Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 1, Article 16, 2003
HILBERT-PACHPATTE TYPE INTEGRAL INEQUALITIES AND THEIR
IMPROVEMENT
S.S. DRAGOMIR AND YOUNG-HO KIM
S CHOOL OF C OMPUTER S CIENCE AND M ATHEMATICS
V ICTORIA U NIVERSITY OF T ECHNOLOGY
PO B OX 14428 , M ELBOURNE C ITY MC
V ICTORIA 8001, AUSTRALIA .
sever.dragomir@vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
D EPARTMENT OF A PPLIED M ATHEMATICS
C HANGWON NATIONAL U NIVERSITY
C HANGWON 641-773, KOREA .
yhkim@sarim.changwon.ac.kr
Received 31 October, 2002; accepted 8 January, 2003
Communicated by P.S. Bullen
A BSTRACT. In this paper, we obtain an extension of multivariable integral inequality of HilbertPachpatte type. By specializing the upper estimate functions in the hypothesis and the parameters, we obtain many special cases.
Key words and phrases: Hilbert’s inequality, Hilbert-Pachpatte type inequality, Hölder’s inequality, Jensen inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTODUCTION
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.
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
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.
114-02
2
S.S. D RAGOMIR AND YOUNG -H O K IM
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
x1
xn
Z
···
0
0
Qn (ki )
u
(s
)
i i
i=1
ds1 · · · dsn
Pn
(αi +1)/(qi ωi )
ω
s
i
i
i=1
n Z
n
1 Y
Y
qi
≤U
xi
i=1
where U = 1
.Q
n
i=1 [(αi
1
qi
i=1
xi
βi +1
(xi − si )
pi
Φi (si ) dsi
p1
i
,
0
1
pi
+ 1) (βi + 1) ] .
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 Hilbert-Pachpatte type inequalities, these
inequalities improve the results obtained by Handley, Koliha and Pečarić [2].
2. M AIN R ESULTS
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}
xi , i ∈ I
xi ∈ R, xi > 0
pi , qi , 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
ai , bi ∈ R+ , ai + bi = 1
Pn
ωi ∈ R, ωi > 0,
i=1 ωi = Ωn
αi , i ∈ I
αi = (ai + bi qi )(mi − ki − 1)
βi , i ∈ I
βi = ai (mi − ki − 1)
ui , i ∈ I
ui ∈ C mi ([0, xi ]) for some m0i ≥ mi
Φi , i ∈ I
Φi ∈ C 1 ([0, xi ]), Φi ≥ mi .
0
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.
J. Inequal. Pure and Appl. Math., 4(1) Art. 16, 2003
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H ILBERT-PACHPATTE T YPE I NTEGRAL I NEQUALITIES
3
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 , si ∈ [0, xi ], i ∈ I,
ui (si ) ≤
0
then
Z
x1
Z
xn
···
(2.2)
0
0
h
1
Ωn
Qn (ki )
u
(s
)
i i=1 i
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
where
V =Q h
n
(2.3)
i=1
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 )
and apply Hölder’s inequality [9, p.106]. Then
q1
Z si
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]
! Ω1
n
n
Y
wi
si
≤
i=1
n
1 X
wi sri
Ωn i=1
! r1
for r > 0, we deduce that
n
Y
i=1
"
ir
sw
i
n
1 X
≤
wi sri
Ωn i=1
# Ωn
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
Ωn i=1
qi
i=1
i=1 (αi + 1)
p1
n Z s i
Y
i
βi
pi
×
(si − τi ) Φi (τi ) dτi
i=1
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4
S.S. D RAGOMIR AND YOUNG -H O K IM
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 ···
h P
iΩn ds1 · · · dsn
(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
=Q
xi
(xi − si )βi +1 Φi (si )pi dsi
.
1
1
n
qi
pi
0
[(α
+
1)
(β
+
1)
]
i=1
i=1
i
i
i=1
This proves the theorem.
Remark 2.2. In Theorem 2.1, setting Ωn = 1, we have Theorem 1.1.
Corollary 2.3. Under the assumptions of Theorem 2.1, if r > 0, we have
Z
x1
Z
···
0
0
Qn (ki ) i=1 ui si xn
h
1
Ωn
(αi +1)/(qi ωi )
i=1 ωi si
Pn
≤p
1
r·p
V
iΩn ds1 · · · dsn
n
Y
1
qi
xi
i=1
"
1
Z xi
r # r·p
n
X
1
(xi − si )βi +1 Φi spi i dsi
,
p
i
0
i=1
where V is defined by (2.3).
Proof. By the inequality of means, for any Ai ≥ 0 and r > 0, we obtain
n
Y
1
pi
Ai
i=1
"
n
X
1 r
Ai
≤ p
p
i
i=1
1
# r·p
.
The corollary then follows from the preceding theorem.
Pn
Lemma 2.4. 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
1 X
ωi γ1 γ2
−γ2
si
≥
ωi si
.
Ωn i=1
i=1
Proof. By the inequality of means, for any γ1 > 0 and γ2 < −1, we have
n
Y
"
sωi i γ1 γ2
i=1
J. Inequal. Pure and Appl. Math., 4(1) Art. 16, 2003
n
1 X
≥
ωi si
Ωn i=1
# γ1 γ2 Ωn
.
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H ILBERT-PACHPATTE T YPE I NTEGRAL I NEQUALITIES
5
− γ1
Using the fact that x
"
is concave and using the Jensen inequality, we have that
# γ1 γ2 Ωn "
# γ 1 γ2 Ω n
n
n
1 X
1 X
−γ2
ωi si
=
ωi f (si )
Ωn i=1
Ωn i=1
!#γ1 γ2 Ωn
"
n
X
1
2
ωi s−γ
≥ f
i
Ωn i=1

!− γ1 γ1 γ2 Ωn
n
2
X
1
2

=
ωi s−γ
i
Ωn i=1
"
#−γ1 Ωn
n
1 X
−γ2
=
ωi si
.
Ωn i=1
2
The proof of the lemma is complete.
Theorem 2.5. Under the assumptions of Theorem 2.1, if γ2 < −1, then
Qn (ki )
Z x1
Z xn
i=1 ui (si )
···
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
h P
n
−γ2
1
0
0
ω
s
i=1 i i
Ωn
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
where V is given by (2.3).
Proof. Using the inequality of Lemma 2.4, for any γ1 > 0 and γ2 < −1, we get
"
#− γ1γΩn
n
n
2
Y
X
1
2
sωi i γ1 ≤
ωi s−γ
.
i
Ωn i=1
i=1
According to above inequality, we deduce that
#−W1
"
n n
Y
1
1 X
(ki )
2
ωi s−γ
ui (si ) ≤ Qn
1
i
Ω
qi
n i=1
i=1
i=1 (αi + 1)
"Z
# p1
n
(si )
i
Y
×
(si − τi )βi Φi (τi )pi dτi
,
i=1
0
where W1 = (αi + 1)Ωn /γ2 qi ωi . The proof of the theorem then follows from the preceding
Theorem 2.1.
Corollary 2.6. Under the assumptions of Theorem 2.5, if r > 0, we have
Qn (ki )
Z x1
Z xn
i=1 ui (si )
···
h P
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
n
−γ2
1
0
0
i=1 ωi si
Ωn
1
" n
r # r·p
n
1
Y
X 1 Z xi
1
qi
(xi − si )βi +1 Φi (si )pi dsi
≤ p r·p V
xi
,
p
i
0
i=1
i=1
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6
S.S. D RAGOMIR AND YOUNG -H O K IM
where V is given by (2.3).
Proof. By the inequality of means, for any Ai ≥ 0 and r > 0, we obtain
1
" n
# r·p
n
1
Y
X
1 r
p
.
Ai i ≤ p
A
p i
i=1
i=1 i
The corollary then follows from the preceding Theorem 2.5.
In the following section we discuss some choice of the functions Φi .
3. T HE VARIOUS I NEQUALITIES
(j)
Theorem 3.1. Let ui ∈ C mi ([0, xi ]) be such that ui (0) = 0 for j ∈ {0, . . . , mi − 1}, i ∈ I.
Then
Qn (ki )
Z x1
Z xn
i=1 ui (si )
(3.1)
···
iΩn ds1 · · · dsn
h P
(αi +1)/(qi ωi )
n
1
0
0
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
0
where
(3.2)
V1 = Q h
n
i=1
1
1
1
(mi − ki − 1)!(αi + 1) qi (βi + 1) pi
i.
Proof. Inequality (3.1) is proved when we set
Φi (si ) =
(mi )
u
(s
)
i
i (mi − ki − 1)!
in Theorem 2.1.
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
···
iΩn ds1 · · · dsn
h P
(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
xi i
(xi − si )βi +1 ui i (si ) dsi
,
≤ p r·p V1
p
0
i=1 i
i=1
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
u
(s
)
i i=1 i
(3.3)
···
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
h P
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).
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H ILBERT-PACHPATTE T YPE I NTEGRAL I NEQUALITIES
7
Proof. Inequality (3.3) is proved when we set
Φi (si ) =
(mi )
ui (si )
(mi − ki − 1)!
in Theorem 2.5.
Corollary 3.4. Under the assumptions of Theorem 3.3, if r > 0, we have
Z
x1
Z
xn
···
0
h
0
1
Ωn
Qn (ki )
(s
)
u
i i
i=1
i−(αi +1)Ωn /γ2 qi ωi ds1 · · · dsn
Pn
−γ2
i=1 ωi si
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
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.
Example 3.1. If ai = 0 and bi = 1 for i ∈ I, then Theorem 3.1 becomes
Z
x1
Z
xn
···
0
Qn (ki )
i=1 ui (si )
h
0
1
Ωn
(qi mi −qi ki −qi +1)/(qi ωi )
i=1 ωi si
Pn
≤ V2
n
Y
iΩn ds1 · · · dsn
1
qi
xi
i=1
n Z
Y
xi
0
i=1
p1
pi
i
(mi )
,
(xi − si ) ui (si ) dsi
where
V2 = Q h
n
i=1
1
1
(mi − ki − 1)!(qi mi − qi ki − qi + 1) qi
i.
Example 3.2. If ai = 0, bi = 1, qi = n, pi = n/(n − 1), mi = m and ki = k for i ∈ I, then
Z
x1
Z
···
0
0
Qn (ki )
i=1 ui (si )
xn
h
1
Ωn
Pn
(nm−nk−n+1)/(nωi )
i=1 ωi si
iΩn ds1 · · · dsn
√
n
x1 · · · xn
n
≤
(m − k − 1)! (nm − nk − n + 1)
n−1
n Z xi
n
Y
n
(m)
n−1
.
×
(xi − si ) ui (si )
dsi
i=1
0
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].
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8
S.S. D RAGOMIR AND YOUNG -H O K IM
Example 3.3. If ai = 0 and bi = 1 for i ∈ I, then Theorem 3.1 becomes
Qn (ki )
Z x1
Z xn
(s
)
u
i i
i=1
···
iΩn ds1 · · · dsn
h P
(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
i=1
0
where
1
.
V3 = Qn i=1 (mi − ki )!
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 )
Z x1
Z 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
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
Z x1 Z x2
|u1 (s1 )| |u2 (s2 )|
h
iΩ2 ds1 ds2
(p1 −1)
(p2 −1)
1
0
0
p 2 s1
+ p 1 s2
p1 p2 Ω2
p1
Z x1
1
p1
(p1 −1)/p1 (p2 −1)/p2
0
≤ x1
x2
(x1 − s1 ) |u1 (s1 )| ds1
0
Z
×
x2
(x2 −
p
s2 ) |u02 (s2 )| 2
p1
ds2
2
.
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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