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R E S E A R C H

Open Access

Some sharp integral inequalities involving partial

derivatives

Chang-Jian Zhao

1*

, Wing-Sum Cheung

2

and Mihály Bencze

3

* Correspondence: chjzhao@163. com

1Department of Mathematics,

China Jiliang University, Hangzhou 310018, P. R. China

Full list of author information is available at the end of the article

Abstract

The main purpose of the present article is to establish some new sharp integral inequalities in 2nindependent variables. Our results in special cases yield some of the recent results on Pachpatter, Agarwal and Sheng’s inequalities and provide some new estimates on such types of inequalities.

Mathematics Subject Classification 2000: 26D15.

Keywords:Cauchy-Schwarz’s inequality, Pachpatte’s inequality, Hölder integral inequality, the arithmetic-geometric means inequality

1 Introduction

Inequalities involving functions ofn 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, Poincaréand 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 article we shall use the same method of Agarwal and Sheng [15], establish some new estimates on these types of inequalities involving 2nindependent variables. We further generalize these inequalities which lead to result sharp than those currently available. An important characteristic of our results is that the constants in the inequalities are explicit.

2 Main results

LetRbe the set of real numbers andℝn then-dimensional Euclidean space. LetE, E’

be a bounded domain inRn defined by E×E=n

i=1[ai,bi]×[ci,di],i= 1, ...,n. For

xi,yiÎ R, i= 1, ...,n, (x,y) = (x1, ...,xn,y1, ...,yn) is a variable point inE×E’and dxdy

=dx1 ...dxndy1 ...dyn. For any continuous real-valued functionu(x,y) defined onE× E’we denote by∫E∫E’u(x,y)dxdythe 2n-fold integral

b1

a1

· · · anbn

c1d1· · ·

cndnu(x1,. . .,xn,y1,. . .,yn)dx1. . .dxndy1. . .dyn,

(2)

and for any (x,y)ÎE×E’,∫E(x)∫E’(x)u(s,t)dsdtis the 2n-fold integral

a1x1· · ·

anxn

c1y1· · ·

cnynu(s1,. . .,sn,t1,. . .,tn)dx1. . .dsndt1. . .dtn,

We represent byF(E×E’) the class of continuous functionsu(x,y) :E×E’®ℝ, for eachi,1≤i≤n,

u(x,y)xi=ai = 0,u(x,y)yi=ci= 0,u(x,y)xi=bi = 0 ,u(x,y)yi=di= 0, (i= 1,. . .,n) the class F(E×E’) is denoted asG(E×E’).

Theorem 2.1.Let l,μ,l≥1,be given real numbers such that 1

μ +

1

λ = 1.Further, let

u(x,y)Î G(E×E’).Then, the following inequality holds

E

Eu(x,y)ldxdy≤ 1 2n

n

i=1

[(biai)(cidi)]μ

1/μ

E

Eu(x,y)(l−1)μdxdy

1/μ

× E

Egradu(x,y)λλdxdy

1/λ

,

(2:1)

where

gradu(x,y)λ=

n

i=1

xi2yiu(x,y) λ1/λ

.

Proof. For each fixedi, 1≤i≤n, in view of

u(x,y)xi=ai = 0,u(x,y)yi=ci = 0,u(x,y)xi=bi = 0,u(x,y)yi=di= 0, (i= 1,. . .,n) we have

ul(x,y) =ul−1(x,y)

aixi

ciyi

2

∂si∂ti

u(x,y;si,ti)dsidti, (2:2)

and

ul(x,y) =ul−1(x,y)

xibi

yidi

2

∂si∂ti

u(x,y;si,ti)dsidti, (2:3)

where

u(x,y;si,ti) =u(x1,. . .,xi−1,si,xi+1,. . .,xn,y1,. . .,yi−1,ti,yi+1,. . .,yn).

Hence, from (2.2) and (2.3) and in view of the arithmetic-geometric means inequality and Hölder inequality with indicesμand l, it follows that

u(x,y)l≤ 1 2u(x,y)

l−1

aibi

cidi

si2tiu(x,y;si,ti)dsidti

≤1 2u(x,y)

l−1

[(biai)(cidi)]1/μ

aixi

ciyi

si2tiu(x,y;si,ti) λ

dsidti

1/λ .

(2:4)

(3)

E

Eu(x,y)ldxdy≤ 1

2n

n

i=1

[(biai)(cidi)]1/μ

×

E

Eu(x,y)l−1

aibi

cidi

si2tiu(x,y;si,ti)

λdsidti

1/λ

dxdy

≤ 1

2n

E

Eu(x,y)(l−1)μdxdy

1/μn i=1

[(biai)(cidi)]1/μ

× E

E

aibi

cidi

si2tiu(x,y;si,ti)

λdsidtidxdy

1/λ

≤ 1

2n

E

Eu(x,y)(l−1)μdxdy

1/μn i=1

[(biai)(cidi)]1/μ+1/λ

×

E

E

2

∂xi∂yi

u(x,y) λ

dxdy

1/λ

≤ 1

2n

E

Eu(x,y)(l−1)μdxdy

1/μn i=1

[(biai)(cidi)]μ

1/μ

×

E

Egradu(x,y)λλdxdy

1/λ ,

where

gradu(x,y)λ=

n

i=1

xi2yiu(x,y) λ1/λ

.

The proof is complete.

Remark 2.1. Letu(x,y) reduce tou(x) in (2.1) and with suitable modifications, then (2.1) becomes

Eu(x)(l)μdx≤ 1

2n

Eu(x)(l−1)μdx

1/μn i=1

(biai)μ

1/μ

× Egradu(x)λλdx

1/λ ,

where

gradu(x)λ=

n

i=1

xiu(x) λ1/λ

.

This is just a important inequality which was given by Agarwal and Sheng [15]. Remark 2.2. For the given real numberslk≥0, 1≤k≤r, such thatrlk≥1, the arith-metic-geometric means inequality and (2.1) gives

E

E

r

k=1

uk(x,y)lkdxdy

1

r

r

k=1

E

Euk(x,y)rlkdxdy

≤ 1

2nr

n

i=1

[(biai)(cidi)]μ

1/μr

k=1

E

Euk(x,y)(rlk−1)μdxdy

1/μ

×

E

Egraduk(x,y)λλdxdy

1/λ .

(4)

This is just a general form of the following result which was given by Agarwal and Sheng [15].

E

r

k=1

uk(x)lkdx

1 2nr

n

i=1

(biai)μ

1/μr

k=1

Euk(x)(rlk−1)μdx

1/μ

× Egraduk(x)λλdx

1/λ

,

where

graduk(x)λ=

n

i=1

xiu(x) λ1/λ

.

Remark 2.3. In particular, for lk = (pk + 2)/(2r), pk ≥ 1,1≤ k ≤ r, μ= l= 2, the inequality (2.5) reduces to

E

E

r

k=1

uk(x,y)

(pk+2)/2 1/r

dxdy

≤ 1

2nr

n

i=1

[(biai)(cidi)]2

1/2r

k=1

E

Eu(x,y)pk dxdy

1/2

×

E

Egraduk(x,y)22dxdy

1/2

.

This is just a general form of the following result which was given by Agarwal and Sheng [15].

E

r

k=1

uk(x)

(pk+2)/2 1/r

dx

≤ 1

2nr

n

i=1

(biai)2

1/2r

k=1

Eu(x)pk dx

1/2

Egraduk(x)

2 2dx

1/2

.

On the other hand, the above inequality with the right-hand side multiplied by

r

k=1((pk+ 2)/2)

1/r

and the term n

i=1(biai)2

1/2

replace by √ has been

proved by Pachpatte [16].

Remark 2.4. Ifu(x,y) reduce tou(x) in (2.1), then the inequality (2.1) and its parti-cular case l ≥2, μ=l = 2 with the right-hand side multiplied byl have been sepa-rately proved by Pachpatte in [17].

Theorem 2.2.Letl≥1and u(x, y)ÎG(E×E’).Then, the following inequality holds

E

Eu(x,y)2λdxdyπλ

2β2α2

128n

E

Eu(x,y)2λdxdy

(λ−1)/λ

×

E

E

n

i=1

s2iuti + (λ−1) 1

u(x,y) ∂u

∂xi

∂u

∂yi

2λdxdy

1/λ ,

(2:6)

(5)

Proof. For each fixedi, 1≤i≤n, we obtain that

(x,y) =λ

aixi

ciyi

−1(x,y;si,ti)

2u

∂si∂ti

+ (λ−1)−2(x,y;si,ti)∂u

∂si

∂u

∂ti

dsidti,

and hence from the Cauchy-Schwarz inequality, it follows that u(x,y)λλ2(xiai)(yici)

× aixi

ciyi

yλ−1(x,y;si,ti)

2u

∂si∂ti + (λ−

1)uλ−2(x,y;si,ti)

u

∂si

∂u

∂ti

2dsidti,

(2:7)

and similarly,

u(x,y)λλ2(bixi)(diyi)

×

xibi

yidi

uλ−1(x,y;si,ti)

2u

∂si∂ti

+ (λ−1)uλ−2(x,y;si,ti)

u

∂si

∂u

∂ti

2dsidti,

(2:8)

Hence, multiplying (2.7) and (2.8) and in view of using the arithmetic-geometric means inequality, summing the resulting inequalities for 1≤i≤n, and then integrating over E×E’, to obtain

E

Eu(x,y)2λdxdyλ

2 2n E E n i=1

[(xiai)(yici)(bixi)(diyi)]1/2

× aibi

cidi

−1(x,y;si,ti) 2u

∂si∂ti

+ (λ−1)−2(x,y;si,ti)∂u

∂si

∂u

∂ti

2dsidti

dxdy =λ 2 2n n i=1

aibi

cidi[(xiai)(yici)(bixi)(diyi)]1/2dxidyi

×

E

Eyλ−1(x,y)

2u

∂si∂ti

+ (λ−1)−2(x,y)∂u

∂xi

∂u

∂yi

2dxdy

πλ2β2α2 128n E E n i=1

−1(x,y) 2u

∂si∂ti

+ (λ−1)−2(x,y)∂u

∂xi

∂u

∂yi

2dxdy,

whereb= max1≤i≤n(bi-ai) anda= max1≤i≤n(di-ci).

Hence, using Hölder inequality with indices land l/(l - 1) in right-hand side of above inequality, we have

E

Eu(x,y)2λdxdyπλ

2β2α2

128n

E

Eu(x,y)2λdxdy

(λ−1)/λ

× E E n i=1

s2iuti+ (λ−1) 1

u(x,y) ∂u

∂xi

∂u

∂yi

2λdxdy

1/λ .

The proof is complete.

Remark 2.5. Letu(x,y) reduce tou(x) in (2.6) and with suitable modifications, then (2.6) becomes the following Agarwal and Sheng [15] inequality.

Eu(x)2λdxπλ

2β2

16n

Eu(x)2λdx

(λ−1)/λ

Egradu(x)22λdx

1/λ

,

(6)

Theorem 2.3.Let l≥0,m≥1be given real numbers, and let u(x,y)Î G(E×E’).

Then, the following inequality holds

E

Eu(x,y)l+mdxdy≤ 1 n

m+l

2m

mn

i=1

[(biai)(dici)]m

×

E

Eul/m(x,y)

2u

∂xi∂yi

+ l

mu

(l/m−1)(x,y)∂u

∂xi

∂u

∂yi

mdxdy.

(2:8a)

Proof. For each fixedi, 1≤i≤n, we obtain that

ul+m(x,y) = m+l

m [u(x,y)]

(m−1)(l+m)/m

×

aixi

ciyi

ul/m(x,y;si,ti)

2u

∂si∂ti

+ l

mu

(l/m−1)(x,y;s

i,ti)

u

∂si

∂u

∂ti

dsidti,

and, hence, it follows that

u(x,y)l+mm+l

m u(x,y)

(m−1)(l+m)/m

×

aixi

ciyi

ul/m(x,y;si,ti)

2u

∂si∂ti

+ l

mu

(l/m−1)(x,y;s

i,ti)

u

∂si

∂u

∂ti

dsidti,

(2:9)

and, similarly,

u(x,y)l+mm+l

m u(x,y)

(m−1)(l+m)/m

×

xibi

yidi

ul/m(x,y;si,ti)

2u

∂si∂ti

+ l

mu

(l/m−1)(x,y;s

i,ti)

u

∂si

∂u

∂ti

dsidti.

(2:10)

Now, adding (2.9) and (2.10) and integrating the resulting inequality from aito bi

and citodi, respectively. Then

aibi

cidiu(x,y)l+mdxidyi

m+l

2m

aibi

cidiu(x,y)(m−1)(l+m)/mdxidyi

×

aibi

cidi

ul/m(x,y)

2u

∂xi∂yi

+ l

mu

(l/m−1)(x,y)∂u

∂xi

∂u

∂yi

dxidyi.

Next in each integral of the right-hand side of the above inequality we apply Hölder inequality with indices mand m/(m- 1), to get

aibi

cidiu(x,y) l+m

dxidyi

m+l

2m

aibi

cidiu(x,y) l+m

dxidyi

(m−1)/m

×[(biai)(dici)]1/m[(biai)(dici)](m−1)/m

×

aibi

cidi

ul/m(x,y)

2u

∂xi∂yi

+ l

mu

(l/m−1)(x,y)∂u

∂xi

∂u

∂yi

mdxidyi

1/m

,

which is unless aibi

cidiu(x,y) l+m

dxidyi= 0 (for which the inequality (2.8) is

(7)

aibi

cidiu(x,y) l+m

dxidyi

1/m

m+l

2m [(biai)(dici)]

×

aibi

cidi

ul/m(x,y)

2u

∂xi∂yi

+ l

mu

(l/m−1)(x,y)∂u

∂xi

∂u

∂yi

mdxidyi

1/m

.

Finally, raisingm-th power both sides of the above inequality, integrating the result-ing inequality fromajto bjandcjto dj, respectively, then summing theninequalities 1 ≤i≤n, we find the desired inequality (2.8).

Remark 2.6. Letu(x,y) reduce tou(x) in (2.8) and with suitable modifications, then (2.8) becomes the following Agarwal and Sheng [15] inequality.

Eu(x)l+mdx≤ 1 n

m+l

2m

mn

i=1

(biai)m

Eu(x)l ∂xi

u(x)

m

dx.

Remark 2.7. The inequality (2.8) for u(x, y) reduce tou(x), with the right-hand sides multiplied bymmand (bi-ai)mreplaced by (ab)mhas been obtained by Pachpatte [18].

Acknowledgements

C.-J. Zhao research was supported by National Natural Sciences Foundation of China (10971205). W.-S. Cheung research was partially supported by a HKU URG grant.

Author details 1

Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R. China2Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong3Str. Hărmanului 6, 505600 Săcele-Négyfalu, Jud. Braşov,

Romania

Authorscontributions

C-JZ, W-SC and MB jointly contributed to the main results Theorems 2.1, 2.2, and 2.3. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 1 February 2012 Accepted: 18 May 2012 Published: 18 May 2012

References

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2. Agarwal, RP, Lakshmikantham, V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore (1993)

3. Agarwal, RP, Thandapani, E: On some new integrodifferential inequalities Anal sti Univ.Al I Cuzadin Iasi.28, 123126 (1982)

4. Bainov, D, Simeonov, P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht (1992) 5. Li, JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl.167, 98–100 (1992).

doi:10.1016/0022-247X(92)90238-9

6. Mitrinovič, DS, Pečarić, JE, Fink, AM: Inequalities involving Functions and Their Integrals ang Derivatives. Kluwer Academic Publishers, Dordrecht (1991)

7. Cheung, WS: On Opial-type inequalities in two variables. Aequationes Math.38, 236244 (1989). doi:10.1007/ BF01840008

8. Cheung, WS: Some new Opial-type inequalities. Mathematika.37, 136142 (1990). doi:10.1112/S0025579300012869 9. Cheung, WS: Some generalized Opial-type inequalities. J Math Anal Appl.162, 317–321 (1991). doi:10.1016/0022-247X

(91)90152-P

10. Cheung, WS: Opial-type inequalities withmfunctions innvariables. Mathematika.39, 319–326 (1992). doi:10.1112/ S0025579300015047

11. Crooke, PS: On two inequalities of the Sobolev type. Appl Anal.3, 345–358 (1974). doi:10.1080/00036817408839076 12. Pachpatte, BG: Opial type inequality in several variables. Tamkang J Math.22, 7–11 (1991)

13. Pachpatte, BG: On some new integral inequalities in two independent variables. J Math Anal Appl.129, 375382 (1988). doi:10.1016/0022-247X(88)90256-9

14. Wang, XJ: Sharp constant in a Sobolev inequality. Nonlinear Anal.20, 261268 (1993). doi:10.1016/0362-546X(93)90162-L 15. Agarwal, RP, Sheng, Q: Sharp integral inequalities innindependent varibles. Nonlinear Anal Theory Methods Appl.

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16. Pachpatte, BG: On Sobolev type integral inequalities. Proc R Soc Edinb.103, 1–14 (1986). doi:10.1017/ S0308210500013986

17. Pachpatte, BG: On two inequalities of the Serrin type. J Math Anal Appl.116, 193–199 (1986). doi:10.1016/0022-247X(86) 90051-X

18. Pachpatte, BG: On some variants of Sobolev’s inequality. Soochow J Math.17, 121–129 (1991)

doi:10.1186/1029-242X-2012-109

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