Journal of Inequalities in Pure and
Applied Mathematics
HILBERT–PACHPATTE TYPE MULTIDIMENSIONAL INTEGRAL
INEQUALITIES
volume 5, issue 2, article 34,
2004.
G.D. HANDLEY, J.J. KOLIHA AND J. PEČARIĆ
Department of Mathematics and Statistics
University of Melbourne
Melbourne, VIC 3010
Australia.
EMail: g.handley@pgrad.unimelb.edu.au
EMail: j.koliha@ms.unimelb.edu.au
Faculty of Textile Engineering
University of Zagreb
10 000 Zagreb
Croatia.
EMail: pecaric@juda.element.hr
Received 24 September, 2003;
accepted 01 March, 2004.
Communicated by: S.S. Dragomir
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper we use a new approach to obtain a class of multivariable integral
inequalities of Hilbert type from which we can recover as special cases integral
inequalities obtained recently by Pachpatte and the present authors.
2000 Mathematics Subject Classification: 26D15
Key words: Hilbert’s inequality, Hilbert-Pachpatte integral inequalities, Hölder’s inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Notation and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4
Applications to Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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1.
Introduction
The integral version of Hilbert’s inequality [7, Theorem 316] has been generalized in several directions (see [1, 3, 4, 7, 8, 9, 20, 21, 22]). Recently, inequalities
similar to those of Hilbert were considered by Pachpatte in [12, 13, 14, 15, 16,
19]. The present authors in [5, 6] established a new class of related inequalities, which were further extended by Dragomir and Kim [2]. Two and higher
dimensional variants were treated by Pachpatte in [17, 18]. In the present paper
we use a new systematic approach to these inequalities based on Theorem 3.1,
which serves as an abstract springboard to classes of concrete inequalities.
To motivate our investigation, we give a typical result of [17]. In this theorem, H(I × J) denotes the class of functions u ∈ C (n−1,m−1) (I × J) such that
D1i u(0, t) = 0, 0 ≤ i ≤ n − 1, t ∈ J, D2j u(s, 0) = 0, 0 ≤ j ≤ m − 1, s ∈ I, and
D1n D2m−1 u(s, t) and D1n−1 D2m u(s, t) are absolutely continuous on I × J. Here
I, J are intervals of the type Iξ = [0, ξ) for some real ξ > 0.
Theorem 1.1 (Pachpatte [17, Theorem 1]). Let u(s, t) ∈ H(Ix × Iy ) and
v(k, r) ∈ H(Iz × Iw ). Then, for 0 ≤ i ≤ n − 1, 0 ≤ j ≤ m − 1, the following
inequality holds:
!
Z xZ y Z zZ w
|D1i D2j u(s, t) D1i D2j v(k, r)|
dk dr ds dt
2n−2i−1 t2m−2j−1 + k 2n−2i−1 r 2m−2j−1
0
0
0
0 s
Z x Z y
21
1
2√
n m
2
(x − s)(y − t)|D1 D2 u(s, t)| ds dt
≤ [Ai,j Bi,j ] xyzw
2
0
0
Z z Z w
21
n m
2
×
(z − k)(w − r)|D1 D2 v(k, r)| dk dr ,
0
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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where
Aij =
1
,
(n − i − 1)!(m − j − 1)!
Bij =
1
.
(2n − 2i − 1)(2m − 2j − 1)
The purpose of the present paper is to obtain a simultaneous generalization
of Pachpatte’s multivariable results [17], and of the results [5, 6] of the present
authors. The single variable results [14, 15, 16, 19] follow as special cases of our
theorems. Our treatment is based on Theorem 3.1, in particular on the abstract
inequality (3.1), which yields a variety of special cases when the functions Φi
are specified.
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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2.
Notation and Preliminaries
By Z (Z+ ) and R (R+ ) we denote the sets of all (nonnegative) integers and (nonnegative) real numbers. We will be working with functions of d variables, where
d is a fixed positive integer, writing the variable as a vector s = (s1 , . . . , sd ) ∈
Rd . A multiindex m is an element m = (m1 , . . . , md ) of Zd+ . As usual, the
factorial of a multiindex m is defined by m! = m1 ! · · · md !. An integer j may
be regarded as the multiindex (j, . . . , j) depending on the context. For vectors
in Rd and multiindices we use the usual operations of vector addition and multiplication of vectors by scalars. We write s ≤ τ (s < τ ) if sj ≤ τ j (sj < τ j )
for 1 ≤ j ≤ d. The same convention will apply to multiindices. In particular,
s ≥ 0 (s > 0) will mean sj ≥ 0 (sj > 0) for 1 ≤ j ≤ d.
If s = (s1 , . . . , sd ) ∈ Rd and s > 0, we define the cell
Q(s) = [0, s1 ] × · · · × [0, sj ] × · · · × [0, sd ];
replacing the factor [0, sj ] by {0} in this product, we get the face ∂j Q(s) of
Q(s).
Let s = (s1 , . . . , sd ), τ = (τ 1 , . . . , τ d ) ∈ Rd , s, τ > 0, let k = (k 1 , . . . , k d )
be a multiindex and let and u : Q(s) → R. Write Dj = ∂s∂ j . We use the
following notation:
τ1
τd
sτ = (s1 ) · · · (sd ) ,
k1
kd
Dk u(s) = D1 · · · Dd u(s),
Z s
Z s1
Z sd
u(τ ) dτ =
···
u(τ ) dτ 1 · · · dτ d .
0
0
0
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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An exponent α ∈ R in the expression sα , where s ∈ Rd , will be regarded as a
multiexponent, that is, sα = s(α,...,α) .
Another positive integer n will be fixed throughout.
The following notation and hypotheses will be used throughout the paper:
I = {1, . . . , n}
n∈N
mi , i ∈ I
mi = (m1i , . . . mdi ) ∈ Zd+
xi , i ∈ I
xi = (x1i , . . . , xdi ) ∈ Rd , xi > 0
pi , qi , i ∈ I
pi , qi ∈ R+ , p1i + q1i = 1
P
P
1
= ni=1 p1i , 1q = ni=1 q1i
p
p, q
ai , b i , i ∈ I
wi , i ∈ I
ai , bi ∈ R+ , ai + bi = 1
P
wi ∈ R, wi > 0, ni=1 wi = 1.
Throughout the paper, ui , vi , Φ will denote functions from [0, xi ] to R of sufficient smoothness. If m is a multiindex and x ∈ Rd , x > 0, then C m [0, x]
will denote the set of all functions u : [0, x] → R which possess continuous
derivatives Dk u, where 0 ≤ k ≤ m.
The coefficients pi , qi are conjugate Hölder exponents used in applications of
Hölder’s inequality, and the coefficients ai , bi are used in exponents to factorize
integrands. The coefficients wi act as weights in applications of the geometricarithmetic mean inequality; this enables us to pass from products to sums of
terms.
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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3.
The Main Result
First we present a theorem that can be regarded as a template for concrete inequalities obtained by selecting suitable functions Φi in (3.1). A special case of
this theorem is given in [6, Theorem 3.1].
Theorem 3.1. Let vi , Φi ∈ C(Q(xi )) and let ci be multiindices for i ∈ I. If
Z si
(3.1)
|vi (si )| ≤
(si − τi )ci Φi (τi ) dτi , si ∈ Q(xi ), i ∈ I,
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
0
then
Z
x1
Z
...
(3.2)
0
0
Qn
xn
i=1
|vi (si )|
ds1 · · · dsn
(α +1)/(qi wi )
wi si i
p1
n
n Z xi
Y
Y
i
1/qi
,
≤U
xi
(xi − si )βi +1 Φi (si )pi dsi
G.D. Handley, J.J. Koliha and
J. Pečarić
Pn
i=1
i=1
0
i=1
i=1 [(αi
1
+
1)1/qi (βi
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where αi = (ai + bi qi )ci , βi = ai ci , and
U = Qn
Title Page
+ 1)1/pi ]
.
Remark 3.1. Remembering our conventions, we observe that, for example,
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1/qi
xi
= (x1i )1/qi . . . (xdi )1/qi ,
n
n Y
d
Y
Y
1/qi
(αi + 1)
=
(αij + 1)1/qi .
i=1
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Proof. Factorize the integrand on the right side of (3.1) as
(si − τi )(ai /qi +bi )ci · (si − τi )(ai /pi )ci Φi (τi )
and apply Hölder’s inequality [10, p. 106] and Fubini’s theorem. Then
q1
si
Z
(ai +bi qi )ci
|vi (si )| ≤
(si − τi )
i
dτi
0
Z
×
=
p1
si
0
(αi +1)/qi
si
(αi + 1)1/qi
a i ci
(si − τi )
pi
Φi (τi ) dτi
si
Z
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
i
(si − τi )βi Φi (τi )pi dτi
p1
i
G.D. Handley, J.J. Koliha and
J. Pečarić
.
0
Using the inequality of means [10, p. 15]
n
Y
i=1
(α +1)/qi
si i
≤
n
X
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(αi +1)/(qi wi )
wi si
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,
JJ
J
i=1
we get
n
Y
i=1
where
|vi (si )| ≤ W
n
X
i=1
(α +1)/(qi wi )
wi si i
n Z
Y
i=1
si
βi
pi
(si − τi ) Φi (τi ) dτi
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p1
i
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,
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0
1
W = Qn
.
1/qi
i=1 (αi + 1)
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In the following estimate we apply Hölder’s inequality, Fubini’s theorem, and,
at the end, change the order of integration:
Qn
Z x1
Z xn
i=1 |vi (si )|
ds1 · · · dsn
...
Pn
(αi +1)/(qi wi )
w
s
0
0
i
i
i=1
"Z #
p1
n
xi Z s i
Y
i
≤W
(si − τi )βi Φi (τi )pi dτi
dsi
≤W
i=1
n
Y
0
1/q
xi i
0
Z
i=1
xi
Z
si
βi
(si − τi ) Φi (τi ) dτi
0
W
= Qn
1/pi
i=1 (βi + 1)
p1
pi
dsi
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
i
0
n
Y
i=1
1/q
xi i
n Z
Y
i=1
xi
βi +1
(xi − τi )
pi
Φi (τi ) dτi
p1
i
.
0
This proves the theorem.
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(k)
If d = 1 and vi are replaced by the derivatives ui , the preceding theorem
reduces to [6, Theorem 3.1].
Corollary 3.2. Under the assumptions of Theorem 3.1,
Qn
Z x1
Z xn
i=1 |vi (si )|
(3.3)
...
ds1 · · · dsn
Pn
(αi +1)/(qi wi )
w
s
0
0
i
i
i=1
! p1
Z xi
n
n
Y 1/q X
1
≤ p1/p U
xi i
(xi − si )βi +1 Φ(τi )pi dsi
,
p
i
0
i=1
i=1
where U is given by (3.2).
G.D. Handley, J.J. Koliha and
J. Pečarić
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Proof. By the inequality of means, for any Ai ≥ 0,
n
Y
i=1
1/pi
Ai
≤ p1/p
n
X
1
Ai
p
i
i=1
! p1
.
The corollary then follows from the preceding theorem.
The preceding corollary reduces to [6, Corollary 3.2] in the special case
(k)
when d = 1 and vi are replaced by ui .
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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4.
Applications to Derivatives
Convention 1. In this section we shall assume that mi , ki are multiindices satisfying 0 ≤ ki ≤ mi − 1, and write
αi = (ai + bi qi )(mi − ki − 1),
(4.1)
βi = ai (mi − ki − 1).
Recall that according to our conventions, mi −ki −1 = (m1i −ki1 −1, . . . , md1 −
kid − 1).
Theorem 4.1. Let ui ∈ C mi (Q(xi )) be such that Djr ui (si ) = 0 for si ∈
∂j Q(xi ), 0 ≤ r ≤ mji − 1, 1 ≤ j ≤ d, i ∈ I. Then
Z
x1
xn
Z
i=1
...
(4.2)
0
Qn
0
≤
G.D. Handley, J.J. Koliha and
J. Pečarić
ki
|D ui (si )|
ds1 · · · dsn
(αi +1)/(qi wi )
i=1 wi si
n
n Z xi
Y
Y
p
1/qi
U1
xi
(xi − si )βi +1 Dmi ui (si ) i
0
i=1
i=1
Pn
Title Page
p1
dsi
i
,
where
(4.3)
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
1
.
1/qi (β + 1)1/pi ]
i
i=1 [(mi − ki − 1)!(αi + 1)
U1 = Q n
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Proof. Under the hypotheses of the theorem we have the following multivariable identities established in [11],
Z si
1
ki
D ui (s) =
(si − τi )mi −ki −1 Dmi ui (τi ) dτi , i ∈ I.
(mi − ki − 1)! 0
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Inequality (4.2) is proved when we set vi (si ) = Dki ui (si ), ci = mi − ki − 1,
and
Φi (si ) =
(4.4)
|Dmi ui (si )|
(mi − ki − 1)!
in Theorem 3.1.
Corollary 4.2. Under the hypotheses of Theorem 4.1,
Z
x1
Z
xn
...
(4.5)
0
0
≤ p1/p U1
Qn
ki
i=1 |D ui (si )|
(αi +1)/(qi wi )
Pn
i=1 wi si
n
Y
1/qi
xi
i=1
ds1 · · · dsn
! p1
Z xi
n
X
1
p
,
(xi − si )βi +1 Dmi ui (si ) i dsi
p
i=1 i 0
where U1 is given by (4.3).
Proof. The result follows by applying the inequality of means to the preceding
theorem.
Single variable analogues of the preceding two results were obtained in [6,
Theorem 4.1] and [6, Corollary 4.2].
We discuss a number of special cases of Theorem 4.1 with similar examples
applying also to Corollary 4.2.
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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Example 4.1. If ai = 0 and bi = 1 for i ∈ I, then (4.2) becomes
Z
x1
Z
xn
...
(4.6)
0
0
Qn
ki
i=1 |D ui (si )|
ds1 · · · dsn
Pn
(qi mi −qi ki −qi +1)/(qi wi )
w
s
i
i
i=1
n
n Z xi
Y
Y
p
1/qi
≤ U1
xi
(xi − si )Dmi ui (si ) i
0
i=1
i=1
p1
dsi
i
,
where
1
.
1/qi ]
i=1 [(mi − ki − 1)!(qi mi − qi ki − qi + 1)
U 1 = Qn
(4.7)
Example 4.2. If ai = 0, bi = 1, qi = n, wi = n1 , pi =
for i ∈ I, then
Z
x1
Z
...
(4.8)
0
0
n
,
n−1
G.D. Handley, J.J. Koliha and
J. Pečarić
mi = m and ki = k
Qn
k
i=1 |D ui (si )|
Pn nm−nk−n+1 ds1 · · · dsn
i=1 si
√
n
x1 · · · xn
1
≤
n
n [(m − k − 1)!] (n(m − k − 1) + 1)
n−1
n Z xi
Y
n
m
n
n−1
×
(xi − si ) D ui (si )
dsi
.
xn
i=1
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
0
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For d = 2 and q = p = n = 2 this is Pachpatte’s theorem [17, Theorem 1] cited
in the Introduction; if d = 1 and q = p = n = 2, we obtain [14, Theorem 1].
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Example 4.3. Let ai = 1 and bi = 0 for i ∈ I. Then (4.2) becomes
Z
x1
Z
...
(4.9)
0
Qn
xn
i=1
|Dki ui (si )|
(m −k )/(qi wi )
Pn
i
i
i=1 wi si
n
n Z
Y
Y
1/qi
e
≤ U1
xi
0
i=1
i=1
xi
ds1 · · · dsn
mi −ki (xi − si )
p
D ui (si ) i dsi
mi
p1
i
,
0
where
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
1
.
i=1 [(mi − ki − 1)!(mi − ki )]
e1 = Qn
U
(4.10)
Example 4.4. Set ai = 0, bi = 1, qi = n, wi =
ki = k for i ∈ I. Then (4.2) becomes
Z
(4.11)
0
x1
1
,
n
pi =
n
,
n−1
G.D. Handley, J.J. Koliha and
J. Pečarić
mi = m and
Qn
k
i=1 |D ui (si )|
...
Pn m−k ds1 · · · dsn
0
i=1 si
√
n
x1 · · · xn
1
≤
n [(m − k − 1)!]n (m − k)n
(n−1)/n
n Z xi
Y
n/(n−1)
m−k m
×
(xi − si )
D ui (si )
dsi
.
Z
xn
i=1
0
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In the following theorem we establish another inequality similar to the integral analogue of Hilbert’s inequality.
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Theorem 4.3. Let ui ∈ C mi +1 (Q(xi )) be such that Dmi ui (si ) = 0 for si ∈
∂j Q(si ), 1 ≤ j ≤ d, i ∈ I. Then
Z x1
Z xn Q n
mi
i=1 |D ui (si )|
(4.12)
...
ds1 · · · dsn
Pn
1/(qi wi )
w
s
0
0
i
i
i=1
p1
n
n Z xi
Y
Y
m +1
pi
i
1/qi
i
≤
xi
(xi − si ) D
ui (si ) dsi
.
i=1
i=1
0
Proof. Under the hypotheses of the theorem we have the following multivariable identities established in [11] for mi = (0, . . . , 0):
Z si
mi
(4.13)
D ui (si ) =
Dmi +1 ui (τi ) dτi , i ∈ I.
0
In Theorem 3.1 set vi (si ) = Dmi ui (si ), ci = 0, Φi (si ) = |Dmi +1 ui (si )|, and the
result follows.
1
,
2
In the special case that d = 2, mi = (0, 0), p = q = n = 2, and wi = the
preceding theorem reduces to [17, Theorem 2].
When we apply the inequality of means to the preceding theorem, we get the
following corollary which generalizes the inequality obtained in [17, Remark 3].
Corollary 4.4. Under the hypotheses of Theorem 4.3,
Z x1
Z xn Q n
mi
i=1 |D ui (si )|
(4.14)
...
ds1 · · · dsn
Pn
1/(qi wi )
0
0
i=1 wi si
! p1
Z xi
n
n
Y
X
1
p
1/q
≤ p1/p
xi i
(xi − si )Dmi +1 ui (si ) i dsi
.
p
i
0
i=1
i=1
Hilbert–Pachpatte Type
Multidimensional Integral
Inequalities
G.D. Handley, J.J. Koliha and
J. Pečarić
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References
[1] Y.C. CHOW, On inequalities of Hilbert and Widder, J. London Math. Soc.,
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