Some sharp integral inequalities involving partial derivatives

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.

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, 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 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.

Let R be the set of real numbers and ℝⁿ the n-dimensional Euclidean space. Let E, E' be a bounded domain in Rⁿ defined by . 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

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

We represent by F(E × E') the class of continuous functions u(x, y) : E × E' → ℝ, for each i,1 ≤ i ≤ 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 . Further, let u(x, y) ∈ G(E × E'). Then, the following inequality holds

where

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

we have

and

where

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

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

where

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

where

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 ≤ k ≤ r, such that rl_k ≥ 1, the arithmetic-geometric means inequality and (2.1) gives

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

where

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

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

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

where β = max_1≤i≤n(b_i - a_i) and α = max_1≤i≤n(d_i - c_i).

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

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

and similarly,

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

where β = max_1≤i≤n(b_i - a_i) and α = max_1≤i≤n(d_i - c_i).

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

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.

where β = max_1≤i≤n(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

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

and, hence, it follows that

and, similarly,

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

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

which is unless (for which the inequality (2.8) is obvious), is the same as

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 ≤ i ≤ n, 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.

Remark 2.7. The inequality (2.8) for u(x, y) reduce to u(x), with the right-hand sides multiplied by m^m and (b_i - a_i)^m replaced by (αβ)^m has been obtained by Pachpatte [18].

The authors declare that they have no competing interests.

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.

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.

1. Agarwal, RP, Pang, PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht (1995)

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 Cuza" din Iasi. 28, 123–126 (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). Publisher Full Text

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, 236–244 (1989). Publisher Full Text

8. Cheung, WS: Some new Opial-type inequalities. Mathematika. 37, 136–142 (1990). Publisher Full Text

9. Cheung, WS: Some generalized Opial-type inequalities. J Math Anal Appl. 162, 317–321 (1991). Publisher Full Text

10. Cheung, WS: Opial-type inequalities with m functions in n variables. Mathematika. 39, 319–326 (1992). Publisher Full Text

11. Crooke, PS: On two inequalities of the Sobolev type. Appl Anal. 3, 345–358 (1974). Publisher Full Text

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, 375–382 (1988). Publisher Full Text

14. Wang, XJ: Sharp constant in a Sobolev inequality. Nonlinear Anal. 20, 261–268 (1993). Publisher Full Text

15. Agarwal, RP, Sheng, Q: Sharp integral inequalities in n independent varibles. Nonlinear Anal Theory Methods Appl. 26(2), 179–210 (1996). Publisher Full Text

16. Pachpatte, BG: On Sobolev type integral inequalities. Proc R Soc Edinb. 103, 1–14 (1986). Publisher Full Text

17. Pachpatte, BG: On two inequalities of the Serrin type. J Math Anal Appl. 116, 193–199 (1986). Publisher Full Text

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