Sharp integral inequalities involving high-order partial derivatives. Journal Of Inequalities And Applications, 2008, v. 2008, article no.

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1 Ttle Sharp ntegral nequaltes nvolvng hgh-order partal dervatves Authors Zhao, CJ; Cheung, WS Ctaton Journal Of Inequaltes And Applcatons, 008, v. 008, artcle no Issued Date 008 URL Rghts Ths work s lcensed under a Creatve Commons Attrbuton- NonCommercal-NoDervatves 4.0 Internatonal Lcense.

2 Hndaw Publshng Corporaton Journal of Inequaltes and Applcatons Volume 008, Artcle ID 5747, 0 pages do:0.55/008/5747 Research Artcle Sharp Integral Inequaltes Involvng Hgh-Order Partal Dervatves C.-J Zhao and W.-S Cheung Department of Informaton and Mathematcs Scences, College of Scence, Chna Jlang Unversty, Hangzhou 3008, Chna Department of Mathematcs, The Unversty of Hong Kong, Pokfulam Road, Hong Kong Correspondence should be addressed to C.-J Zhao, chjzhao@63.com Receved 8 November 007; Accepted 0 Aprl 008 Recommended by Peter Pang The man purpose of the present paper s to establsh some new sharp ntegral nequaltes nvolvng hgher-order partal dervatves. Our results n specal cases yeld some of the recent results on Agarwal, Wrtnger, Poncaré, Pachpatte, Smth, and Stredulnsky s nequaltes and provde some new estmates on such types of nequaltes. Copyrght q 008 C.-J Zhao and W.-S Cheung. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.. Introducton Inequaltes nvolvng functons of n ndependent varables, ther partal dervatves, ntegrals play a fundamental role n establshng the exstence and unqueness of ntal and boundary value problems for ordnary and partal dfferental equatons as well as dfference equatons 0. specally, n vew of wder applcatons, nequaltes due to Agarwal, Opal, Pachpatte, Wrtnger, Poncaré and et al. have been generalzed and sharpened from the very day of ther dscover. As a matter of fact, these now have become research topc n ther own rght 4. In the present paper, we wll use the same method of Agarwal and Sheng 5, establsh some new estmates on these types of nequaltes nvolvng hgher-order partal dervatves. We further generalze these nequaltes whch lead to result sharper than those currently avalable. An mportant characterstc of our results s that the constant n the nequaltes are explct.. Man results Let R be the set of real numbers and R n the n-dmensonal ucldean space. Let, be a bounded doman n R n defned by n a,b c,d,,...,n.for x,y R,,...,n, x, y x,...,x n,y,...,y n s a varable pont n and

3 Journal of Inequaltes and Applcatons dxdy dx dx n dy dy n. For any contnuous real-valued functon u x, y defned on,wedenoteby u x, y dxdy the n-fold ntegral b bn d dn a and for any x, y, x a n u x,...,x n,y,...,y n dx dx n dy dy n,. c c n x u s, t dsdt s the n-fold ntegral x xn y yn a a n u s,...,s n,t,...,t n dx ds n dt dt n.. c c n We represent by F the class of contnuous functons u x, y : R for whch D n u x, y D D n u x, y, D x,...,d n x n,d n y,...,d n y n.3 exstsandthatforeach, n, u x, y x a 0, u x, y y c 0, u x, y x b 0, u x, y y d 0,,...,n.4 the class F s denoted as G. Theorem.. Let μ 0, be gven real numbers, and let p x, y 0, x, y be a contnuous functon. Further, let u x, y G. Then, the followng nequalty holds p x, y u x, y μ dx dy p x, y q x, y,, μ dx dy D n u x, y μ/ dx dy,.5 q x, y,, μ n μ/ [ ] / x a b x y c d y..6 Proof. For the set {,...,n},letπ A B, π A B be parttons, A j,...,j k,b j,...,j n,a,..., k, and B,..., n are such that card A card A k and card B card B n k, 0 k n. It s clear that there are n such parttons. The set of all such parttons we wll denote as Z and Z, respectvely. For fxed partton π, π and x, y, we defne π x π y u s, t ds dt A x B x A y B y u s, t ds dt,.7

4 C.-J Zhao and W.-S Cheung 3 A x, A y denote the k-fold ntegral, B x, represent the n k -fold ntegral. Thus B y from the assumptons t s clear that for each π Z, π Z u x, y D n u s, t ds dt..8 π x In vew of Hölder ntegral nequalty, we have u x, y A π y x a b x B π x π y y c A D n u s, t / ds dt. / d y B A multplcaton of these n nequaltes and an applcaton of the Arthmetc-Geometrc mean nequalty gve u x, y μ n [ x a b x μ/ ] / y c d y n π Z, π Z π x π y D n u s, t / n μ/ ds dt μ/ [ ] / x a b x y c d y π Z, π Z π x π y D n u s, t μ/ ds dt q x, y,, μ D n u s, t μ/ ds dt. Now, multplyng both the sdes of.0 by p x, y and ntegratng the resultng nequalty on,wehave p x, y u x, y μ dx dy p x, y q x, y,, μ dx dy D n u s, t μ/ ds dt,. μ/ [ ] / q x, y,, μ x a b x y c d y.. n.9.0 Remark.. Takng for p x, y n.5,.5 reduces to u x, y μ dx dy K D n 0 u x, y μ/ dx dy,.3 μ/ K 0 B μ μ, μ μ n n [ ] μ μ/, b a d c.4 and B s the Beta functon.

5 4 Journal of Inequaltes and Applcatons Takng for μ n.3 reduces to u x, y π n dx dy M D n u x, y dx dy,.5 8 b M a d c..6 4 Let u x, y reduce to u x n.5 and wth sutable modfcatons, then.5 becomes the followng two Wrtng type nequaltes: u x π n dx M D n u x dx,.7 4 Smlarly u x 4 dx M b a..8 3π n M 4 D n u x 4 dx,.9 6 M s as n.7. For n, the nequaltes.7 and.9 have been obtaned by Smth and Stredulnsky 6, however, wth the rght-hand sdes, respectvely, multples 4/π and 6/3π 4. Hence, t s clear that nequaltes.7 and.9 are more strengthed. Remark.3. Let u x, y reduce to u x n.5 and wth sutable modfcatons, then.5 becomes the followng result: p x u x μ dx q x,, μ n p x q x,, μ dx D n u x μ/ dx,.0 μ/ [ ] / x a b x.. Ths s just a new result whch was gven by Agarwal and Sheng 5. Theorem.4. Let p x, y 0, x, y be a contnuous functon. Further, let for k,...,r, 0, k, be gven real numbers such that r / k,andu k x, y G. Then the followng nequalty holds p x, y uk x, y dx dy p x, y q r x, y, k, dx dy D n u k x, y. k dx dy. k

6 C.-J Zhao and W.-S Cheung 5 Proof. Settng μ, k and u x, y u k x, y, k r n.0, multplyng the r nequaltes, and applyng the extended Arthmetc-Geometrc means nequalty, a / k k r k a k, a k 0,.3 to obtan uk x, y μk / k q x, y, k,μ k D n u k s, t k ds dt q r x, y, k, D n u k s, t k ds dt. k.4 Now multplyng both sdes of.4 by p x, y and then ntegratng over, we obtan.. Corollary.5. Let the condtons of Theorem.4 be satsfed. Then the followng nequalty holds p x, y u k x, y r dx dy < K p x, y dx dy D n u k x, y k dx dy, k.5 r K n [ ] r b a b a..6 Ths s just a general form of the followng nequalty whch was establshed by Agarwal and Sheng 5 : p x u k x r dx < K p x dx K r n k D n u k x k dx,.7 r b a..8 Remark.6. For p x, y, the nequalty. becomes u k x, y r dx dy K k D n u k x, y k dx dy,.9 K B r, r n n [ ] r b a d c..30 For u x, y u x, the nequalty.9 has been obtaned by Agarwal and Sheng 5.

7 6 Journal of Inequaltes and Applcatons Theorem.7. Let and u x, y be as n Theorem., μ be a gven real number. Then the followng nequalty holds u x, y dx dy K 3, μ grad u x, y dx dy, μ.3 K 3, μ n B, n [ ] /n, K b a d c μ grad u x, y n μ μ /μ.3 u x, y x y, and K /μ f μ,andk /μ n /μ f 0 /μ. Proof. For each fxed, n, n vew of u x, y x a 0, u x, y y c 0, u x, y x b 0, u x, y y d 0,,...,n,.33 we have u x, y u x, y x y a b d x s t u x, y; s,t ds dt, c u x, y; s,t ds dt, y s t u x, y; s,t u x,...,x,s,x,...,x n,y,...,y,t,y,...,y n Hence from Hölder nequalty wth ndces and /, t follows that u x, y [ ] x y x a y d u x, y; s,t s t ds dt, u x, y [ ] b b x d y a c d x y u x, y; s,t s t ds dt..36 Multplyng.36, and then applyng the Arthmetc-Geometrc means nequalty, to obtan u x, y [ ] b d / x a y c b x d y u x, y; s,t a c s t ds dt,.37 and now ntegratng.37 on, we arrve at b d u x, y [ ] /dx dx dy x a y c b x d y dy a c u x, y x y dx dy..38

8 C.-J Zhao and W.-S Cheung 7 Next, multplyng the nequalty.38 for n, and usng the Arthmetc-Geometrc means nequalty, and n vew of the followng nequalty: n n α a α K α a, a > 0,.39 K α fα, and K α n α f 0 α, we get b d u x, y [ ] /n /dx dx dy x a y c b x d y dy a c /n u x, y x y dx dy n b d a c [ x a y c b x d y ] /dx dy /n n u x, y x y dx dy n B, n [ ] /n b a d c K 3, μ grad u x, y dx dy, grad u x, y dx dy, μ.40 K 3, μ n B, n [ ] /n, K b a d c μ grad u x, y n μ μ /μ u x, y x y,.4 and K /μ f μ, andk /μ n /μ f 0 /μ. Remark.8. Let u x, y reduce to u x n.3 and wth sutable modfcatons, and let, μ, then.3 becomes u x dx K 3, grad u x μ dx..4 Ths s just a better nequalty than the followng nequalty whch was gven by Pachpatte 7 u x dx β grad u x n μ dx..43 Because for, t s clear that K 3, < /n β/,β max n b a.

9 8 Journal of Inequaltes and Applcatons On the other hand, takng for μ, orμ, 4n.3 and let u x, y reduce to u x wth sutable modfcatons, t follows the followng Poncaré-type nequaltes: u x dx π 6n β grad u x dx, u x 4 dx 3π 56n β4.44 grad u x 4 dx. The nequaltes.44 have been dscussed n 8 wth the rght-hand sdes, respectvely, multpled by 4/π and 6/3π. Hence nequaltes.44 are more strong results on these types of nequaltes. If μ, n the rght sdes of.3 we can apply Hölder nequalty wth ndces μ/ and μ/ μ, to obtan the followng corollary. Corollary.9. Let the condtons of Theorem.7 be satsfed and μ. Then u x, y dx dy K 4, μ grad u x, y /μ μ dx dy,.45 μ K 4, μ K 3, μ [ ] μ /μ. b a d c.46 Remark.0. Takng u x, y u x and wth sutable modfcatons, the nequalty.45 reduces to the followng result whch was gven by Agarwal and Sheng 5 : u x dx K 6, μ K 6, μ K 5, μ grad u x μ μ dx /μ, μ /μ, b a.47 K 5, μ n B, n.48 /n, K b a μ and K /μ s as n Theorem.7. Takng, μ the nequalty.45,.45 reduces to u x, y dx dy K 4, grad u x, y dx dy..49 Ths s just a general form of the followng nequalty whch was gven by Agarwal and Sheng 5. u x dx [ K 6, ] grad u x Smlar to the proof of Theorem.7, we have the followng theorem. dx dy..50

10 C.-J Zhao and W.-S Cheung 9 Theorem.. For u k x, y G,, k r. Then the followng nequalty holds n uk x, y /r k μ dx dy K 5 r grad uk x, y dx dy,.5 K 5 /r r nr B, /r r n [ ] r b a d c /nr..5 Remark.. Takng u x, y u x and wth sutable modfcatons, the nequalty.5 reduces to the followng result: n u k x /r dx K 9 r grad u x dx,.53 K 9 /r r nr B, /r r n r b a /nr..54 In 9, Pachpatte proved the nequalty.53 for, k r wth K 9 replaced by /nr β/ r μk/r, β s as n Remark.8. It s clear that K 9 < /nr β/ r μk/r, and hence.53 s a better nequalty than a result of Pachpatte. Smlarly, all other results n 5 also can be generalzed by the same way. Here, we omt the detals. Acknowledgments Research s supported by Zhejang Provncal Natural Scence Foundaton of Chna Y605065, Foundaton of the ducaton Department of Zhejang Provnce of Chna Research s partally supported by the Research Grants Councl of the Hong Kong SAR, Chna Project No.:HKU706/07P. References R. P. Agarwal and P. Y. H. Pang, Opal Inequaltes wth Applcatons n Dfferental and Dfference quatons, vol. 30 of Mathematcs and Its Applcatons, Kluwer Academc Publshers, Dordrecht, The Netherlands, 995. R. P. Agarwal and V. Lakshmkantham, Unqueness and Nonunqueness Crtera for Ordnary Dfferental quatons, vol.6ofseres n Real Analyss, World Scentfc, Sngapore, R. P. Agarwal and. Thandapan, On some new ntegro-dfferental nequaltes, Analele Ştnţfce ale Unverstăţ Al. I. Cuza dn Iaş, vol. 8, no., pp. 3 6, D. Baĭnov and P. Smeonov, Integral Inequaltes and Applcatons, vol. 57 of Mathematcs and Its Applcatons, Kluwer Academc Publshers, Dordrecht, The Netherlands, J. D. L, Opal-type ntegral nequaltes nvolvng several hgher order dervatves, Journal of Mathematcal Analyss and Applcatons, vol. 67, no., pp. 98 0, D. S. Mtrnovč, J.. Pečarć, and A. M. Fnk, Inequaltes Involvng Functons and Ther Integrals ang Dervatves, Kluwer Academc Publshers, Dordrecht, The Netherlands, 99.

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