Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 5, Issue 2, Article 34, 2004
HILBERT–PACHPATTE TYPE MULTIDIMENSIONAL INTEGRAL INEQUALITIES
G. D. HANDLEY, J. J. KOLIHA, AND J. PEČARIĆ
D EPARTMENT OF M ATHEMATICS AND S TATISTICS
U NIVERSITY OF M ELBOURNE
M ELBOURNE , VIC 3010
AUSTRALIA .
g.handley@pgrad.unimelb.edu.au
j.koliha@ms.unimelb.edu.au
FACULTY OF T EXTILE E NGINEERING
U NIVERSITY OF Z AGREB
10 000 Z AGREB
C ROATIA .
pecaric@juda.element.hr
Received 24 September, 2003; accepted 01 March, 2004
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Hilbert’s inequality, Hilbert-Pachpatte integral inequalities, Hölder’s inequality.
2000 Mathematics Subject Classification. 26D15.
1. I NTRODUCTION
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
ISSN (electronic): 1443-5756
c 2004 Victoria University. All rights reserved.
129-03
2
G. D. H ANDLEY , J. J. KOLIHA , AND J. P E ČARI Ć
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
≤ [Ai,j Bi,j ] xyzw
(x − s)(y − t)|D1 D2 u(s, t)| ds dt
2
0
0
Z z Z w
21
n m
2
×
(z − k)(w − r)|D1 D2 v(k, r)| dk dr ,
0
0
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.
2. N OTATION AND P RELIMINARIES
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:
1
d
sτ = (s1 )τ · · · (sd )τ ,
1
d
Dk u(s) = D1k · · · Ddk u(s),
Z s
Z s1
Z sd
u(τ ) dτ =
···
u(τ ) dτ 1 · · · dτ d .
0
0
0
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.
J. Inequal. Pure and Appl. Math., 5(2) Art. 34, 2004
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H ILBERT T YPE M ULTIDIMENSIONAL I NEQUALITIES
3
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
a i , b i ∈ R + , ai + b i = 1
P
wi ∈ R, wi > 0, ni=1 wi = 1.
wi , i ∈ I
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 geometric-arithmetic mean inequality; this enables
us to pass from products to sums of terms.
3. T HE M AIN R ESULT
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
(si − τi )ci Φi (τi ) dτi , si ∈ Q(xi ), i ∈ I,
(3.1)
|vi (si )| ≤
0
then
Z
x1
Z
xn
0
|vi (si )|
(αi +1)/(qi wi )
i=1 wi si
i=1
...
(3.2)
Qn
Pn
0
ds1 · · · dsn
≤U
n
Y
1/q
xi i
i=1
n Z
Y
i=1
xi
(xi − si )βi +1 Φi (si )pi dsi
p1
i
,
0
where αi = (ai + bi qi )ci , βi = ai ci , and
U = Qn
i=1 [(αi
1
+
1)1/qi (βi
+ 1)1/pi ]
.
Remark 3.2. Remembering our conventions, we observe that, for example,
1/q
xi i
=
(x1i )1/qi
. . . (xdi )1/qi ,
n
n Y
d
Y
Y
1/qi
(αi + 1)
=
(αij + 1)1/qi .
i=1
i=1 j=1
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 )
J. Inequal. Pure and Appl. Math., 5(2) Art. 34, 2004
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4
G. D. H ANDLEY , J. J. KOLIHA , AND J. P E ČARI Ć
and apply Hölder’s inequality [10, p. 106] and Fubini’s theorem. Then
Z s i
q1
i
(ai +bi qi )ci
|vi (si )| ≤
(si − τi )
dτi
0
Z
p1
si
a i ci
×
(si − τi )
pi
Φi (τi ) dτi
i
0
(α +1)/qi
si i
(αi + 1)1/qi
=
si
Z
(si − τi )βi Φi (τi )pi dτi
p1
i
.
0
Using the inequality of means [10, p. 15]
n
Y
(αi +1)/qi
si
n
X
≤
i=1
(αi +1)/(qi wi )
wi si
,
i=1
we get
n
Y
|vi (si )| ≤ W
i=1
n
X
(α +1)/(qi wi )
wi si i
i=1
n Z
Y
p1
si
βi
pi
i
(si − τi ) Φi (τi ) dτi
,
0
i=1
where
1
.
1/qi
i=1 (αi + 1)
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 )
0
0
i=1 wi si
" Z Z
#
p1
n
xi
si
Y
i
≤W
(si − τi )βi Φi (τi )pi dτi
dsi
W = Qn
i=1
≤W
n
Y
0
1/q
xi i
0
Z
xi
βi
W
= Qn
1/pi
i=1 (βi + 1)
p1
pi
(si − τi ) Φi (τi ) dτi
0
i=1
si
Z
dsi
i
0
n
Y
1/q
xi i
i=1
n Z
Y
i=1
xi
βi +1
(xi − τi )
pi
Φi (τi ) dτi
p1
i
.
0
This proves the theorem.
(k)
If d = 1 and vi are replaced by the derivatives ui , the preceding theorem reduces to [6,
Theorem 3.1].
Corollary 3.3. 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/p U
n
Y
i=1
1/qi
xi
! p1
Z
n
X
1 xi
(xi − si )βi +1 Φ(τi )pi dsi
,
p
i
0
i=1
where U is given by (3.2).
J. Inequal. Pure and Appl. Math., 5(2) Art. 34, 2004
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H ILBERT T YPE M ULTIDIMENSIONAL I NEQUALITIES
5
Proof. By the inequality of means, for any Ai ≥ 0,
n
Y
1/p
Ai i
≤ p1/p
i=1
n
X
1
Ai
p
i=1 i
! p1
.
The corollary then follows from the preceding theorem.
The preceding corollary reduces to [6, Corollary 3.2] in the special case when d = 1 and vi
(k)
are replaced by ui .
4. A PPLICATIONS TO D ERIVATIVES
Convention 1. In this section we shall assume that mi , ki are multiindices satisfying 0 ≤ ki ≤
mi − 1, and write
(4.1)
αi = (ai + bi qi )(mi − ki − 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
Z xn Q n
ki
i=1 |D ui (si )|
ds1 · · · dsn
(4.2)
...
Pn
(αi +1)/(qi wi )
w
s
0
0
i
i
i=1
p1
n
n Z xi
Y
Y
pi
i
1/qi
βi +1 mi
≤ U1
xi
(xi − si )
D ui (si ) dsi
,
i=1
i=1
0
where
(4.3)
1
.
1/qi (β + 1)1/pi ]
i
i=1 [(mi − ki − 1)!(αi + 1)
U1 = Qn
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
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 Q n
ki
i=1 |D ui (si )|
(4.5)
...
ds1 · · · dsn
Pn
(αi +1)/(qi wi )
0
0
i=1 wi si
≤ p1/p U1
n
Y
i=1
1/qi
xi
! 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.
J. Inequal. Pure and Appl. Math., 5(2) Art. 34, 2004
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6
G. D. H ANDLEY , J. J. KOLIHA , AND J. P E ČARI Ć
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.
Example 4.1. If ai = 0 and bi = 1 for i ∈ I, then (4.2) becomes
Qn
Z x1
Z xn
ki
i=1 |D ui (si )|
(4.6)
...
ds1 · · · dsn
Pn
(qi mi −qi ki −qi +1)/(qi wi )
0
0
i=1 wi si
p1
n
n Z xi
Y
Y
m
pi
i
1/qi
≤ U1
xi
(xi − si )D i ui (si ) dsi
,
i=1
0
i=1
where
(4.7)
1
.
1/qi ]
i=1 [(mi − ki − 1)!(qi mi − qi ki − qi + 1)
U 1 = Qn
n
Example 4.2. If ai = 0, bi = 1, qi = n, wi = n1 , pi = n−1
, mi = m and ki = k for i ∈ I, then
Z x1
Z xn Q n
|Dk ui (si )|
(4.8)
...
Pni=1 nm−nk−n+1 ds1 · · · dsn
0
0
i=1 si
√
n
x1 · · · xn
1
≤
n [(m − k − 1)!]n (n(m − k − 1) + 1)
n−1
n Z xi
Y
n
m
n
×
(xi − si )D ui (si ) n−1 dsi
.
0
i=1
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].
Example 4.3. Let ai = 1 and bi = 0 for i ∈ I. Then (4.2) becomes
Z x1
Z xn Q n
ki
i=1 |D ui (si )|
(4.9)
...
ds1 · · · dsn
Pn
(mi −ki )/(qi wi )
w
s
0
0
i
i
i=1
p1
n
n Z xi
Y
Y
pi
i
1/qi
mi −ki mi
e
≤ U1
xi
(xi − si )
D ui (si ) dsi
,
i=1
i=1
0
where
(4.10)
1
.
i=1 [(mi − ki − 1)!(mi − ki )]
e1 = Qn
U
n
Example 4.4. Set ai = 0, bi = 1, qi = n, wi = n1 , pi = n−1
, mi = m and ki = k for i ∈ I.
Then (4.2) becomes
Z x1
Z xn Q n
k
i=1 |D ui (si )|
(4.11)
...
Pn m−k ds1 · · · dsn
0
0
i=1 si
(n−1)/n
√
n Z xi
n
Y
n/(n−1)
x1 · · · xn
1
m−k m
≤
(xi − si )
D ui (si )
dsi
.
n [(m − k − 1)!]n (m − k)n i=1
0
In the following theorem we establish another inequality similar to the integral analogue of
Hilbert’s inequality.
J. Inequal. Pure and Appl. Math., 5(2) Art. 34, 2004
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H ILBERT T YPE M ULTIDIMENSIONAL I NEQUALITIES
7
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 )
0
0
i=1 wi si
p1
n
n Z xi
Y
Y
m +1
pi
i
1/qi
xi
≤
(xi − si )D i 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
mi
In Theorem 3.1 set vi (si ) = D ui (si ), ci = 0, Φi (si ) = |Dmi +1 ui (si )|, and the result follows.
In the special case that d = 2, mi = (0, 0), p = q = n = 2, and wi = 21 , 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 )
w
s
0
0
i=1 i i
≤ p1/p
n
Y
1/qi
xi
i=1
! p1
Z
n
X
m +1
pi
1 xi
(xi − si )D i ui (si ) dsi
.
p
i
0
i=1
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