Journal of Inequalities in Pure and
Applied Mathematics
HILBERT-PACHPATTE TYPE INTEGRAL INEQUALITIES AND
THEIR IMPROVEMENT
volume 4, issue 1, article 16,
2003.
S.S. DRAGOMIR AND YOUNG-HO KIM
School of Computer Science and Mathematics
Victoria University of Technology
PO Box 14428 , Melbourne City MC
Victoria 8001, Australia.
EMail: sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Department of Applied Mathematics
Changwon National University
Changwon 641-773, Korea.
EMail: yhkim@sarim.changwon.ac.kr
Received 31 October, 2002;
accepted 8 January, 2003.
Communicated by: P.S. Bullen
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
114-02
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Abstract
In this paper, we obtain an extension of multivariable integral inequality of
Hilbert-Pachpatte type. By specializing the upper estimate functions in the hypothesis and the parameters, we obtain many special cases.
2000 Mathematics Subject Classification: 26D15.
Key words: Hilbert’s inequality, Hilbert-Pachpatte type inequality, Hölder’s inequality,
Jensen inequality.
The authors would like to thank Professor P.S. Bullen, University of British Columbia,
Canada, for the careful reading of the manuscript which led to a considerable improvement in the presentation of this paper.
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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Contents
1
Intoduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Various Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
References
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1.
Intoduction
Hilbert’s double series theorem [3, p. 226] was proved first by Hilbert in his
lectures on integral equations. The determination of the constant, the integral
analogue, the extension, other proofs of the whole or of parts of the theorems
and generalizations in different directions have been given by several authors
(cf. [3, Chap. 9]). Specifically, in [10] – [14] the author has established some
new inequalities similar to Hilbert’s double-series inequality and its integral
analogue which we believe will serve as a model for further investigation. Recently, G.D. Handley, J.J. Koliha and J.E. Pečarić [2] established a new class of
related integral inequalities from which the results of Pachpatte [12] – [14] are
obtained by specializing the parameters and the functions Φi . A representative
sample is the following.
Theorem 1.1 (Handley, Koliha and Pečarić [2, Theorem 3.1]). Let ui ∈
C mi ([0, xi ]) for i ∈ I. If
Z si
(ki )
(si − τi )mi −ki −1 Φi (τi ) dτi , si ∈ [0, xi ], i ∈ I,
ui (si ) ≤
0
then
Z
S.S. Dragomir and Young-Ho Kim
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x1
Z
···
0
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
0
xn
Qn (ki )
u
(s
)
i i=1 i
ds1 · · · dsn
Pn
(αi +1)/(qi ωi )
i=1 ωi si
p1
n
n Z xi
1 Y
Y
i
qi
βi +1
pi
≤U
xi
(xi − si )
Φi (si ) dsi
,
i=1
i=1
0
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where U = 1
.Q
n
i=1 [(αi
1
1
+ 1) qi (βi + 1) pi ] .
The purpose of the present paper is to derive an extension of the inequality
given in Theorem 1.1. In addition, we obtain some new inequalities as HilbertPachpatte type inequalities, these inequalities improve the results obtained by
Handley, Koliha and Pečarić [2].
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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2.
Main Results
In what follows we denote by R the set of real numbers; R+ denotes the interval
[0, ∞). The symbols N, Z have their usual meaning. The following notation and
hypotheses will be used throughout the paper:
I = {1, ..., n}
n∈N
mi , i ∈ I
mi ∈ N
ki , i ∈ I
ki ∈ {0, 1, . . . , mi − 1}
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
xi , i ∈ I
xi ∈ R, xi > 0
S.S. Dragomir and Young-Ho Kim
pi , q i , i ∈ I
pi , qi ∈ R, pi , qi > 0, p1i + q1i = 1
Pn 1 1 Pn 1 1
=
i=1 pi , q =
i=1 qi
p
p, q
ai , bi , i ∈ I
ωi , i ∈ I
a i , bi ∈ R + , a i + b i = 1
P
ωi ∈ R, ωi > 0, ni=1 ωi = Ωn
αi , i ∈ I
αi = (ai + bi qi )(mi − ki − 1)
βi , i ∈ I
βi = ai (mi − ki − 1)
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m0i
([0, xi ]) for some m0i ≥ mi
ui , i ∈ I
ui ∈ C
Φi , i ∈ I
Φi ∈ C 1 ([0, xi ]), Φi ≥ mi .
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Here the ui are given functions of sufficient smoothness, and the Φi are subject
to choice. The coefficients pi , qi are conjugate Hölder exponents to be used
in applications of Hölder’s inequality, and the coefficients ai , bi will be used
in exponents to factorize integrands. The coefficients ωi will act as weights in
applications of the geometric-arithmetic mean inequality. The coefficients αi
and βi arise naturally in the derivation of the inequalities. Our main results are
given in the following theorems.
Theorem 2.1. Let ui ∈ C mi ([0, xi ]) for i ∈ I. If
Z si
(ki )
(2.1)
(si − τi )mi −ki −1 Φi (τi ) dτi ,
ui (si ) ≤
si ∈ [0, xi ], i ∈ I,
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
0
S.S. Dragomir and Young-Ho Kim
then
Z
x1
Z
···
(2.2)
0
0
xn
h
1
Ωn
Qn (ki )
u
(s
)
i i
i=1
iΩn ds1 · · · dsn
Pn
(αi +1)/(qi ωi )
i=1 ωi si
p1
n
n Z xi
1 Y
Y
i
qi
βi +1
pi
≤V
xi
(xi − si )
Φi (si ) dsi
,
i=1
0
i=1
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where
(2.3)
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V =Q h
n
i=1
Close
1
1
qi
(αi + 1) (βi + 1)
1
pi
i.
Proof. Factorize the integrand on the right side of (2.1) as
(si − τi )(ai /qi +bi )(mi −ki −1) × (si − τi )(ai /pi )(mi −ki −1) Φi (τi )
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and apply Hölder’s inequality [9, p.106]. Then
q1
Z s i
i
(ki )
(ai +bi qi )(mi −ki −1)
(si − τi )
dτi
ui (si ) ≤
0
Z
si
×
ai (mi −ki −1)
(si − τi )
p1
pi
Φi (τi ) dτi
i
0
=
(α +1)/qi
si i
(αi + 1)
si
Z
1
qi
βi
pi
(si − τi ) Φi (τi ) dτi
p1
i
.
0
Using the inequality of means [9, p. 15]
! r1
! Ω1
n
n
n
X
Y
1
i
≤
wi sri
sw
i
Ωn i=1
i=1
for r > 0, we deduce that
n
Y
i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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"
ir
sw
≤
i
1
Ωn
n
X
# Ωn
wi sri
i=1
for r > 0. According to above inequality, we have
"
# Ωn
n n
Y
X
1
1
(ki )
(α +1)/(qi ωi )
ωi si i
ui (si ) ≤ Qn
1
Ω
qi
n i=1
i=1
i=1 (αi + 1)
p1
n Z si
Y
i
βi
pi
×
(si − τi ) Φi (τi ) dτi
i=1
0
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for r = (αi + 1)/qi ωi . In the following estimate we apply Hölder’s inequality
and, at the end, change the order of integration:
Qn (ki ) Z x1
Z xn
i=1 ui si ···
iΩn ds1 · · · dsn
h P
(αi +1)/(qi ωi )
n
1
0
0
i=1 ωi si
Ωn
" Z Z
#
p1
n
xi
si
Y
i
1
≤Q
(si − τi )βi Φi (τi )pi dτi ,
dsi
1
n
qi
0
0
(α
+
1)
i=1
i
i=1
Z xi Z si
p1
n
1
Y
i
1
qi
βi
pi
≤Q
xi
(si − τi ) Φi (τi ) dτi , dsi
1
n
qi
0
0
i=1
i=1 (αi + 1)
p1
n
n Z xi
1 Y
Y
i
1
qi
βi +1
pi
xi
(xi − si )
Φi (si ) dsi
.
=Q
1
1
n
qi
pi
0
i=1
i=1 [(αi + 1) (βi + 1) ] i=1
This proves the theorem.
Remark 2.1. In Theorem 2.1, setting Ωn = 1, we have Theorem 1.1.
Corollary 2.2. Under the assumptions of Theorem 2.1, if r > 0, we have
Qn (ki ) Z x1
Z xn
i=1 ui si ···
h P
iΩn ds1 · · · dsn
(αi +1)/(qi ωi )
n
1
0
0
ω
s
i=1 i i
Ωn
1
" n
Z xi
r # r·p
n
1
Y
X
1
1
q
≤ p r·p V
xi i
(xi − si )βi +1 Φi spi i dsi
,
p
i
0
i=1
i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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where V is defined by (2.3).
Proof. By the inequality of means, for any Ai ≥ 0 and r > 0, we obtain
1
# r·p
" n
n
1
X
Y
1 r
p
Ai i ≤ p
A
.
p i
i=1
i=1 i
The corollary then follows from the preceding theorem.
Pn
Lemma 2.3. Let γ1 > 0 and γ2 < −1. Let ωi > 0,
i=1 ωi = Ωn and let
si > 0, i = 1, . . . , n be real numbers. Then
"
#−γ1 Ωn
n
n
Y
X
1
2
sωi i γ1 γ2 ≥
ωi s−γ
.
i
Ω
n
i=1
i=1
Proof. By the inequality of means, for any γ1 > 0 and γ2 < −1, we have
"
# γ1 γ2 Ω n
n
n
Y
X
1
sωi i γ1 γ2 ≥
ωi si
.
Ω
n
i=1
i=1
−
1
Using the fact that x γ2 is concave and using the Jensen inequality, we have
that
"
# γ1 γ2 Ωn "
# γ1 γ2 Ωn
n
n
1 X
1 X
2
ωi si
=
ωi f (s−γ
)
i
Ωn i=1
Ωn i=1
"
!#γ1 γ2 Ωn
n
1 X
−γ2
ωi si
≥ f
Ωn i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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
=
1
Ωn
n
X
!− γ1 γ1 γ2 Ωn
2
2
ωi s−γ
i

i=1
"
n
1 X
2
=
ωi s−γ
i
Ωn i=1
#−γ1 Ωn
.
The proof of the lemma is complete.
Theorem 2.4. Under the assumptions of Theorem 2.1, if γ2 < −1, then
Qn (ki )
Z x1
Z xn
u
(s
)
i i
i=1
···
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
h P
n
−γ2
1
0
0
i=1 ωi si
Ωn
p1
n Z xi
n
1 Y
Y
i
qi
βi +1
pi
(xi − si )
Φi (si ) dsi
,
≤V
xi
i=1
i=1
0
where V is given by (2.3).
Proof. Using the inequality of Lemma 2.3, for any γ1 > 0 and γ2 < −1, we get
#− γ1γΩn
"
n
n
2
Y
X
1
2
ωi s−γ
sωi i γ1 ≤
.
i
Ω
n
i=1
i=1
According to above inequality, we deduce that
#−W1
"
n
n X
Y
1
1
(ki )
2
ωi s−γ
ui (si ) ≤ Qn
1
i
(αi + 1) qi Ωn i=1
i=1
i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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×
"Z
n
Y
i=1
# p1
(si )
i
(si − τi )βi Φi (τi )pi dτi
,
0
where W1 = (αi + 1)Ωn /γ2 qi ωi . The proof of the theorem then follows from
the preceding Theorem 2.1.
Corollary 2.5. Under the assumptions of Theorem 2.4, if r > 0, we have
Z
x1
Z
···
0
0
Qn (ki )
u
(s
)
i i
i=1
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
h P
n
−γ2
1
i=1 ωi si
Ωn
1
" n
Z xi
r # r·p
n
1
Y
X
1
1
q
,
≤ p r·p V
xi i
(xi − si )βi +1 Φi (si )pi dsi
p
i
0
i=1
i=1
xn
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
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where V is given by (2.3).
Proof. By the inequality of means, for any Ai ≥ 0 and r > 0, we obtain
n
Y
i=1
1
pi
Ai
"
n
X
1 r
≤ p
Ai
p
i
i=1
1
# r·p
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The corollary then follows from the preceding Theorem 2.4.
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In the following section we discuss some choice of the functions Φi .
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3.
The Various Inequalities
(j)
Theorem 3.1. Let ui ∈ C mi ([0, xi ]) be such that ui (0) = 0 for j ∈ {0, . . . , mi −
1}, i ∈ I. Then
Z
x1
Z
···
(3.1)
0
0
xn
Qn (ki )
u
(s
)
i i=1 i
iΩn ds1 · · · dsn
h P
(αi +1)/(qi ωi )
n
1
i=1 ωi si
Ωn
p1
n
n Z xi
pi
1 Y
Y
i
qi
βi +1 (mi )
≤ V1
xi
(xi − si )
,
ui (si ) dsi
i=1
i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
0
S.S. Dragomir and Young-Ho Kim
where
(3.2)
V1 = Q h
n
i=1
1
1
qi
(mi − ki − 1)!(αi + 1) (βi + 1)
Proof. Inequality (3.1) is proved when we set
(mi )
ui (si )
Φi (si ) =
(mi − ki − 1)!
in Theorem 2.1.
1
pi
i.
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Corollary 3.2. Under the assumptions of Theorem 3.1, if r > 0, we have
Qn (ki )
Z x1
Z xn
u
(s
)
i i=1 i
···
h P
iΩn ds1 · · · dsn
(αi +1)/(qi ωi )
n
1
0
0
ω
s
i=1 i i
Ωn
1
" n
Z xi
r # r·p
n
pi
1
Y
X
1
1
(m )
q
≤ p r·p V1
xi i
(xi − si )βi +1 ui i (si ) dsi
,
p
i
0
i=1
i=1
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
where V1 is given by (3.2).
Theorem 3.3. Under the assumptions of Theorem 3.1, if γ2 < −1, then
Qn (ki )
Z x1
Z xn
i=1 ui (si )
(3.3)
···
h P
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
n
−γ2
1
0
0
ω
s
i=1 i i
Ωn
p1
n
n Z xi
pi
1 Y
Y
i
(m
)
qi
≤ V1
xi
(xi − si )βi +1 ui i (si ) dsi
,
i=1
i=1
0
where V1 is given by (3.2).
Proof. Inequality (3.3) is proved when we set
(mi )
ui (si )
Φi (si ) =
(mi − ki − 1)!
in Theorem 2.4.
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Corollary 3.4. Under the assumptions of Theorem 3.3, if r > 0, we have
Z
x1
0
Qn (ki )
u
(s
)
i i
i=1
···
h P
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
n
−γ2
1
0
i=1 ωi si
Ωn
1
" n
Z xi
r # r·p
n
pi
1
Y
X
1
1
(m )
q
≤ p r·p V1
xi i
(xi − si )βi +1 ui i (si ) dsi
.
p
i
0
i=1
i=1
xn
Z
We discuss a number of special cases of Theorem 3.1. Similar examples
apply also to Corollary 3.2, Theorem 3.3 and Corollary 3.4.
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
Example 3.1. If ai = 0 and bi = 1 for i ∈ I, then Theorem 3.1 becomes
Z
x1
···
0
0
Title Page
Qn (ki )
i=1 ui (si )
xn
Z
h
1
Ωn
(qi mi −qi ki −qi +1)/(qi ωi )
i=1 ωi si
Pn
≤ V2
n
Y
i=1
1
qi
xi
n Z
Y
0
i=1
xi
Contents
iΩn ds1 · · · dsn
1
pi
pi
(mi )
(xi − si ) ui (si ) dsi
,
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where
V2 = Q h
n
i=1
1
1
(mi − ki − 1)!(qi mi − qi ki − qi + 1) qi
i.
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Example 3.2. If ai = 0, bi = 1, qi = n, pi = n/(n − 1), mi = m and ki = k
for i ∈ I, then
Qn (ki )
Z x1
Z xn
u
(s
)
i i
i=1
···
h P
iΩn ds1 · · · dsn
(nm−nk−n+1)/(nωi )
n
1
0
0
i=1 ωi si
Ωn
√
n
x1 · · · xn
n
≤
(m − k − 1)! (nm − nk − n + 1)
n−1
n Z xi
n
Y
n
(m)
n−1
dsi
.
×
(xi − si ) ui (si )
0
i=1
For q = p = n = 2 and ωi = n1 this is [12, Theorem 1]. Setting q = p = 2,
k = 0, n = 1 and ωi = n1 , we recover the result of [14].
Example 3.3. If ai = 0 and bi = 1 for i ∈ I, then Theorem 3.1 becomes
Qn (ki )
Z x1
Z xn
u
(s
)
i i
i=1
···
h P
iΩn ds1 · · · dsn
(mi −ki )/(qi ωi )
n
1
0
0
i=1 ωi si
Ωn
p1
n
n Z xi
pi
1 Y
Y
i
qi
mi −ki (mi )
≤ V3
xi
(xi − si )
,
ui (si ) dsi
i=1
where
i=1
0
1
.
V3 = Qn i=1 (mi − ki )!
Hilbert-Pachpatte Type Integral
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S.S. Dragomir and Young-Ho Kim
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Example 3.4. If ai = 1, bi = 0, qi = n, pi = n/(n − 1), mi = m and ki = k
for i ∈ I. Then (3.1) becomes
Qn (ki )
x1
xn
i=1 ui (si )
···
iΩn ds1 · · · dsn
h P
(m−k)/(nωi )
n
1
0
0
ω
s
i=1 i i
Ωn
(n−1)
√
n Z xi
n/(n−1)
n
n
x1 · · · xn Y
m−k (m)
n
≤
(xi − si )
dsi
.
ui (si )
(m − k)! i=1 0
Z
Z
Example 3.5. Let p1 , p2 ∈ R+ . If we set n = 2, ω1 = p11 , ω2 = p12 , mi = 1 and
ki = 0 for i = 1, 2 in Theorem 3.1, then by our assumptions q1 = p1 /(p1 − 1),
q2 = p2 /(p2 − 1), and we obtain
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
Title Page
Z
0
x1
Z
0
x2
|u1 (s1 )| |u2 (s2 )|
h
1
p1 p2 Ω2
(p −1)
p 2 s1 1
+
(p −1)/p1 (p2 −1)/p2
x2
≤ x1 1
(p −1)
p 1 s2 2
Z
x1
Contents
iΩ2 ds1 ds2
p1
(x1 − s1 ) |u01 (s1 )|
JJ
J
p1
1
ds1
0
Z
×
x2
p2
(x2 − s2 ) |u02 (s2 )|
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p1
ds2
2
II
I
.
0
If we set ω1 + ω2 = 1 in Example 3.5, then we have [13, Theorem 2]. (The
values of ai and bi are irrelevant.)
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References
[1] BICHENG YANG, On Hilbert’s integral inequality, J. Math. Anal. Appl.,
220 (1988), 778–785.
[2] G.D. HANDLEY, J.J. KOLIHA AND PEČARIĆ, New Hilbert-Pachpatte
type integral inequalities, J. Math. Anal. Appl., 257 (2001), 238–250.
[3] G.H. HARDY, J.E. LITTLEWOOD
bridge Univ. Press, London, 1952.
AND
G. POLYA, Inequalities, Cam-
[4] YOUNG-HO KIM, Refinements and Extensions of an inequality, J. Math.
Anal. Appl., 245 (2000), 628–632.
[5] V. LEVIN, On the two-parameter extension and analogue of Hilbert’s inequality, J. London Math. Soc., 11 (1936), 119–124.
[6] G. MINGZE, On Hilbert’s inequality and its applications, J. Math. Anal.
Appl., 212 (1997), 316–323.
[7] D.S. MITRINOVIĆ, Analytic inequalities, Springer-Verlag, Berlin, New
York, 1970.
Hilbert-Pachpatte Type Integral
Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
Title Page
Contents
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J
II
I
[8] D.S. MITRINOVIĆ AND J.E. PEČARIĆ, On inequalities of Hilbert and
Widder, Proc. Edinburgh Math. Soc., 34 (1991), 411–414.
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[9] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic, Dordrecht, 1993.
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[10] B.G. PACHPATTE, A note on Hilbert type inequality, Tamkang J. Math.,
29 (1998), 293–298.
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[11] B.G. PACHPATTE, On some new inequalities similar to Hilbert’s inequality, J. Math. Anal. Appl., 226 (1998), 166–179.
[12] B.G. PACHPATTE, Inequalities similar to the integral analogue of
Hilbert’s Inequality, Tamkang J. Math., 30 (1999), 139–146.
[13] B.G. PACHPATTE, Inequalities similar to certain extensions of Hilbert’s
inequality, J. Math. Anal. Appl., 243 (2000), 217–227.
[14] B.G. PACHPATTE, A note on inequality of Hilbert type, Demonstratio
Math., in press.
[15] D.V. WIDDER, An inequality related to one of Hilbert’s, J. London Math.
Soc., 4 (1929), 194–198.
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Inequalities and their
Improvement
S.S. Dragomir and Young-Ho Kim
Title Page
Contents
JJ
J
II
I
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