b]→R, n ≥ 1, by
Z
1
µ̌n (t) =
(t − s)n−1 dµ(s).
(n − 1)! [a,t]
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
Note that
1
µ̌n (t) =
(n − 2)!
and
Z
t
(t − s)n−2 µ̌1 (s)ds,
n≥2
a
(t − a)n−1
|µ̌n (t)| ≤
kµk ,
(n − 1)!
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t ∈ [a, b], n ≥ 1.
The function µ̌n is differentiable, µ̌0n (t) = µ̌n−1 (t) and µ̌n (a) = 0, for every n ≥ 2,
while for n = 1
Z
µ̌1 (t) =
dµ(s) = µ([a, t]),
[a,t]
which means that µ̌1 (t) is equal to the distribution function of µ. A sequence of
functions Pn : [a, b] → R, n ≥ 1, is called a µ-harmonic sequence of functions on
[a, b] if
Pn0 (t) = Pn−1 (t), n ≥ 2; P1 (t) = c + µ̌1 (t), t ∈ [a, b],
for some c ∈ R. The sequence (µ̌n , n ≥ 1) is an example of a µ-harmonic sequence
of functions on [a, b]. The notion of a µ-harmonic sequence of functions has been
introduced in [2]. See also [1].
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Remark 1. Let w0 : [a, b] → R be an absolutely integrable function and let µ ∈
M [a, b] be defined by
dµ(t) = w0 (t)dt.
If (Pn , n ≥ 1) is a µ-harmonic sequence of functions on [a, b], then w0 , P1 , . . . , Pn
is a w0 -Appell type sequence of functions on [a, b].
For µ ∈ M [a, b] let µ = µ+ − µ− be the Jordan-Hahn decomposition of µ, where
µ+ and µ− are orthogonal and positive measures. Then we have |µ| = µ+ + µ− and
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
kµk = |µ| ([a, b]) = kµ+ k + kµ− k = µ+ ([a, b]) + µ− ([a, b]).
vol. 8, iss. 4, art. 93, 2007
The measure µ ∈ M [a, b] is said to be balanced if µ([a, b]) = 0. This is equivalent to
kµ+ k = kµ− k =
1
kµk .
2
Measure µ ∈ M [a, b] is called n-balanced if µ̌n (b) = 0. We see that a 1-balanced
measure is the same as a balanced measure. We also write
Z
mk (µ) =
tk dµ(t), k ≥ 0
[a,b]
for the k-th moment of µ.
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Lemma 2.1. For every f ∈ C[a, b] and µ ∈ M [a, b] we have
Z
Z
f (t)dµ̌1 (t) =
f (t)dµ(t) − µ({a})f (a).
[a,b]
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[a,b]
Proof. Define I, J : C[a, b] × M [a, b] → R by
Z
I(f, µ) =
f (t)dµ̌1 (t)
[a,b]
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and
Z
f (t)dµ(t) − µ({a})f (a).
J(f, µ) =
[a,b]
Then I and J are continuous bilinear functionals, since
|I(f, µ)| ≤ kf k kµk ,
|J(f, µ)| ≤ 2 kf k kµk .
Let us prove that I(f, µ) = J(f, µ) for every f ∈ C[a, b] and every discrete
measure µ ∈ M [a, b].
For x ∈ [a, b] let µ = δx be the Dirac measure at x, i.e. the measure defined by
R
f (t)dδx (t) = f (x).
[a,b]
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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If a < x ≤ b, then
(
µ̌1 (t) = δx ([a, t]) =
0, a ≤ t < x
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1, x ≤ t ≤ b
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and by a simple calculation we have
Z
R
f (t)dµ̌1 (t) = f (x) =
I(f, δx ) =
f (t)dδx (t) − 0
[a,b]
=
R
[a,b]
f (t)dδx (t) − δx ({a})f (a) = J(f, δx ).
[a,b]
Similarly, if x = a, then
µ̌1 (t) = δa ([a, t]) = 1,
a≤t≤b
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and by a similar calculation we have
Z
I(f, δa ) =
f (t)dµ̌1 (t) = 0 = f (a) − f (a)
[a,b]
R
=
f (t)dδa (t) − δa ({a})f (a) = J(f, δx ).
[a,b]
Therefore, for every f ∈ C[a, b] and every x ∈ [a, b] we have I(f, δx ) = J(f, δx ).
Every discrete measure µ ∈ M [a, b] has the form
X
µ=
ck δxk ,
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
k≥1
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where (ck , k ≥ 1) is a sequence in R such that
X
|ck | < ∞,
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k≥1
and {xk ; k ≥ 1} is a subset of [a, b].
By using the continuity of I and J, for every f ∈ C[a, b] and every discrete
measure µ ∈ M [a, b] we have
!
X
X
I(f, µ) = I f,
ck δxk =
ck I(f, δxk )
k≥1
X
ck J(f, δxk ) = J
k≥1
= J(f, µ).
f,
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k≥1
!
=
Page 8 of 26
X
k≥1
ck δxk
Since the Banach subspace M [a, b]d of all discrete measures is weakly∗ dense
in M [a, b] and the functionals I(f, ·) and J(f, ·) are also weakly∗ continuous we
conclude that I(f, µ) = J(f, µ) for every f ∈ C[a, b] and µ ∈ M [a, b].
Theorem 2.2. Let f : [a, b] → R be such that f (n−1) has bounded variation for some
n ≥ 1. Then for every µ-harmonic sequence (Pn , n ≥ 1) we have
Z
(2.1)
f (t)dµ(t) = µ({a})f (a) + Sn + Rn ,
A. Čivljak, Lj. Dedić and M. Matić
[a,b]
vol. 8, iss. 4, art. 93, 2007
where
(2.2)
Integration-by-parts Formula
Sn =
n
X
(−1)k−1 Pk (b)f (k−1) (b) − Pk (a)f (k−1) (a)
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k=1
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and
(2.3)
Rn = (−1)n
Z
Pn (t)df (n−1) (t).
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[a,b]
Proof. By partial integration, for n ≥ 2, we have
Z
n
Rn = (−1)
Pn (t)df (n−1) (t)
[a,b]
= (−1)n Pn (b)f (n−1) (b) − Pn (a)f (n−1) (a)
Z
n
− (−1)
Pn−1 (t)f (n−1) (t)dt
[a,b]
n
= (−1) Pn (b)f (n−1) (b) − Pn (a)f (n−1) (a) + Rn−1 .
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By Lemma 2.1 we have
Z
R1 = −
P1 (t)df (t)
[a,b]
Z
= − [P1 (b)f (b) − Pn (a)f (a)] +
f (t)dP1 (t)
[a,b]
Z
= − [P1 (b)f (b) − Pn (a)f (a)] +
f (t)dµ̌1 (t)
A. Čivljak, Lj. Dedić and M. Matić
[a,b]
Z
= − [P1 (b)f (b) − Pn (a)f (a)] +
Integration-by-parts Formula
f (t)dµ(t) − µ({a})f (a).
vol. 8, iss. 4, art. 93, 2007
[a,b]
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Therefore, by iteration, we have
Rn =
n
X
k
(−1) Pk (b)f (k−1) (b) − Pk (a)f (k−1) (a) +
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Z
f (t)dµ(t)−µ({a})f (a),
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which proves our assertion.
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Remark 2. By Remark 1 we see that identity (2.1) is a generalization of the integrationby-parts formula (1.1).
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[a,b]
k=1
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(n−1)
Corollary 2.3. Let f : [a, b] → R be such that f
has bounded variation for
some n ≥ 1. Then for every µ ∈ M [a, b] we have
Z
f (t)dµ(t) = Šn + Řn ,
[a,b]
where
Šn =
n
X
k=1
(−1)k−1 µ̌k (b)f (k−1) (b)
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and
n
Z
µ̌n (t)df (n−1) (t).
Řn = (−1)
[a,b]
Proof. Apply the theorem above for the µ-harmonic sequence (µ̌n , n ≥ 1) and note
that µ̌n (a) = 0, for n ≥ 2.
Corollary 2.4. Let f : [a, b] → R be such that f (n−1) has bounded variation for
some n ≥ 1. Then for every x ∈ [a, b] we have
f (x) =
n
X
k=1
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
k−1
(x − b)
f (k−1) (b) + Rn (x),
(k − 1)!
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where
(−1)n
Rn (x) =
(n − 1)!
Z
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(t − x)n−1 df (n−1) (t).
[x,b]
Proof. Apply Corollary 2.3 for µ = δx and note that in this case
µ̌k (t) =
(t − x)k−1
,
(k − 1)!
x ≤ t ≤ b,
and
µ̌k (t) = 0,
a ≤ t < x,
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for k ≥ 1.
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Corollary 2.5. Let f : [a, b] → R be such that f (n−1) has bounded variation for
some n ≥ 1. Further, let (cm , m ≥ 1) be a sequence in R such that
X
|cm | < ∞
m≥1
Close
and let {xm ; m ≥ 1} ⊂ [a, b]. Then
X
m≥1
cm f (xm ) =
n
XX
m≥1 k=1
cm
X
(xm − b)k−1 (k−1)
cm Rn (xm ),
f
(b) +
(k − 1)!
m≥1
where Rn (xm ) is from Corollary 2.4.
Proof. Apply Corollary 2.3 for the discrete measure µ =
P
m≥1 cm δxm .
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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3.
Some Ostrowski-type Inequalities
In this section we shall use the same notations as above.
Theorem 3.1. Let f : [a, b] → R be such that f (n−1) is L-Lipschitzian for some
n ≥ 1. Then for every µ-harmonic sequence (Pn , n ≥ 1) we have
Z
Z b
(3.1)
f (t)dµ(t) − µ({a})f (a) − Sn ≤ L
|Pn (t)| dt,
[a,b]
a
where Sn is defined by (2.2).
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a
which proves our assertion.
Corollary 3.2. If f is L-Lipschitzian, then for every c ∈ R and µ ∈ M [a, b] we have
Z
Z b
f (t)dµ(t) − µ([a, b])f (b) − c [f (b) − f (a)] ≤ L
|c + µ̌1 (t)| dt.
[a,b]
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
Proof. By Theorem 2.2 we have
Z
Z b
(n−1)
|Rn | = Pn (t)df
(t) ≤ L
|Pn (t)| dt,
[a,b]
Integration-by-parts Formula
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a
Proof. Put n = 1 in the theorem above and note that P1 (t) = c + µ̌1 (t), for some
c ∈ R.
Corollary 3.3. If f is L-Lipschitzian, then for every c ≥ 0 and µ ≥ 0 we have
Z
f (t)dµ(t) − µ([a, b])f (b) − c [f (b) − f (a)]
[a,b]
≤ L [c(b − a) + µ̌2 (b)]
≤ L(b − a)(c + kµk).
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Proof. Apply Corollary 3.2 and note that in this case
Z b
Z b
|c + µ̌1 (t)| dt =
[c + µ̌1 (t)] dt
a
a
= c(b − a) + µ̌2 (b)
≤ c(b − a) + (b − a) kµk
= (b − a)(c + kµk).
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
Corollary 3.4. Let f be L-Lipschitzian, (cm , m ≥ 1) a sequence in [0, ∞) such that
X
cm < ∞,
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m≥1
Contents
and let {xm ; m ≥ 1} ⊂ [a, b]. Then for every c ≥ 0 we have
#
"
X
X
cm [f (b) − f (xm )] + c [f (b) − f (a)] ≤ L c(b − a) +
cm (b − xm )
m≥1
m≥1
"
#
X
≤ L(b − a) c +
cm .
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m≥1
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Proof. Apply Corollary 3.3 for the discrete measure µ =
P
m≥1 cm δxm .
Corollary 3.5. If f is L-Lipschitzian and µ ≥ 0, then
Z
f
(t)dµ(t)
−
µ([a,
x])f
(a)
−
µ((x,
b])f
(b)
[a,b]
≤ L [(2x − a − b)µ̌1 (x) − 2µ̌2 (x) + µ̌2 (b)] ,
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for every x ∈ [a, b].
Proof. Apply Corollary 3.2 for c = −µ̌1 (x). Then
c + µ̌1 (b) = µ((x, b]),
µ̌1 (x) = µ([a, x])
and
Z
b
Z
|−µ̌1 (x) + µ̌1 (t)| dt =
x
Z
(µ̌1 (x) − µ̌1 (t)) dt +
a
a
b
(µ̌1 (t) − µ̌1 (x)) dt
x
= (2x − a − b)µ̌1 (x) − 2µ̌2 (x) + µ̌2 (b).
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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(n−1)
Corollary 3.6. Let f : [a, b] → R be such that f
is L-Lipschitzian for some
n ≥ 1. Then for every µ ∈ M [a, b] we have
Z
Z b
(b − a)n
f
(t)dµ(t)
−
Š
≤
L
|µ̌
(t)|
dt
≤
L kµk ,
n
n
n!
[a,b]
a
where Šn is from Corollary 2.3.
Proof. Apply the theorem above for the µ-harmonic sequence (µ̌n , n ≥ 1).
Corollary 3.7. Let f : [a, b] → R be such that f (n−1) is L-Lipschitzian for some
n ≥ 1. Then for every x ∈ [a, b] we have
n
(b − x)n
k−1
X
(x
−
b)
(k−1)
f
(b)
L.
f
(x)
−
≤
(k
−
1)!
n!
k=1
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Proof. Apply Corollary 3.6 for µ = δx and note that in this case
µ̌k (t) =
(t − x)k−1
, x ≤ t ≤ b,
(k − 1)!
µ̌k (t) = 0, a ≤ t < x,
and
for k ≥ 1.
Corollary 3.8. Let f : [a, b] → R be such that f (n−1) is L-Lipschitzian, for some
n ≥ 1. Further, let (cm , m ≥ 1) be a sequence in R such that
X
|cm | < ∞
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
m≥1
and let {xm ; m ≥ 1} ⊂ [a, b]. Then
n
X
XX
(xm − b)k−1 (k−1) cm
f
(b)
cm f (xm ) −
(k − 1)!
m≥1
m≥1 k=1
L X
≤
|cm | (b − xm )n
n! m≥1
X
L
≤ (b − a)n
|cm | .
n!
m≥1
P
Proof. Apply Corollary 3.6 for the discrete measure µ = m≥1 cm δxm .
(n−1)
Theorem 3.9. Let f : [a, b] → R be such that f
has bounded variation for some
n ≥ 1. Then for every µ-harmonic sequence (Pn , n ≥ 1) we have
Z
b
_
f (t)dµ(t) − µ({a})f (a) − Sn ≤ max |Pn (t)| (f (n−1) ),
[a,b]
t∈[a,b]
a
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where
Wb
a (f
(n−1)
) is the total variation of f (n−1) on [a, b].
Proof. By Theorem 2.2 we have
Z
b
_
(n−1)
|Rn | = Pn (t)df
(t) ≤ max |Pn (t)| (f (n−1) ),
[a,b]
t∈[a,b]
a
which proves our assertion.
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
Corollary 3.10. If f is a function of bounded variation, then for every c ∈ R and
µ ∈ M [a, b] we have
Z
[a,b]
b
_
f (t)dµ(t) − µ([a, b])f (b) − c [f (b) − f (a)] ≤ max |c + µ̌1 (t)| (f ).
t∈[a,b]
vol. 8, iss. 4, art. 93, 2007
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a
Proof. Put n = 1 in the theorem above.
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Corollary 3.11. If f is a function of bounded variation, then for every c ≥ 0 and
µ ≥ 0 we have
Z
b
_
f (t)dµ(t) − µ([a, b])f (b) − c [f (b) − f (a)] ≤ [c + kµk] (f ).
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[a,b]
a
Proof. In this case we have
max |c + µ̌1 (t)| = c + µ̌1 (b) = c + kµk .
t∈[a,b]
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Corollary 3.12. Let f be a function of bounded variation, (cm , m ≥ 1) a sequence
in [0, ∞) such that
X
cm < ∞
m≥1
and let {xm ; m ≥ 1} ⊂ [a, b]. Then for every c ≥ 0 we have
"
# b
X
X
_
cm [f (b) − f (xm )] + c [f (b) − f (a)] ≤ c +
cm
(f ).
m≥1
m≥1
Proof. Apply Corollary 3.11 for the discrete measure µ =
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
a
vol. 8, iss. 4, art. 93, 2007
P
m≥1 cm δxm .
Corollary 3.13. If f is a function of bounded variation and µ ≥ 0, then we have
Z
f (t)dµ(t) − µ([a, x])f (a) − µ((x, b])f (b)
[a,b]
b
_
1
≤ [µ̌1 (b) − µ̌1 (a) + |µ̌1 (a) + µ̌1 (b) − 2µ̌1 (x)|] (f ).
2
a
Proof. Apply Corollary 3.11 for c = −µ̌1 (x). Then
max |c + µ̌1 (t)| = max |µ̌1 (t) − µ̌1 (x)|
t∈[a,b]
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t∈[a,b]
= max{µ̌1 (x) − µ̌1 (a), µ̌1 (b) − µ̌1 (x)}
1
= [µ̌1 (b) − µ̌1 (a) + |µ̌1 (a) + µ̌1 (b) − 2µ̌1 (x)|] .
2
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Corollary 3.14. Let f : [a, b] → R be such that f (n−1) has bounded variation for
some n ≥ 1. Then for every µ ∈ M [a, b] we have
Z
b
_
f (t)dµ(t) − Šn ≤ max |µ̌n (t)| (f (n−1) )
t∈[a,b]
[a,b]
a
b
_
(b − a)n−1
kµk (f (n−1) ),
≤
(n − 1)!
a
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
where Šn is from Corollary 2.3.
Proof. Apply the theorem above for the µ-harmonic sequence (µ̌n , n ≥ 1).
Corollary 3.15. Let f : [a, b] → R be such that f (n−1) has bounded variation for
some n ≥ 1. Then for every x ∈ [a, b] we have
b
n
(b − x)n−1 _
k−1
X
(x
−
b)
(k−1)
f
(b)
≤
(f (n−1) ).
f
(x)
−
(k
−
1)!
(n
−
1)!
a
k=1
Proof. Apply Corollary 3.14 for µ = δx and note that in this case
(b − x)n−1
max |µ̌n (t)| =
.
t∈[a,b]
(n − 1)!
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Corollary 3.16. Let f : [a, b] → R be such that f (n−1) has bounded variation for
some n ≥ 1. Further, let (cm , m ≥ 1) be a sequence in R such that
X
|cm | < ∞
m≥1
and let {xm ; m ≥ 1} ⊂ [a, b]. Then
n
X
k−1
X
X
(xm − b)
(k−1)
cm f (xm ) −
cm
f
(b)
(k − 1)!
m≥1
≤
≤
m≥1 k=1
1
(n − 1)!
b
_
(f (n−1) )
a
|cm | (b − xm )n−1
m≥1
b
n−1 _
(b − a)
(n − 1)!
X
(f (n−1) )
a
Integration-by-parts Formula
X
A. Čivljak, Lj. Dedić and M. Matić
|cm |
vol. 8, iss. 4, art. 93, 2007
m≥1
Proof. Apply Corollary 3.14 for the discrete measure µ =
P
m≥1 cm δxm .
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Theorem 3.17. Let f : [a, b] → R be such that f (n) ∈ Lp [a, b] for some n ≥ 1. Then
for every µ-harmonic sequence (Pn , n ≥ 1) we have
Z
f (t)dµ(t) − µ({a})f (a) − Sn ≤ kPn kq kf (n) kp ,
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where p, q ∈ [1, ∞] and 1/p + 1/q = 1.
Page 20 of 26
[a,b]
Proof. By Theorem 2.2 and the Hölder inequality we have
Z
Z
(n−1)
(n)
|Rn | = Pn (t)df
(t) = Pn (t)f (t)dt
[a,b]
Z
≤
[a,b]
1q Z b
p1
b
(n) p
q
f (t) dt
|Pn (t)| dt
a
= kPn kq kf
a
(n)
kp .
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Remark 3. We see that the inequality of the theorem above is a generalization of
inequality (1.2).
Corollary 3.18. Let f : [a, b] → R be such that f (n) ∈ Lp [a, b] for some n ≥ 1, and
µ ∈ M [a,b]. Then
Z
≤ kµ̌n k kf (n) kp
f
(t)dµ(t)
−
Š
n
q
[a,b]
≤
(b − a)n−1+1/q
(n − 1)! [(n − 1)q + 1]1/q
Integration-by-parts Formula
kµk kf
(n)
kp ,
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
where p, q ∈ [1, ∞] and 1/p + 1/q = 1.
Proof. Apply the theorem above for the µ-harmonic sequence (µ̌n , n ≥ 1).
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(n)
Corollary 3.19. Let f : [a, b] → R be such that f
∈ Lp [a, b], for some n ≥ 1.
Further, let (cm , m ≥ 1) be a sequence
X in R such that
|cm | < ∞
m≥1
and let {xm ; m ≥ 1} ⊂ [a, b]. Then
n
X
k−1
X
X
(xm − b)
(k−1)
cm f (xm ) −
cm
f
(b)
(k − 1)!
m≥1
m≥1 k=1
X
kf (n) kp
≤
|cm | (b − xm )n−1+1/q
(n − 1)! [(n − 1)q + 1]1/q m≥1
(b − a)n−1+1/q kf (n) kp X
≤
|cm | ,
(n − 1)! [(n − 1)q + 1]1/q m≥1
where p, q ∈ [1, ∞] and 1/p + 1/q = 1.
Proof. Apply the theorem above for the discrete measure µ =
P
m≥1 cm δxm .
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4.
Some Grüss-type Inequalities
Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b], for some n ≥ 1. Then
mn ≤ f (n) (t) ≤ Mn ,
t ∈ [a, b], a.e.
for some real constants mn and Mn .
Theorem 4.1. Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b], for some n ≥ 1.
Further, let (Pk , k ≥ 1) be a µ-harmonic sequence such that
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
Pn+1 (a) = Pn+1 (b) ,
for that particular n. Then
Z
Z
Mn − m n b
|Pn (t)| dt.
f (t)dµ(t) − µ({a})f (a) − Sn ≤
2
a
[a,b]
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Proof. Apply Theorem 2.2 for the special case when f (n−1) is absolutely continuous
and its derivative f (n) , existing a.e., is bounded a.e. Define the measure νn by
Page 22 of 26
dνn (t) = −Pn (t) dt.
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Then
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Z
νn ([a, b]) = −
b
Pn (t) dt = Pn+1 (a) − Pn+1 (b) = 0,
a
which means that νn is balanced. Further,
Z b
kνn k =
|Pn (t)| dt
a
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and by [1, Theorem 2]
Z b
(n)
|Rn | = Pn (t) f (t)dt
a
Mn − m n
kνn k
2
Z b
Mn − m n
|Pn (t)| dt,
=
2
a
≤
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
which proves our assertion.
Corollary 4.2. Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b], for some n ≥ 1.
Then for every (n + 1)-balanced measure µ ∈ M [a, b] we have
Z
Z
Mn − m n b
f (t)dµ(t) − Šn ≤
|µ̌n (t)| dt
2
[a,b]
a
Mn − mn (b − a)n
kµk ,
≤
2
n!
where Šn is from Corollary 2.3.
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Proof. Apply Theorem 4.1 for the µ-harmonic sequence (µ̌k , k ≥ 1) and note that
the condition Pn+1 (a) = Pn+1 (b) reduces to µ̌n+1 (b) = 0, which means that µ is
(n + 1)-balanced.
Corollary 4.3. Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b] for some n ≥ 1.
Further, let (cm , m ≥ 1) be a sequence in R such that
X
|cm | < ∞
m≥1
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and let {xm ; m ≥ 1} ⊂ [a, b] satisfy the condition
X
cm (b − xm )n = 0.
m≥1
Then
n
X
k−1
X
X
(x
−
b)
m
cm f (xm ) −
f (k−1) (b)
cm
(k
−
1)!
m≥1
m≥1 k=1
Mn − m n X
≤
|cm | (b − xm )n
2n!
m≥1
X
Mn − m n
≤
(b − a)n
|cm | .
2n!
m≥1
Proof. Apply Corollary 4.2 for the discrete measure µ =
P
m≥1 cm δxm .
Corollary 4.4. Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b] for some n ≥ 1. Then
for every µ ∈ M [a, b], such that all k-moments of µ are zero for k = 0, . . . , n, we
have
Z
Z
Mn − m n b
f (t)dµ(t) ≤
|µ̌n (t)| dt
2
[a,b]
a
Mn − mn (b − a)n
≤
kµk .
2
n!
Proof. By [1, Theorem 5], the condition mk (µ) = 0, k = 0, . . . , n is equivalent
to µ̌k (b) = 0, k = 1, . . . , n + 1. Apply Corollary 4.2 and note that in this case
Šn = 0.
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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Corollary 4.5. Let f : [a, b] → R be such that f (n) ∈ L∞ [a, b] for some n ≥ 1.
Further, let (cm , m ≥ 1) be a sequence in R such that
X
|cm | < ∞
m≥1
and let {xm ; m ≥ 1} ⊂ [a, b]. If
X
X
X
cm =
cm xm = · · · =
cm xnm = 0,
m≥1
m≥1
m≥1
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
then
X
M −m X
n
n
c
f
(x
)
|cm | (b − xm )n
m
m ≤
2n!
m≥1
m≥1
X
Mn − m n
≤
(b − a)n
|cm | .
2n!
m≥1
Proof. Apply Corollary 4.4 for the discrete measure µ =
P
m≥1 cm δxm .
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References
[1] A. ČIVLJAK, LJ. DEDIĆ AND M. MATIĆ, Euler-Grüss type inequalities involving measures, submitted.
[2] A.ČIVLJAK, LJ. DEDIĆ AND M. MATIĆ, Euler harmonic identities for measures, Nonlinear Functional Anal. & Applics., 12(1) (2007).
[3] Lj. DEDIĆ, M. MATIĆ, J. PEČARIĆ AND A. AGLIĆ ALJINOVIĆ, On
weighted Euler harmonic identities with applications, Math. Inequal. & Appl.,
8(2), (2005), 237–257.
[4] S.S. DRAGOMIR, The generalised integration by parts formula for Appell sequences and related results, RGMIA Res. Rep. Coll., 5(E) (2002), Art. 18. [ONLINE: http://rgmia.vu.edu.au/v5(E).html].
[5] P. CERONE, Generalised Taylor’s formula with estimates of the remainder,
RGMIA Res. Rep. Coll., 5(2) (2002), Art. 8. [ONLINE: http://rgmia.vu.
edu.au/v5n2.html].
[6] P. CERONE, Perturbated generalised Taylor’s formula with sharp bounds,
RGMIA Res. Rep. Coll., 5(2) (2002), Art. 6. [ONLINE: http://rgmia.vu.
edu.au/v5n2.html].
Integration-by-parts Formula
A. Čivljak, Lj. Dedić and M. Matić
vol. 8, iss. 4, art. 93, 2007
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[7] S.S. DRAGOMIR AND A. SOFO, A perturbed version of the generalised Taylor’s formula and applications, RGMIA Res. Rep. Coll., 5(2) (2002), Art. 16.
[ONLINE: http://rgmia.vu.edu.au/v5n2.html].
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