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Some sharp integral inequalities involving partial derivatives

Abstract

The main purpose of the present article is to establish some new sharp integral inequalities in 2n independent 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.

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 [110]. 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 [1114]. 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 2n independent 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

Let R be the set of real numbers and nthe n-dimensional Euclidean space. Let E, E' be a bounded domain in Rndefined by E × E = i = 1 n [ a i , b i ] × [ c i , d i ] , i = 1 , . . . , n . For x i , y i R, i = 1, ..., n, (x, y) = (x1, ..., x n , y1, ..., y n ) is a variable point in E × E' and dxdy = dx1 ... dx n dy1 ... dy n . For any continuous real-valued function u(x, y) defined on E × E' we denote by ∫ E E'u(x, y) dxdy the 2n-fold integral

a 1 b 1 a n b n c 1 d 1 c n d n u ( x 1 , , x n , y 1 , , y n ) d x 1 d x n d y 1 d y n ,

and for any (x, y) E × E', ∫E(x) ∫E'(x)u(s, t) dsdt is the 2n-fold integral

a 1 x 1 a n x n c 1 y 1 c n y n u ( s 1 , , s n , t 1 , , t n ) d x 1 d s n d t 1 d t n ,

We represent by F(E × E') the class of continuous functions u(x, y) : E × E' → , for each i,1 ≤ in,

u ( x , y ) | x i = a i = 0 , u ( x , y ) | y i = c i = 0 , u ( x , y ) | x i = b i = 0 , u ( x , y ) | y i = d i = 0 , ( i = 1 , , n )

the class F(E × E') is denoted as G(E × E').

Theorem 2.1. Let l, μ, λ ≥ 1, be given real numbers such that 1 μ + 1 λ =1. Further, let u(x, y) G(E × E'). Then, the following inequality holds

E E u ( x , y ) l d x d y 1 2 n i = 1 n [ ( b i - a i ) ( c i - d i ) ] μ 1 / μ E E u ( x , y ) ( l - 1 ) μ d x d y 1 / μ × E E grad u ( x , y ) λ λ d x d y 1 / λ ,
(2.1)

where

grad u ( x , y ) λ = i = 1 n 2 x i y i u ( x , y ) λ 1 / λ .

Proof. For each fixed i, 1 ≤ in, in view of

u ( x , y ) | x i = a i = 0 , u ( x , y ) | y i = c i = 0 , u ( x , y ) | x i = b i = 0 , u ( x , y ) | y i = d i = 0 , ( i = 1 , , n )

we have

u l ( x , y ) = u l - 1 ( x , y ) a i x i c i y i 2 s i t i u ( x , y ; s i , t i ) d s i d t i ,
(2.2)

and

u l ( x , y ) = u l - 1 ( x , y ) x i b i y i d i 2 s i t i u ( x , y ; s i , t i ) d s i d t i ,
(2.3)

where

u ( x , y ; s i , t i ) = u ( x 1 , , x i - 1 , s i , x i + 1 , , x n , y 1 , , y i - 1 , t i , y i + 1 , , y n ) .

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

u ( x , y ) l 1 2 u ( x , y ) l - 1 a i b i c i d i 2 s i t i u ( x , y ; s i , t i ) d s i d t i 1 2 u ( x , y ) l - 1 [ ( b i - a i ) ( c i - d i ) ] 1 / μ a i x i c i y i 2 s i t i u ( x , y ; s i , t i ) λ d s i d t i 1 / λ .
(2.4)

Now, summing the inequalities (2.4) for 1 ≤ in, integrating over E × E' and applying Holder inequality with indices μ and λ two times, we get

E E u ( x , y ) l d x d y 1 2 n i = 1 n [ ( b i - a i ) ( c i - d i ) ] 1 / μ × E E u ( x , y ) l - 1 a i b i c i d i 2 s i t i u ( x , y ; s i , t i ) λ d s i d t i 1 / λ d x d y 1 2 n E E u ( x , y ) ( l - 1 ) μ d x d y 1 / μ i = 1 n [ ( b i - a i ) ( c i - d i ) ] 1 / μ × E E a i b i c i d i 2 s i t i u ( x , y ; s i , t i ) λ d s i d t i d x d y 1 / λ 1 2 n E E u ( x , y ) ( l - 1 ) μ d x d y 1 / μ i = 1 n [ ( b i - a i ) ( c i - d i ) ] 1 / μ + 1 / λ × E E 2 x i y i u ( x , y ) λ d x d y 1 / λ 1 2 n E E u ( x , y ) ( l - 1 ) μ d x d y 1 / μ i = 1 n [ ( b i - a i ) ( c i - d i ) ] μ 1 / μ × E E grad u ( x , y ) λ λ d x d y 1 / λ ,

where

grad u ( x , y ) λ = i = 1 n 2 x i y i u ( x , y ) λ 1 / λ .

The proof is complete.

Remark 2.1. Let u(x, y) reduce to u(x) in (2.1) and with suitable modifications, then (2.1) becomes

E u ( x ) ( l ) μ d x 1 2 n E u ( x ) ( l - 1 ) μ d x 1 / μ i = 1 n ( b i - a i ) μ 1 / μ × E grad u ( x ) λ λ d x 1 / λ ,

where

grad u ( x ) λ = i = 1 n 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 numbers l k ≥ 0, 1 ≤ kr, such that rl k ≥ 1, the arithmetic-geometric means inequality and (2.1) gives

E E k = 1 r u k ( x , y ) l k d x d y 1 r k = 1 r E E u k ( x , y ) r l k d x d y 1 2 n r i = 1 n [ ( b i - a i ) ( c i - d i ) ] μ 1 / μ k = 1 r E E u k ( x , y ) ( r l k - 1 ) μ d x d y 1 / μ × E E grad u k ( x , y ) λ λ d x d y 1 / λ .
(2.5)

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

E k = 1 r u k ( x ) l k d x 1 2 n r i = 1 n ( b i - a i ) μ 1 / μ k = 1 r E u k ( x ) ( r l k - 1 ) μ d x 1 / μ × E grad u k ( x ) λ λ d x 1 / λ ,

where

grad u k ( x ) λ = i = 1 n x i u ( x ) λ 1 / λ .

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

E E k = 1 r u k ( x , y ) ( p k + 2 ) / 2 1 / r d x d y 1 2 n r i = 1 n [ ( b i - a i ) ( c i - d i ) ] 2 1 / 2 k = 1 r E E u ( x , y ) p k d x d y 1 / 2 × E E grad u k ( x , y ) 2 2 d x d y 1 / 2 .

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

E k = 1 r u k ( x ) ( p k + 2 ) / 2 1 / r d x 1 2 n r i = 1 n ( b i - a i ) 2 1 / 2 k = 1 r E u ( x ) p k d x 1 / 2 E grad u k ( x ) 2 2 d x 1 / 2 .

On the other hand, the above inequality with the right-hand side multiplied by k = 1 r ( ( p k + 2 ) / 2 ) 1 / r and the term i = 1 n ( b i - a i ) 2 1 / 2 replace by n β has been proved by Pachpatte [16].

Remark 2.4. If u(x, y) reduce to u(x) in (2.1), then the inequality (2.1) and its particular case l ≥ 2, μ = λ = 2 with the right-hand side multiplied by l have been separately proved by Pachpatte in [17].

Theorem 2.2. Let λ ≥ 1 and u(x, y) G(E × E'). Then, the following inequality holds

E E u ( x , y ) 2 λ d x d y π λ 2 β 2 α 2 1 2 8 n E E u ( x , y ) 2 λ d x d y ( λ - 1 ) / λ × E E i = 1 n 2 u s i t i + ( λ - 1 ) 1 u ( x , y ) u x i u y i 2 λ d x d y 1 / λ ,
(2.6)

where β = max1≤in(b i - a i ) and α = max1≤in(d i - c i ).

Proof. For each fixed i, 1 ≤ in, we obtain that

u λ ( x , y ) = λ a i x i c i y i u λ - 1 ( x , y ; s i , t i ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ; s i , t i ) u s i u t i d s i d t i ,

and hence from the Cauchy-Schwarz inequality, it follows that

u ( x , y ) λ λ 2 ( x i - a i ) ( y i - c i ) × a i x i c i y i y λ - 1 ( x , y ; s i , t i ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ; s i , t i ) u s i u t i 2 d s i d t i ,
(2.7)

and similarly,

u ( x , y ) λ λ 2 ( b i - x i ) ( d i - y i ) × x i b i y i d i u λ - 1 ( x , y ; s i , t i ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ; s i , t i ) u s i u t i 2 d s i d t i ,
(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 ≤ in, and then integrating over E × E', to obtain

E E u ( x , y ) 2 λ d x d y λ 2 2 n E E i = 1 n [ ( x i - a i ) ( y i - c i ) ( b i - x i ) ( d i - y i ) ] 1 / 2 × a i b i c i d i u λ - 1 ( x , y ; s i , t i ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ; s i , t i ) u s i u t i 2 d s i d t i d x d y = λ 2 2 n i = 1 n a i b i c i d i [ ( x i - a i ) ( y i - c i ) ( b i - x i ) ( d i - y i ) ] 1 / 2 d x i d y i × E E y λ - 1 ( x , y ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ) u x i u y i 2 d x d y π λ 2 β 2 α 2 1 2 8 n E E i = 1 n u λ - 1 ( x , y ) 2 u s i t i + ( λ - 1 ) u λ - 2 ( x , y ) u x i u y i 2 d x d y ,

where β = max1≤in(b i - a i ) and α = max1≤in(d i - c i ).

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

E E u ( x , y ) 2 λ d x d y π λ 2 β 2 α 2 1 2 8 n E E u ( x , y ) 2 λ d x d y ( λ - 1 ) / λ × E E i = 1 n 2 u s i t i + ( λ - 1 ) 1 u ( x , y ) u x i u y i 2 λ d x d y 1 / λ .

The proof is complete.

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

E u ( x ) 2 λ d x π λ 2 β 2 1 6 n E u ( x ) 2 λ d x ( λ - 1 ) / λ E grad u ( x ) 2 2 λ d x 1 / λ ,

where β = max1≤in(b i - a i ).

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

Then, the following inequality holds

E E u ( x , y ) l + m d x d y 1 n m + l 2 m m i = 1 n [ ( b i - a i ) ( d i - c i ) ] m × E E u l / m ( x , y ) 2 u x i y i + l m u ( l / m - 1 ) ( x , y ) u x i u y i m d x d y .
(2.8a)

Proof. For each fixed i, 1 ≤ in, we obtain that

u l + m ( x , y ) = m + l m [ u ( x , y ) ] ( m - 1 ) ( l + m ) / m × a i x i c i y i u l / m ( x , y ; s i , t i ) 2 u s i t i + l m u ( l / m - 1 ) ( x , y ; s i , t i ) u s i u t i d s i d t i ,

and, hence, it follows that

u ( x , y ) l + m m + l m u ( x , y ) ( m - 1 ) ( l + m ) / m × a i x i c i y i u l / m ( x , y ; s i , t i ) 2 u s i t i + l m u ( l / m - 1 ) ( x , y ; s i , t i ) u s i u t i d s i d t i ,
(2.9)

and, similarly,

u ( x , y ) l + m m + l m u ( x , y ) ( m - 1 ) ( l + m ) / m × x i b i y i d i u l / m ( x , y ; s i , t i ) 2 u s i t i + l m u ( l / m - 1 ) ( x , y ; s i , t i ) u s i u t i d s i d t i .
(2.10)

Now, adding (2.9) and (2.10) and integrating the resulting inequality from a i to b i and c i to d i , respectively. Then

a i b i c i d i u ( x , y ) l + m d x i d y i m + l 2 m a i b i c i d i u ( x , y ) ( m - 1 ) ( l + m ) / m d x i d y i × a i b i c i d i u l / m ( x , y ) 2 u x i y i + l m u ( l / m - 1 ) ( x , y ) u x i u y i d x i d y i .

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

a i b i c i d i u ( x , y ) l + m d x i d y i m + l 2 m a i b i c i d i u ( x , y ) l + m d x i d y i ( m - 1 ) / m × [ ( b i - a i ) ( d i - c i ) ] 1 / m [ ( b i - a i ) ( d i - c i ) ] ( m - 1 ) / m × a i b i c i d i u l / m ( x , y ) 2 u x i y i + l m u ( l / m - 1 ) ( x , y ) u x i u y i m d x i d y i 1 / m ,

which is unless a i b i c i d i u ( x , y ) l + m d x i d y i =0 (for which the inequality (2.8) is obvious), is the same as

a i b i c i d i u ( x , y ) l + m d x i d y i 1 / m m + l 2 m [ ( b i - a i ) ( d i - c i ) ] × a i b i c i d i u l / m ( x , y ) 2 u x i y i + l m u ( l / m - 1 ) ( x , y ) u x i u y i m d x i d y i 1 / m .

Finally, raising m-th power both sides of the above inequality, integrating the resulting inequality from a j to b j and c j to d j , respectively, then summing the n inequalities 1 ≤ in, we find the desired inequality (2.8).

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

E u ( x ) l + m d x 1 n m + l 2 m m i = 1 n ( b i - a i ) m E u ( x ) l x i u ( x ) m d x .

Remark 2.7. The inequality (2.8) for u(x, y) reduce to u(x), with the right-hand sides multiplied by mmand (b i - a i )mreplaced by (αβ)mhas been obtained by Pachpatte [18].

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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.

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Correspondence to Chang-Jian Zhao.

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The authors declare that they have no competing interests.

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.

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Zhao, C., Cheung, W. & Bencze, M. Some sharp integral inequalities involving partial derivatives. J Inequal Appl 2012, 109 (2012). https://doi.org/10.1186/1029-242X-2012-109

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Keywords

  • Cauchy-Schwarz's inequality
  • Pachpatte's inequality
  • Hölder integral inequality
  • the arithmetic-geometric means inequality