### 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

1_{Department 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

*b*1

*a*1

· · · *anbn*

*c*1*d*1· · ·

*cndnu*(*x*1,*. . .*,*xn*,*y*1,*. . .*,*yn*)*dx*1*. . .dxndy*1*. . .dyn*,

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

*a*1*x*1· · ·

*anxn*

*c*1*y*1· · ·

*cnynu*(*s*1,*. . .*,*sn*,*t*1,*. . .*,*tn*)*dx*1*. . .dsndt*1*. . .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
2*n*

_{}_{n}

*i*=1

[(*bi*−*ai*)(*ci*−*di*)]*μ*

1/*μ*_{}

*E*

*Eu*(*x*,*y*)(*l*−1)*μdxdy*

1/*μ*

× *E*

*E*grad*u*(*x*,*y*)*λ _{λ}dxdy*

1/*λ*

,

(2:1)

where

grad*u*(*x*,*y*)* _{λ}*=

_{n}

*i*=1

* _{∂}_{x}∂_{i}_{∂}*2

*(*

_{y}_{i}u*x*,

*y*)

*λ*1/

*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*(*x*1,*. . .*,*xi*−1,*si*,*xi*+1,*. . .*,*xn*,*y*1,*. . .*,*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
2*u*(*x*,*y*)

*l*−1

*aibi*

*cidi*

* _{∂}_{s}∂_{i}_{∂}*2

*(*

_{t}_{i}u*x*,

*y*;

*si,ti)dsidti*

≤1
2*u*(*x*,*y*)

*l*−1

[(*bi*−*ai)(ci*−*di)]*1/*μ*

*aixi*

*ciyi*

* _{∂}_{s}∂_{i}*2

*(*

_{∂}_{t}_{i}u*x*,

*y*;

*si,ti)*

*λ*

*dsidti*

1/*λ*
.

(2:4)

*E*

*Eu*(*x*,*y*)*ldxdy*≤ 1

2*n*

*n*

*i*=1

[(*bi*−*ai*)(*ci*−*di*)]1/*μ*

×

*E*

*Eu*(*x*,*y*)*l*−1

*aibi*

*cidi*

* _{∂}_{s}∂_{i}*2

*(*

_{∂}_{t}_{i}u*x*,

*y*;

*si*,

*ti*)

*λdsidti*

1/*λ*

*dxdy*

≤ 1

2*n*

*E*

*Eu*(*x*,*y*)(*l*−1)*μdxdy*

1/*μ*_{}*n*
*i*=1

[(*bi*−*ai*)(*ci*−*di*)]1/*μ*

× *E*

*E*

*aibi*

*cidi*

* _{∂}_{s}∂_{i}*2

*(*

_{∂}_{t}_{i}u*x*,

*y*;

*si*,

*ti*)

*λdsidtidxdy*

1/*λ*

≤ 1

2*n*

*E*

*Eu*(*x*,*y*)(*l*−1)*μdxdy*

1/*μ*_{}*n*
*i*=1

[(*bi*−*ai*)(*ci*−*di*)]1/*μ*+1/*λ*

×

*E*

*E* *∂*

2

*∂xi∂yi*

*u*(*x*,*y*)
*λ*

*dxdy*

1/*λ*

≤ 1

2*n*

*E*

*Eu*(*x*,*y*)(*l*−1)*μdxdy*

1/*μ*_{}*n*
*i*=1

[(*bi*−*ai*)(*ci*−*di*)]*μ*

1/*μ*

×

*E*

*E*grad*u*(*x*,*y*)*λ _{λ}dxdy*

1/*λ*
,

where

grad*u*(*x*,*y*)* _{λ}*=

_{n}

*i*=1

* _{∂}_{x}∂_{i}_{∂}*2

*(*

_{y}_{i}u*x*,

*y*)

*λ*1/

*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

2*n*

*Eu*(*x*)(*l*−1)*μdx*

1/*μ*_{}*n*
*i*=1

(*bi*−*ai*)*μ*

1/*μ*

× *E*grad*u*(*x*)*λ _{λ}dx*

1/*λ*
,

where

grad*u*(*x*)* _{λ}*=

_{n}

*i*=1

* _{∂}∂_{x}_{i}u*(

*x*)

*λ*1/

*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

2*nr*

_{n}

*i*=1

[(*bi*−*ai*)(*ci*−*di*)]*μ*

1/*μ*_{}_{r}

*k*=1

*E*

*Euk*(*x*,*y*)(*rlk*−1)*μdxdy*

1/*μ*

×

*E*

*E*grad*uk*(*x*,*y*)*λ _{λ}dxdy*

1/*λ*
.

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
2*nr*

_{n}

*i*=1

(*bi*−*ai*)*μ*

1/*μ*_{}*r*

*k*=1

*Euk*(*x*)(*rlk*−1)*μdx*

1/*μ*

× *E*grad*uk*(*x*)*λ _{λ}dx*

1/*λ*

,

where

grad*uk*(*x*)* _{λ}*=

_{n}

*i*=1

* _{∂}∂_{x}_{i}u*(

*x*)

*λ*1/

*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

2*nr*

_{n}

*i*=1

[(*bi*−*ai*)(*ci*−*di*)]2

1/2_{}_{r}

*k*=1

*E*

*Eu*(*x*,*y*)*pk*
*dxdy*

1/2

×

*E*

*E*grad*uk*(*x*,*y*)2_{2}*dxdy*

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

2*nr*

_{n}

*i*=1

(*bi*−*ai*)2

1/2_{}_{r}

*k*=1

*Eu*(*x*)*pk*
*dx*

1/2

*E*grad*uk*(*x*)

2
2*dx*

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(*bi*−*ai*)2

1/2

replace by √*nβ* 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*)2*λdxdy*≤ *πλ*

2* _{β}*2

*2*

_{α}128*n*

*E*

*Eu*(*x*,*y*)2*λdxdy*

(*λ−*1)/*λ*

×

*E*

*E*

*n*

*i*=1

* _{∂}∂_{s}*2

*+ (λ−1) 1*

_{i}_{∂}u_{t}_{i}*u*(*x*,*y*)
*∂u*

*∂xi*

*∂u*

*∂yi*

2*λdxdy*

1/*λ*
,

(2:6)

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

*uλ*(*x*,*y*) =*λ*

*aixi*

*ciyi*

*uλ*−1(*x*,*y*;*si,ti)* *∂*

2_{u}

*∂si∂ti*

+ (*λ*−1)*uλ*−2(*x*,*y*;*si,ti)∂u*

*∂si*

*∂u*

*∂ti*

*dsidti,*

and hence from the Cauchy-Schwarz inequality, it follows that
*u*(*x*,*y*)*λ*≤*λ*2(*xi*−*ai*)(*yi*−*ci*)

× *aixi*

*ciyi*

*yλ−*1(*x*,*y*;*si*,*ti*) *∂*

2_{u}

*∂si∂ti* + (λ−

1)*uλ−*2(*x*,*y*;*si*,*ti*)*∂*

*u*

*∂si*

*∂u*

*∂ti*

2*dsidti*,

(2:7)

and similarly,

*u*(*x*,*y*)*λ*≤*λ*2(*bi*−*xi*)(*di*−*yi*)

×

*xibi*

*yidi*

*uλ−*1(*x*,*y*;*si*,*ti*) *∂*

2_{u}

*∂si∂ti*

+ (*λ*−1)*uλ−*2(*x*,*y*;*si*,*ti*)*∂*

*u*

*∂si*

*∂u*

*∂ti*

2*dsidti*,

(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*)2*λdxdy*≤ *λ*

2
2*n*
*E*
*E*
_{n}*i*=1

[(*xi*−*ai)(yi*−*ci)(bi*−*xi)(di*−*yi)]*1/2

× *aibi*

*cidi*

*uλ*−1(*x*,*y*;*si,ti)* *∂*
2_{u}

*∂si∂ti*

+ (*λ*−1)*uλ*−2(*x*,*y*;*si,ti)∂u*

*∂si*

*∂u*

*∂ti*

2*dsidti*

*dxdy*
=*λ*
2
2*n*
*n*
*i*=1

*aibi*

*cidi*[(*xi*−*ai)(yi*−*ci)(bi*−*xi)(di*−*yi)]*1/2*dxidyi*

×

*E*

*Eyλ*−1(*x*,*y*) *∂*

2_{u}

*∂si∂ti*

+ (*λ*−1)*uλ*−2(*x*,*y*)*∂u*

*∂xi*

*∂u*

*∂yi*

2*dxdy*

≤*πλ*2*β*2*α*2
128*n*
*E*
*E*
*n*
*i*=1

*uλ*−1(*x*,*y*) *∂*
2_{u}

*∂si∂ti*

+ (*λ*−1)*uλ*−2(*x*,*y*)*∂u*

*∂xi*

*∂u*

*∂yi*

2*dxdy*,

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*)2*λdxdy*≤ *πλ*

2* _{β}*2

*2*

_{α}128*n*

*E*

*Eu*(*x*,*y*)2*λdxdy*

(*λ−*1)/*λ*

×
*E*
*E*
*n*
*i*=1

* _{∂}∂_{s}*2

*+ (*

_{i}_{∂}u_{t}_{i}*λ*−1) 1

*u*(*x*,*y*)
*∂u*

*∂xi*

*∂u*

*∂yi*

2*λdxdy*

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*)2*λdx*≤*πλ*

2* _{β}*2

16*n*

*Eu*(*x*)2*λdx*

(*λ−*1)/*λ*

*E*grad*u*(*x*)2_{2}*λdx*

1/*λ*

,

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*

2*m*

*m*_{}*n*

*i*=1

[(*bi*−*ai*)(*di*−*ci*)]*m*

×

*E*

*Eul*/*m*(*x*,*y*) *∂*

2_{u}

*∂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*) *∂*

2_{u}

*∂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*+*m*≤ *m*+*l*

*m* *u*(*x*,*y*)

(*m*−1)(*l*+*m*)/*m*

×

*aixi*

*ciyi*

*ul*/*m*(*x*,*y*;*si*,*ti*) *∂*

2_{u}

*∂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*+*m*≤ *m*+*l*

*m* *u*(*x*,*y*)

(*m*−1)(*l*+*m*)/*m*

×

*xibi*

*yidi*

*ul*/*m*(*x*,*y*;*si*,*ti*) *∂*

2_{u}

*∂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*

2*m*

*aibi*

*cidiu*(*x*,*y*)(*m*−1)(*l*+*m*)/*mdxidyi*

×

*aibi*

*cidi*

*ul*/*m*(*x*,*y*) *∂*

2_{u}

*∂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*

2*m*

*aibi*

*cidiu*(*x*,*y*)
*l*+*m*

*dxidyi*

(*m*−1)/*m*

×[(*bi*−*ai*)(*di*−*ci*)]1/*m*[(*bi*−*ai*)(*di*−*ci*)](*m*−1)/*m*

×

*aibi*

*cidi*

*ul*/*m*(*x*,*y*) *∂*

2_{u}

*∂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

*aibi*

*cidiu*(*x*,*y*)
*l*+*m*

*dxidyi*

1/*m*

≤ *m*+*l*

2*m* [(*bi*−*ai*)(*di*−*ci*)]

×

*aibi*

*cidi*

*ul*/*m*(*x*,*y*) *∂*

2_{u}

*∂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*

2*m*

*m*_{}*n*

*i*=1

(*bi*−*ai*)*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 Kong3_{Str. H}_{ă}_{rmanului 6, 505600 S}_{ă}_{cele-Négyfalu, Jud. Bra}_{ş}_{ov,}

Romania

Authors_{’}contributions

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

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

Cite this article as:Zhaoet al.:Some sharp integral inequalities involving partial derivatives.Journal of Inequalities and Applications20122012:109.

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