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Hindawi Publishing 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.-J Zhao1 and W.-S Cheung2 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, [email protected] Received 28 November 2007; Accepted 10 April 2008 Recommended by 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 results on Agarwal, Wirtinger, Poincaré, 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 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, Poincaré 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 inequalities involving 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 Rn the n-dimensional Euclidean space. Let E, E n be a bounded domain in Rn defined by E × E i1 ai , bi × ci , di , i 1, . . . , n. For xi , yi ∈ R, i 1, . . . , n, x, y x1 , . . . , xn , y1 , . . . , yn is a variable point in E × E and 2 Journal of Inequalities and Applications dxdy dx1 · · · dxn dy 1 · · · dyn . For any continuous real-valued function ux, y defined on E × E , we denote by E E ux, ydxdy the 2n-fold integral b1 ··· a1 bn d1 an c1 and for any x, y ∈ E × E , x1 ··· a1 ··· u x1 , . . . , xn , y1 , . . . , yn dx1 · · · dxn dy1 · · · dyn , 2.1 cn Ex E x xn y 1 an dn ··· c1 yn us, tdsdt is the 2n-fold integral u s1 , . . . , sn , t1 , . . . , tn dx1 · · · dsn dt1 · · · dtn . 2.2 cn We represent by FE × E the class of continuous functions ux, y : E × E → R for which D2n ux, y D1 · · · D2n ux, y, where D1 ∂ ∂ ∂ ∂ , . . . , Dn , Dn1 , . . . , D2n ∂x1 ∂xn ∂y1 ∂yn 2.3 exists and that for each i, 1 ≤ i ≤ n, ux, yxi ai 0, ux, yyi ci 0, ux, yxi bi 0, ux, yyi di 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 μ px, yux, y dx dy E μ/λ 2n D ux, yλ dx dy ≤ px, yqx, y, λ, μdx dy , E E 2.5 E E where qx, y, λ, μ n 1 2n1 λ−1/2 xi − ai bi − xi yi − ci di − yi μ/λ . 2.6 i1 Proof. For the set {1, . . . , n}, let π A ∪ B, π A ∪ B be partitions, where A j1 , . . . , jk , B jk1 , . . . , jn , A i1 , . . . , ik , and B ik1 , . . . , in are such that card A card A k and card B card B n − k, 0 ≤ k ≤ n. It is clear that there are 2n1 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 Eπ x Eπ 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 2n ux, y ≤ D us, tds dt. 2.8 Eπ x Eπ y In view of Hölder integral inequality, we have λ−1/λ ux, y ≤ xi − ai bi − x i yi − c i di − yi i∈A × Eπ y Eπ x i∈A i∈B i∈B 2.9 1/λ 2n D us, tλ ds dt . A multiplication of these 2n1 inequalities and an application of the Arithmetic-Geometric mean inequality give μ/λ n μ λ−1/2 ux, y ≤ xi − ai bi − xi yi − ci di − yi i1 × ≤ n 1 Eπ x Eπ y π∈Z, π ∈Z 2n1 × xi − ai bi − x i i1 1/2n1 μ/λ 2n D us, tλ ds dt π∈Z, π ∈Z Eπ x Eπ y yi − c i di − yi λ−1/2 μ/λ 2.10 μ/λ 2n D us, tλ ds dt μ/λ 2n λ D us, t ds dt qx, y, λ, μ . E E Now, multiplying both the sides of 2.10 by px, y and integrating the resulting inequality on E × E , we have μ/λ 2n μ D us, tλ ds dt px, yux, y dx dy ≤ px, yqx, y, λ, μdx dy , E E E E where qx, y, λ, μ n 1 2n1 E E λ−1/2 xi − ai bi − xi yi − ci di − yi 2.11 μ/λ Remark 2.2. Taking for px, y 1 in 2.5, 2.5 reduces to μ/λ 2n ux, yμ dx dy ≤ K D ux, yλ dx dy , 0 E E where K0 E E n μ/λ n 1μ−μ/λ μ μ μ μ 1 2 B 1 − , bi − ai di − ci ,1 − 2 2 2λ 2 2λ i1 and B is the Beta function. . 2.12 i1 2.13 2.14 4 Journal of Inequalities and Applications Taking for λ μ 2 in 2.13 reduces to 2 n 2n 2 2 ux, y2 dx dy ≤ π D ux, y dx dy , M 8 E E E E where M n bi − ai di − ci . 4 i1 2.15 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: n n ux2 dx ≤ π M2 D ux2 dx , 2.17 4 E E where n bi − ai M . 2 i1 2.18 Similarly ux4 dx ≤ E 3π 16 n M4 n D ux4 dx , 2.19 E 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/π2 and 16/3π4 . 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: μ/λ n μ λ D ux dx px ux dx ≤ pxqx, λ, μdx , 2.20 E E E where qx, λ, μ n λ−1/2 1 x − ai bi − xi n 2 i1 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, μk ≥ 0, λk ≥ 1, be given real numbers such that rk1 μk /λk 1, and uk x, y ∈ GE × E . Then the following inequality holds r μ px, y uk x, y k dx dy E E k1 2.22 r r 2n μk D uk x, yλk dx dy. ≤ px, y q x, y, λk , μk dx dy λ E E k1 k E E k1 C.-J Zhao and W.-S Cheung 5 Proof. Setting μ μk , λ λk and ux, y uk x, y, 1 ≤ k ≤ r in 2.10, multiplying the r inequalities, and applying the extended Arithmetic-Geometric means inequality, r k1 μ /λk ak k ≤ r μk k1 λk ak , ak ≥ 0, 2.23 to obtain μk /λk r r 2n uk x, yμk ≤ D uk s, tλk ds dt q x, y, λk , μk k1 E E k1 r r 2n μk D uk s, tλk ds dt. ≤ q x, y, λk , μk λ k1 k E E k1 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 r r 2n μk μk D uk x, yλk dx dy, uk x, y dx dy < K1 px, y px, ydx dy λ E E E k1 k E E k1 2.25 where K1 1 rk1 μk n 2n1 −1rk1 μk . bi − ai bi − ai 2.26 i1 This is just a general form of the following inequality which was established by Agarwal and Sheng 15: r r μ λ μk n D uk x k dx, px uk x k dx < K1 pxdx λ E E k1 k E k1 2.27 where K1 1 2n rk1 μk n bi − ai −1rk1 μk . 2.28 i1 Remark 2.6. For px, y 1, the inequality 2.22 becomes r r 2n μk uk x, yμk dx dy ≤ K D uk x, yλk dx dy, 2 λ E E k1 k1 k E E where K2 r r n n rk1 μk 1 2 1 k1 μk 1 k1 μk B , . bi − ai di − ci 2 2 2 i1 For ux, y ux, the inequality 2.29 has been obtained by Agarwal and Sheng 15. 2.29 2.30 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 ux, yλ dx dy ≤ K λ, μ grad ux, yλ dx dy, 2.31 3 μ E E where K3 λ, μ E E λ/n λ n 1 2 λ1 λ1 B , K , bi − ai di − ci 2n 2 2 μ i1 μ 1/μ n 2 grad ux, y ∂ ux, y , ∂x ∂y μ i1 i 2.32 i and where Kλ/μ 1 if λ ≥ μ, and Kλ/μ n1−λ/μ if 0 ≤ λ/μ ≤ 1. Proof. For each fixed i, 1 ≤ i ≤ n, in view of ux, yxi ai 0, ux, yyi ci 0, ux, yxi bi 0, we have ux, y xi y i ai ux, y ci ∂2 u x, y; si , ti dsi dti , ∂si ∂ti yi ∂2 u x, y; si , ti dsi dti , ∂si ∂ti bi di xi ux, yyi di 0, i 1, . . . , n, 2.33 2.34 where u x, y; si , ti u x1 , . . . , xi−1 , si , xi1 , . . . , xn , y1 , . . . , yi−1 , ti , yi1 , . . . , yn . Hence from Hölder inequality with indices λ and λ/1 − λ, it follows that xi y i 2 ∂ λ ux, yλ ≤ xi − ai yi − di λ−1 u x, y; s , t i i dsi dti , ai ci ∂si ∂ti bi di 2 ∂ λ ux, yλ ≤ bi − xi di − yi λ−1 u x, y; s , t i i dsi dti . xi yi ∂si ∂ti 2.35 2.36 Multiplying 2.36, and then applying the Arithmetic-Geometric means inequality, to obtain bi di 2 ∂ λ ux, yλ ≤ 1 xi − ai yi − ci bi − xi di − yi λ−1/2 × dsi dti , u x, y; s , t i i 2 ai ci ∂si ∂ti 2.37 and now integrating 2.37 on E × E , we arrive at bi di λ−1/2 1 ux, yλ dx dy ≤ dxi dyi xi − ai yi − ci bi − xi di − yi 2 E E ai c i λ ∂2 × ux, y dx dy. ∂xi ∂yi E E 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: α n n α ai ≤ Kα ai , ai > 0, 2.39 i1 i1 where Kα 1 if α ≥ 1, and Kα n1−α if 0 ≤ α ≤ 1, we get E E bi di 1/n λ−1/2 1 dxi dyi xi − ai yi − ci bi − xi di − yi ai c i 2 i1 λ 1/n n ∂2 × ux, y dx dy E E ∂xi ∂yi i1 1/n n bi di λ−1/2 1 1 ≤ dxi dyi xi − ai yi − ci bi − xi di − yi 2n i1 ai ci 2 λ n ∂2 × ux, y dx dy ∂xi ∂yi i1 E E n λ/n 1 2 λ1 λ1 B , bi − ai di − ci ≤ 2n 2 2 i1 grad ux, yλ dx dy, × λ n ux, yλ dx dy ≤ E E grad ux, yλ dx dy, ≤ K3 λ, μ μ E E 2.40 where λ/n λ n 1 2 λ1 λ1 K , bi − ai di − ci B , 2n 2 2 μ i1 μ 1/μ n 2 ∂ grad ux, y , ∂x ∂y ux, y μ K3 λ, μ i i1 2.41 i and where Kλ/μ 1 if λ ≥ μ, and Kλ/μ n1−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 uxλ dx ≤ K λ, 2 grad uxλ dx. 2.42 3 μ E E This is just a better inequality than the following inequality which was given by Pachpatte 17 λ uxλ dx ≤ 1 β grad uxλ dx. μ n 2 E E Because for λ ≥ 2, it is clear that K3 λ, 2 < 1/nβ/2λ , where β max1≤i≤n bi − ai . 2.43 8 Journal of Inequalities and Applications On the other hand, taking for μ 2, λ 2 or μ 2, λ 4 in 2.31 and let ux, y reduce to ux with suitable modifications, it follows the following Poincaré-type inequalities: ux2 dx ≤ π β2 grad ux2 dx, 2 16n E E 2.44 ux4 dx ≤ 3π β4 grad ux4 dx. 2 256n E E 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 Hölder 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 λ/μ ux, yλ dx dy ≤ K λ, μ grad ux, yμ dx dy , 4 μ E E 2.45 where K4 λ, μ K3 λ, μ n bi − ai di − ci μ−λ/μ 2.46 . i1 Remark 2.10. Taking ux, y ux and with suitable modifications, the inequality 2.45 reduces to the following result which was given by Agarwal and Sheng 15: uxλ dx ≤ K6 λ, μ E grad uxμ dx μ E λ/μ , 2.47 where K6 λ, μ K5 λ, μ n μ−λ/μ , bi − ai i1 λ/n 1λ 1λ λ n 1 B , K K5 λ, μ , bi − ai 2n 2 2 μ i1 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, ydx dy 2 ≤ K4 1, 2 E E grad ux, y2 dx dy. 2 2.49 This is just a general form of the following inequality which was given by Agarwal and Sheng 15. 2 uxdx ≤ K6 1, 2 2 grad ux2 dx dy. 2.50 2 E E 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 uk x, y ∈ GE × E , μk ≥ 1, 1 ≤ k ≤ r. Then the following inequality holds 1/r n r μk uk x, y grad uk x, yμk dx dy, dx dy ≤ K5 μk E E E E k1 i1 2.51 where K5 r r n rk1 μk /nr 1 2 1 1/r k1 μk 1 1/r k1 μk . bi − ai di − ci B , 2nr 2 2 i1 2.52 Remark 2.12. Taking ux, y ux and with suitable modifications, the inequality 2.51 reduces to the following result: 1/r n r uk xμk grad uxμk dx, dx ≤ K9 μk E i1 2.53 E k1 where n r μ /nr 1 1/r rk1 μk 1 1/r rk1 μk 1 K9 bi − ai k1 k . B , 2nr 2 2 i1 2.54 In 19, Pachpatte proved the inequality 2.53 for μk ≥ 2, 1 ≤ k ≤ r with K9 replaced by r r μk /r k1 1/nrβ/2 , where β is as in Remark 2.8. It is clear that K9 < 1/nrβ/2 k1 μk /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. 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