Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 571417, 10 pages
doi:10.1155/2008/571417
Research Article
Sharp Integral Inequalities Involving High-Order
Partial Derivatives
C.-J Zhao1 and W.-S Cheung2
1
Department of Information and Mathematics Sciences, College of Science, China Jiliang University,
Hangzhou 310018, China
2
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Correspondence should be addressed to C.-J Zhao, chjzhao@163.com
Received 28 November 2007; Accepted 10 April 2008
Recommended by Peter Pang
The main purpose of the present paper is to establish some new sharp integral inequalities involving
higher-order partial derivatives. Our results in special cases yield some of the recent results on
Agarwal, Wirtinger, Poincaré, Pachpatte, Smith, and Stredulinsky’s inequalities and provide some
new estimates on such types of inequalities.
Copyright q 2008 C.-J Zhao and W.-S Cheung. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
Inequalities involving functions of n independent variables, their partial derivatives, integrals
play a fundamental role in establishing the existence and uniqueness of initial and boundary
value problems for ordinary and partial differential equations as well as difference equations
1–10. Especially, in view of wider applications, inequalities due to Agarwal, Opial, Pachpatte,
Wirtinger, Poincaré and et al. have been generalized and sharpened from the very day of their
discover. As a matter of fact, these now have become research topic in their own right 11–14.
In the present paper, we will use the same method of Agarwal and Sheng 15, establish some
new estimates on these types of inequalities involving higher-order partial derivatives. We
further generalize these inequalities which lead to result sharper than those currently available.
An important characteristic of our results is that the constant in the inequalities are explicit.
2. Main results
Let R be the set of real numbers and Rn the n-dimensional Euclidean space. Let E, E
n
be a bounded domain in Rn defined by E × E i1 ai , bi × ci , di , i 1, . . . , n. For
xi , yi ∈ R, i 1, . . . , n, x, y x1 , . . . , xn , y1 , . . . , yn is a variable point in E × E and
2
Journal of Inequalities and Applications
dxdy dx1 · · · dxn dy
1 · · · dyn . For any continuous real-valued function ux, y defined on
E × E , we denote by E E ux, ydxdy the 2n-fold integral
b1
···
a1
bn d1
an
c1
and for any x, y ∈ E × E ,
x1
···
a1
···
u x1 , . . . , xn , y1 , . . . , yn dx1 · · · dxn dy1 · · · dyn ,
2.1
cn
Ex E x
xn y 1
an
dn
···
c1
yn
us, tdsdt is the 2n-fold integral
u s1 , . . . , sn , t1 , . . . , tn dx1 · · · dsn dt1 · · · dtn .
2.2
cn
We represent by FE × E the class of continuous functions ux, y : E × E → R for which
D2n ux, y D1 · · · D2n ux, y, where
D1 ∂
∂
∂
∂
, . . . , Dn , Dn1 , . . . , D2n ∂x1
∂xn
∂y1
∂yn
2.3
exists and that for each i, 1 ≤ i ≤ n,
ux, yxi ai 0,
ux, yyi ci 0,
ux, yxi bi 0,
ux, yyi di 0,
i 1, . . . , n
2.4
the class FE × E is denoted as GE × E .
Theorem 2.1. Let μ ≥ 0, λ ≥ 1 be given real numbers, and let px, y ≥ 0, x, y ∈ E × E be a
continuous function. Further, let ux, y ∈ GE × E . Then, the following inequality holds
E
μ
px, yux, y dx dy
E μ/λ
2n
D ux, yλ dx dy
≤
px, yqx, y, λ, μdx dy
,
E E
2.5
E E
where
qx, y, λ, μ n
1 2n1
λ−1/2
xi − ai bi − xi yi − ci di − yi
μ/λ
.
2.6
i1
Proof. For the set {1, . . . , n}, let π A ∪ B, π A ∪ B be partitions, where A j1 , . . . , jk , B jk1 , . . . , jn , A i1 , . . . , ik , and B ik1 , . . . , in are such that card A card A k and
card B card B n − k, 0 ≤ k ≤ n. It is clear that there are 2n1 such partitions. The set
of all such partitions we will denote as Z and Z , respectively. For fixed partition π, π and
x ∈ E, y ∈ E , we define
Eπ x Eπ y
us, tds dt Ax Bx A y B y
us, tds dt,
2.7
C.-J Zhao and W.-S Cheung
3
where Ax , A y denote the k-fold integral, Bx , B y represent the n − k-fold integral. Thus
from the assumptions it is clear that for each π ∈ Z, π ∈ Z 2n
ux, y ≤
D us, tds dt.
2.8
Eπ x Eπ y
In view of Hölder integral inequality, we have
λ−1/λ
ux, y ≤
xi − ai
bi − x i
yi − c i
di − yi
i∈A
×
Eπ y
Eπ x
i∈A
i∈B
i∈B 2.9
1/λ
2n
D us, tλ ds dt
.
A multiplication of these 2n1 inequalities and an application of the Arithmetic-Geometric
mean inequality give
μ/λ
n
μ
λ−1/2
ux, y ≤
xi − ai bi − xi yi − ci di − yi
i1
×
≤
n
1 Eπ x Eπ y
π∈Z, π ∈Z 2n1
×
xi − ai
bi − x i
i1
1/2n1 μ/λ
2n
D us, tλ ds dt
π∈Z, π ∈Z Eπ x Eπ y
yi − c i
di − yi
λ−1/2
μ/λ
2.10
μ/λ
2n
D us, tλ ds dt
μ/λ
2n
λ
D us, t ds dt
qx, y, λ, μ
.
E E
Now, multiplying both the sides of 2.10 by px, y and integrating the resulting inequality on
E × E , we have
μ/λ
2n
μ
D us, tλ ds dt
px, yux, y dx dy ≤
px, yqx, y, λ, μdx dy
,
E E
E E
where
qx, y, λ, μ n
1 2n1
E E
λ−1/2
xi − ai bi − xi yi − ci di − yi
2.11
μ/λ
Remark 2.2. Taking for px, y 1 in 2.5, 2.5 reduces to
μ/λ
2n
ux, yμ dx dy ≤ K D ux, yλ dx dy
,
0
E E
where
K0
E E
n μ/λ n
1μ−μ/λ
μ
μ
μ
μ
1
2
B 1 −
,
bi − ai di − ci
,1 −
2
2 2λ
2 2λ
i1
and B is the Beta function.
.
2.12
i1
2.13
2.14
4
Journal of Inequalities and Applications
Taking for λ μ 2 in 2.13 reduces to
2 n
2n
2
2
ux, y2 dx dy ≤ π
D ux, y dx dy ,
M
8
E E
E E
where
M n
bi − ai di − ci
.
4
i1
2.15
2.16
Let ux, y reduce to ux in 2.15 and with suitable modifications, then 2.15 becomes the
following two Wirting type inequalities:
n
n
ux2 dx ≤ π M2
D ux2 dx ,
2.17
4
E
E
where
n
bi − ai
M
.
2
i1
2.18
Similarly
ux4 dx ≤
E
3π
16
n
M4
n
D ux4 dx ,
2.19
E
where M is as in 2.17.
For n 2, the inequalities 2.17 and 2.19 have been obtained by Smith and
Stredulinsky 16, however, with the right-hand sides, respectively, multiplies 4/π2 and
16/3π4 . Hence, it is clear that inequalities 2.17 and 2.19 are more strengthed.
Remark 2.3. Let ux, y reduce to ux in 2.5 and with suitable modifications, then 2.5
becomes the following result:
μ/λ
n
μ
λ
D ux dx
px ux dx ≤ pxqx, λ, μdx
,
2.20
E
E
E
where
qx, λ, μ n
λ−1/2
1
x − ai bi − xi
n
2 i1
μ/λ
.
2.21
This is just a new result which was given by Agarwal and Sheng 15.
Theorem 2.4. Let px, y ≥ 0, x, y ∈ E × E be a continuous function. Further, let for k 1, . . . , r, μk ≥ 0, λk ≥ 1, be given real numbers such that rk1 μk /λk 1, and uk x, y ∈ GE × E .
Then the following inequality holds
r μ
px, y uk x, y k dx dy
E E
k1
2.22
r
r
2n
μk
D uk x, yλk dx dy.
≤
px, y q x, y, λk , μk dx dy
λ
E E
k1 k E E
k1
C.-J Zhao and W.-S Cheung
5
Proof. Setting μ μk , λ λk and ux, y uk x, y, 1 ≤ k ≤ r in 2.10, multiplying the r
inequalities, and applying the extended Arithmetic-Geometric means inequality,
r
k1
μ /λk
ak k
≤
r
μk
k1
λk
ak ,
ak ≥ 0,
2.23
to obtain
μk /λk
r r
2n
uk x, yμk ≤
D uk s, tλk ds dt
q x, y, λk , μk
k1
E E
k1
r
r
2n
μk
D uk s, tλk ds dt.
≤
q x, y, λk , μk
λ
k1 k E E
k1
2.24
Now multiplying both sides of 2.24 by px, y and then integrating over E × E , we obtain
2.22.
Corollary 2.5. Let the conditions of Theorem 2.4 be satisfied. Then the following inequality holds
E
r r
2n
μk
μk
D uk x, yλk dx dy,
uk x, y dx dy < K1
px, y
px, ydx dy
λ
E
E E
k1 k E E
k1
2.25
where
K1
1
rk1 μk n
2n1
−1rk1 μk
.
bi − ai bi − ai
2.26
i1
This is just a general form of the following inequality which was established by Agarwal
and Sheng 15:
r r
μ
λ
μk n
D uk x k dx,
px uk x k dx < K1 pxdx
λ
E
E
k1 k E
k1
2.27
where
K1 1
2n
rk1 μk n
bi − ai
−1rk1 μk
.
2.28
i1
Remark 2.6. For px, y 1, the inequality 2.22 becomes
r r
2n
μk
uk x, yμk dx dy ≤ K D uk x, yλk dx dy,
2
λ
E E k1
k1 k E E
where
K2
r
r
n n
rk1 μk
1 2 1 k1 μk 1 k1 μk
B
,
.
bi − ai di − ci
2
2
2
i1
For ux, y ux, the inequality 2.29 has been obtained by Agarwal and Sheng 15.
2.29
2.30
6
Journal of Inequalities and Applications
Theorem 2.7. Let λ and ux, y be as in Theorem 2.1, μ ≥ 1 be a given real number. Then the following
inequality holds
ux, yλ dx dy ≤ K λ, μ
grad ux, yλ dx dy,
2.31
3
μ
E E
where
K3 λ, μ
E E
λ/n
λ n 1 2 λ1 λ1
B
,
K
,
bi − ai di − ci
2n
2
2
μ i1
μ 1/μ
n 2
grad ux, y ∂
ux,
y
,
∂x ∂y
μ
i1
i
2.32
i
and where Kλ/μ 1 if λ ≥ μ, and Kλ/μ n1−λ/μ if 0 ≤ λ/μ ≤ 1.
Proof. For each fixed i, 1 ≤ i ≤ n, in view of
ux, yxi ai 0, ux, yyi ci 0, ux, yxi bi 0,
we have
ux, y xi y i
ai
ux, y ci
∂2
u x, y; si , ti dsi dti ,
∂si ∂ti
yi
∂2
u x, y; si , ti dsi dti ,
∂si ∂ti
bi di
xi
ux, yyi di 0,
i 1, . . . , n,
2.33
2.34
where
u x, y; si , ti u x1 , . . . , xi−1 , si , xi1 , . . . , xn , y1 , . . . , yi−1 , ti , yi1 , . . . , yn .
Hence from Hölder inequality with indices λ and λ/1 − λ, it follows that
xi y i 2
∂
λ
ux, yλ ≤ xi − ai yi − di λ−1
u
x,
y;
s
,
t
i i dsi dti ,
ai ci ∂si ∂ti
bi di 2
∂
λ
ux, yλ ≤ bi − xi di − yi λ−1
u
x,
y;
s
,
t
i i dsi dti .
xi yi ∂si ∂ti
2.35
2.36
Multiplying 2.36, and then applying the Arithmetic-Geometric means inequality, to obtain
bi di 2
∂
λ
ux, yλ ≤ 1 xi − ai yi − ci bi − xi di − yi λ−1/2 ×
dsi dti ,
u
x,
y;
s
,
t
i
i
2
ai ci ∂si ∂ti
2.37
and now integrating 2.37 on E × E , we arrive at
bi di
λ−1/2
1 ux, yλ dx dy ≤
dxi dyi
xi − ai yi − ci bi − xi di − yi
2
E E
ai c i
λ
∂2
×
ux, y dx dy.
∂xi ∂yi
E E
2.38
C.-J Zhao and W.-S Cheung
7
Next, multiplying the inequality 2.38 for 1 ≤ i ≤ n, and using the Arithmetic-Geometric
means inequality, and in view of the following inequality:
α
n
n
α
ai ≤ Kα
ai , ai > 0,
2.39
i1
i1
where Kα 1 if α ≥ 1, and Kα n1−α if 0 ≤ α ≤ 1, we get
E E
bi di
1/n
λ−1/2
1 dxi dyi
xi − ai yi − ci bi − xi di − yi
ai c i 2
i1
λ
1/n
n
∂2
×
ux, y dx dy
E E ∂xi ∂yi
i1
1/n
n bi di
λ−1/2
1 1 ≤
dxi dyi
xi − ai yi − ci bi − xi di − yi
2n i1 ai ci 2
λ
n ∂2
×
ux, y dx dy
∂xi ∂yi
i1 E E
n
λ/n
1 2 λ1 λ1 B
,
bi − ai di − ci
≤
2n
2
2
i1
grad ux, yλ dx dy,
×
λ
n
ux, yλ dx dy ≤
E E
grad ux, yλ dx dy,
≤ K3 λ, μ
μ
E E
2.40
where
λ/n
λ n 1 2 λ1 λ1
K
,
bi − ai di − ci
B
,
2n
2
2
μ i1
μ 1/μ
n 2
∂
grad ux, y ,
∂x ∂y ux, y
μ
K3 λ, μ i
i1
2.41
i
and where Kλ/μ 1 if λ ≥ μ, and Kλ/μ n1−λ/μ if 0 ≤ λ/μ ≤ 1.
Remark 2.8. Let ux, y reduce to ux in 2.31 and with suitable modifications, and let λ ≥ 2,
μ 2, then 2.31 becomes
uxλ dx ≤ K λ, 2 grad uxλ dx.
2.42
3
μ
E
E
This is just a better inequality than the following inequality which was given by Pachpatte 17
λ uxλ dx ≤ 1 β
grad uxλ dx.
μ
n 2
E
E
Because for λ ≥ 2, it is clear that K3 λ, 2 < 1/nβ/2λ , where β max1≤i≤n bi − ai .
2.43
8
Journal of Inequalities and Applications
On the other hand, taking for μ 2, λ 2 or μ 2, λ 4 in 2.31 and let ux, y reduce
to ux with suitable modifications, it follows the following Poincaré-type inequalities:
ux2 dx ≤ π β2 grad ux2 dx,
2
16n
E
E
2.44
ux4 dx ≤ 3π β4 grad ux4 dx.
2
256n
E
E
The inequalities 2.44 have been discussed in 18 with the right-hand sides, respectively,
multiplied by 4/π and 16/3π. Hence inequalities 2.44 are more strong results on these types
of inequalities.
If μ ≥ λ, in the right sides of 2.31 we can apply Hölder inequality with indices μ/λ and
μ/μ − λ, to obtain the following corollary.
Corollary 2.9. Let the conditions of Theorem 2.7 be satisfied and μ ≥ λ. Then
E E
λ/μ
ux, yλ dx dy ≤ K λ, μ
grad ux, yμ dx dy
,
4
μ
E E
2.45
where
K4 λ, μ K3 λ, μ
n
bi − ai
di − ci
μ−λ/μ
2.46
.
i1
Remark 2.10. Taking ux, y ux and with suitable modifications, the inequality 2.45
reduces to the following result which was given by Agarwal and Sheng 15:
uxλ dx ≤ K6 λ, μ
E
grad uxμ dx
μ
E
λ/μ
,
2.47
where
K6 λ, μ K5 λ, μ
n
μ−λ/μ
,
bi − ai
i1
λ/n
1λ 1λ
λ n 1
B
,
K
K5 λ, μ ,
bi − ai
2n
2
2
μ i1
2.48
and Kλ/μ is as in Theorem 2.7.
Taking λ 1, μ 2 the inequality 2.45, 2.45 reduces to
E E
ux, ydx dy
2
≤ K4 1, 2
E E
grad ux, y2 dx dy.
2
2.49
This is just a general form of the following inequality which was given by Agarwal and Sheng
15.
2
uxdx ≤ K6 1, 2 2 grad ux2 dx dy.
2.50
2
E
E
Similar to the proof of Theorem 2.7, we have the following theorem.
C.-J Zhao and W.-S Cheung
9
Theorem 2.11. For uk x, y ∈ GE × E , μk ≥ 1, 1 ≤ k ≤ r. Then the following inequality holds
1/r
n r μk
uk x, y
grad uk x, yμk dx dy,
dx dy ≤ K5
μk
E E
E E k1
i1
2.51
where
K5
r
r
n
rk1 μk /nr
1 2 1 1/r k1 μk 1 1/r k1 μk .
bi − ai di − ci
B
,
2nr
2
2
i1
2.52
Remark 2.12. Taking ux, y ux and with suitable modifications, the inequality 2.51
reduces to the following result:
1/r
n r uk xμk
grad uxμk dx,
dx ≤ K9
μk
E
i1
2.53
E k1
where
n
r μ /nr
1 1/r rk1 μk 1 1/r rk1 μk 1
K9 bi − ai k1 k .
B
,
2nr
2
2
i1
2.54
In 19,
Pachpatte proved the inequality 2.53 for μk ≥ 2, 1 ≤ k ≤ r with K9 replaced by
r
r
μk /r
k1
1/nrβ/2
, where β is as in Remark 2.8. It is clear that K9 < 1/nrβ/2 k1 μk /r , and
hence 2.53 is a better inequality than a result of Pachpatte.
Similarly, all other results in 15 also can be generalized by the same way. Here, we omit
the details.
Acknowledgments
Research is supported by Zhejiang Provincial Natural Science Foundation of ChinaY605065,
Foundation of the Education Department of Zhejiang Province of China 20050392. Research
is partially supported by the Research Grants Council of the Hong Kong SAR, China Project
No.:HKU7016/07P.
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