Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 571417, 10 pages doi:10.1155/2008/571417 ResearchArticleSharpIntegralInequalitiesInvolvingHigh-OrderPartial Derivatives C J Zhao 1 and W S Cheung 2 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 sharpintegralinequalitiesinvolving higher-order partial derivatives. Our results in special cases yield some of the recent results on Agarwal, Wirtinger, Poincar ´ e, 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 Inequalitiesinvolving 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, Poincar ´ e 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 inequalitiesinvolving 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 R n the n-dimensional Euclidean space. Let E, E be a bounded domain in R n defined by E × E n i1 a i ,b i × c i ,d i ,i 1, ,n.For x i ,y i ∈ R, i 1, ,n, x, yx 1 , ,x n ,y 1 , ,y n is a variable point in E × E and 2 Journal of Inequalities and Applications dxdy dx 1 ···dx n dy 1 ···dy n . For any continuous real-valued function ux, y defined on E × E ,wedenoteby E E ux, ydxdy the 2n-fold integral b 1 a 1 ··· b n a n d 1 c 1 ··· d n c n u x 1 , ,x n ,y 1 , ,y n dx 1 ···dx n dy 1 ···dy n , 2.1 and for any x, y ∈ E × E , Ex E x us, tdsdt is the 2n-fold integral x 1 a 1 ··· x n a n y 1 c 1 ··· y n c n u s 1 , ,s n ,t 1 , ,t n dx 1 ···ds n dt 1 ···dt n . 2.2 We represent by FE × E the class of continuous functions ux, y : E × E → R for which D 2n ux, yD 1 ···D 2n ux, y,where D 1 ∂ ∂x 1 , ,D n ∂ ∂x n ,D n1 ∂ ∂y 1 , ,D 2n ∂ ∂y n 2.3 existsandthatforeachi,1≤ i ≤ n, ux, y x i a i 0,ux, y y i c i 0,ux, y x i b i 0,ux, y y i d i 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 E px, y ux, y μ dx dy ≤ E E px, yqx, y, λ, μdx dy E E D 2n ux, y λ dx dy μ/λ , 2.5 where qx, y, λ, μ 1 2 n1 n i1 x i − a i b i − x i y i − c i d i − y i λ−1/2 μ/λ . 2.6 Proof. For the set {1, ,n},letπ A ∪ B, π A ∪ B be partitions, where A j 1 , ,j k ,B j k1 , ,j n ,A i 1 , ,i k , and B i k1 , ,i n are such that card A card A k and card B card B n − k, 0 ≤ k ≤ n. It is clear that there are 2 n1 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 E π x E π 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 ux, y ≤ E π x E π y D 2n us, t ds dt. 2.8 In view of H ¨ older integral inequality, we have ux, y ≤ i∈A x i − a i i∈B b i − x i i∈A y i − c i i∈B d i − y i λ−1/λ × E π x E π y D 2n us, t λ ds dt 1/λ . 2.9 A multiplication of these 2 n1 inequalities and an application of the Arithmetic-Geometric mean inequality give ux, y μ ≤ n i1 x i − a i b i − x i y i − c i d i − y i λ−1/2 μ/λ × π∈Z, π ∈Z E π x E π y D 2n us, t λ ds dt 1/2 n1 μ/λ ≤ 1 2 n1 n i1 x i − a i b i − x i y i − c i d i − y i λ−1/2 μ/λ × π∈Z, π ∈Z E π x E π y D 2n us, t λ ds dt μ/λ qx, y, λ, μ E E D 2n us, t λ ds dt μ/λ . 2.10 Now, multiplying both the sides of 2.10 by px, y and integrating the resulting inequality on E × E ,wehave E E px, y ux, y μ dx dy ≤ E E px, yqx, y, λ, μdx dy E E D 2n us, t λ ds dt μ/λ , 2.11 where qx, y, λ, μ 1 2 n1 n i1 x i − a i b i − x i y i − c i d i − y i λ−1/2 μ/λ . 2.12 Remark 2.2. Taking for px, y1in2.5, 2.5 reduces to E E ux, y μ dx dy ≤ K 0 E E D 2n ux, y λ dx dy μ/λ , 2.13 where K 0 1 2 μ/λ B 2 1 μ 2 − μ 2λ , 1 μ 2 − μ 2λ n n i1 b i − a i d i − c i 1μ−μ/λ , 2.14 and B is the Beta function. 4 Journal of Inequalities and Applications Taking for λ μ 2in2.13 reduces to E E ux, y 2 dx dy ≤ π 2 8 n M 2 E E D 2n ux, y 2 dx dy , 2.15 where M n i1 b i − a i d i − c i 4 . 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: E ux 2 dx ≤ π 4 n M 2 E D n ux 2 dx , 2.17 where M n i1 b i − a i 2 . 2.18 Similarly E ux 4 dx ≤ 3π 16 n M 4 E D n ux 4 dx , 2.19 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/π 2 and 16/3π 4 . 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: E px ux μ dx ≤ E pxqx, λ, μdx E D n ux λ dx μ/λ , 2.20 where qx, λ, μ 1 2 n n i1 x − a i b i − x i λ−1/2 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, μ k ≥ 0,λ k ≥ 1, be given real numbers such that r k1 μ k /λ k 1,andu k x, y ∈ GE × E . Then the following inequality holds E E px, y r k1 u k x, y μ k dx dy ≤ E E px, y r k1 q x, y, λ k ,μ k dx dy r k1 μ k λ k E E D 2n u k x, y λ k dx dy. 2.22 C J Zhao and W S Cheung 5 Proof. Setting μ μ k ,λ λ k and ux, yu k x, y,1≤ k ≤ r in 2.10, multiplying the r inequalities, and applying the extended Arithmetic-Geometric means inequality, r k1 a μ k /λ k k ≤ r k1 μ k λ k a k ,a k ≥ 0, 2.23 to obtain r k1 u k x, y μ k ≤ r k1 q x, y, λ k ,μ k E E D 2n u k s, t λ k ds dt μ k /λ k ≤ r k1 q x, y, λ k ,μ k r k1 μ k λ k E E D 2n u k s, t λ k ds dt. 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 E px, y r k1 u k x, y μ k dx dy < K 1 E E px, ydx dy r k1 μ k λ k E E D 2n u k x, y λ k dx dy, 2.25 where K 1 1 2 n1 r k1 μ k n i1 b i − a i b i − a i −1 r k1 μ k . 2.26 This is just a general form of the following inequality which was established by Agarwal and Sheng 15: E px r k1 u k x μ k dx < K 1 E pxdx r k1 μ k λ k E D n u k x λ k dx, 2.27 where K 1 1 2 n r k1 μ k n i1 b i − a i −1 r k1 μ k . 2.28 Remark 2.6. For px, y1, the inequality 2.22 becomes E E r k1 u k x, y μ k dx dy ≤ K 2 r k1 μ k λ k E E D 2n u k x, y λ k dx dy, 2.29 where K 2 1 2 B 2 1 r k1 μ k 2 , 1 r k1 μ k 2 n n i1 b i − a i d i − c i r k1 μ k . 2.30 For ux, yux, the inequality 2.29 has been obtained by Agarwal and Sheng 15. 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 E E ux, y λ dx dy ≤ K 3 λ, μ E E grad ux, y λ μ dx dy, 2.31 where K 3 λ, μ 1 2n B 2 λ 1 2 , λ 1 2 K λ μ n i1 b i − a i d i − c i λ/n , grad ux, y μ n i1 ∂ 2 ∂x i ∂y i ux, y μ 1/μ , 2.32 and where Kλ/μ1 if λ ≥ μ,andKλ/μn 1−λ/μ if 0 ≤ λ/μ ≤ 1. Proof. For each fixed i,1≤ i ≤ n, in view of ux, y x i a i 0,ux, y y i c i 0,ux, y x i b i 0,ux, y y i d i 0, i 1, ,n, 2.33 we have ux, y x i a i y i c i ∂ 2 ∂s i ∂t i u x, y; s i ,t i ds i dt i , ux, y b i x i d i y i ∂ 2 ∂s i ∂t i u x, y; s i ,t i ds i dt i , 2.34 where u x, y; s i ,t i u x 1 , ,x i−1 ,s i ,x i1 , ,x n ,y 1 , ,y i−1 ,t i ,y i1 , ,y n . 2.35 Hence from H ¨ older inequality with indices λ and λ/1 − λ, it follows that ux, y λ ≤ x i − a i y i − d i λ−1 x i a i y i c i ∂ 2 ∂s i ∂t i u x, y; s i ,t i λ ds i dt i , ux, y λ ≤ b i − x i d i − y i λ−1 b i x i d i y i ∂ 2 ∂s i ∂t i u x, y; s i ,t i λ ds i dt i . 2.36 Multiplying 2.36, and then applying the Arithmetic-Geometric means inequality, to obtain ux, y λ ≤ 1 2 x i − a i y i − c i b i − x i d i − y i λ−1/2 × b i a i d i c i ∂ 2 ∂s i ∂t i u x, y; s i ,t i λ ds i dt i , 2.37 and now integrating 2.37 on E × E , we arrive at E E ux, y λ dx dy ≤ b i a i d i c i 1 2 x i − a i y i − c i b i − x i d i − y i λ−1/2 dx i dy i × E E ∂ 2 ∂x i ∂y i ux, y λ dx dy. 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: n i1 a α i ≤ Kα n i1 a i α ,a i > 0, 2.39 where Kα1ifα ≥ 1, and Kαn 1−α if 0 ≤ α ≤ 1, we get E E ux, y λ dx dy ≤ n i1 b i a i d i c i 1 2 x i − a i y i − c i b i − x i d i − y i λ−1/2 dx i dy i 1/n × n i1 E E ∂ 2 ∂x i ∂y i ux, y λ dx dy 1/n ≤ 1 2n n i1 b i a i d i c i 1 2 x i − a i y i − c i b i − x i d i − y i λ−1/2 dx i dy i 1/n × n i1 E E ∂ 2 ∂x i ∂y i ux, y λ dx dy ≤ 1 2n B 2 λ 1 2 , λ 1 2 n i1 b i − a i d i − c i λ/n × E E grad ux, y λ λ dx dy, ≤ K 3 λ, μ E E grad ux, y λ μ dx dy, 2.40 where K 3 λ, μ 1 2n B 2 λ 1 2 , λ 1 2 K λ μ n i1 b i − a i d i − c i λ/n , grad ux, y μ n i1 ∂ 2 ∂x i ∂y i ux, y μ 1/μ , 2.41 and where Kλ/μ1ifλ ≥ μ,andKλ/μn 1−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 E ux λ dx ≤ K 3 λ, 2 E grad ux λ μ dx. 2.42 This is just a better inequality than the following inequality which was given by Pachpatte 17 E ux λ dx ≤ 1 n β 2 λ E grad ux λ μ dx. 2.43 Because for λ ≥ 2, it is clear that K 3 λ, 2 < 1/nβ/2 λ ,whereβ max 1≤i≤n b i − a i . 8 Journal of Inequalities and Applications On the other hand, taking for μ 2,λ 2orμ 2,λ 4in2.31 and let ux, y reduce to ux with suitable modifications, it follows the following Poincar ´ e-type inequalities: E ux 2 dx ≤ π 16n β 2 E grad ux 2 2 dx, E ux 4 dx ≤ 3π 256n β 4 E grad ux 4 2 dx. 2.44 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 H ¨ older 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 ux, y λ dx dy ≤ K 4 λ, μ E E grad ux, y μ μ dx dy λ/μ , 2.45 where K 4 λ, μK 3 λ, μ n i1 b i − a i d i − c i μ−λ/μ . 2.46 Remark 2.10. Taking ux, yux and with suitable modifications, the inequality 2.45 reduces to the following result which was given by Agarwal and Sheng 15: E ux λ dx ≤ K 6 λ, μ E grad ux μ μ dx λ/μ , 2.47 where K 6 λ, μK 5 λ, μ n i1 b i − a i μ−λ/μ , K 5 λ, μ 1 2n B 1 λ 2 , 1 λ 2 K λ μ n i1 b i − a i λ/n , 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, y dx dy 2 ≤ K 4 1, 2 E E grad ux, y 2 2 dx dy. 2.49 This is just a general form of the following inequality which was given by Agarwal and Sheng 15. E ux dx 2 ≤ K 6 1, 2 2 E grad ux 2 2 dx dy. 2.50 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 u k x, y ∈ GE × E , μ k ≥ 1, 1 ≤ k ≤ r. Then the following inequality holds E E n i1 u k x, y μ k 1/r dx dy ≤ K 5 E E r k1 grad u k x, y μ k μ k dx dy, 2.51 where K 5 1 2nr B 2 1 1/r r k1 μ k 2 , 1 1/r r k1 μ k 2 n i1 b i − a i d i − c i r k1 μ k /nr . 2.52 Remark 2.12. Taking ux, yux and with suitable modifications, the inequality 2.51 reduces to the following result: E n i1 u k x μ k 1/r dx ≤ K 9 E r k1 grad ux μ k μ k dx, 2.53 where K 9 1 2nr B 1 1/r r k1 μ k 2 , 1 1/r r k1 μ k 2 n i1 b i − a i r k1 μ k /nr . 2.54 In 19, Pachpatte proved the inequality 2.53 for μ k ≥ 2, 1 ≤ k ≤ r with K 9 replaced by 1/nrβ/2 r k1 μ k /r , where β is as in Remark 2.8. It is clear that K 9 < 1/nrβ/2 r k1 μ k /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. References 1 R. P. Agarwal and P. Y. H. Pang, Opial Inequalities with Applications in Differential and Difference Equations, vol. 320 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. 2 R. P. Agarwal and V. Lakshmikantham, Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations,vol.6ofSeries in Real Analysis, World Scientific, Singapore, 1993. 3 R. P. Agarwal and E. Thandapani, “On some new integro-differential inequalities,” Analele S¸tiint¸ifice ale Universit ˘ at¸ii “Al. I. 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D. Smith and E. W. Stredulinsky, “Nonlinear elliptic systems with certain unbounded coefficients,” Communications on Pure and Applied Mathematics, vol. 37, no. 4, pp. 495–510, 1984. 17 B. G. Pachpatte, “A note on Poincar ´ e and Sobolev type integral inequalities,” Tamkang Journal of Mathematics, vol. 18, no. 1, pp. 1–7, 1987. 18 B. G. Pachpatte, “On multidimensional integralinequalitiesinvolving three functions,” Soochow Journal of Mathematics, vol. 12, pp. 67–78, 1986. 19 B. G. Pachpatte, “On Sobolev type integral inequalities,” Proceedings of the Royal Society of Edinburgh A, vol. 103, no. 1-2, pp. 1–14, 1986. . 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. 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. 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