Ttle O verse Hlbert-tye equaltes Authors Chagja, Z; Cheug, WS Ctato Joural Of Iequaltes Ad Alcatos, 2008, v. 2008,. 693248 Issued Date 2008 URL htt://hdl.hadle.et/0722/56208 Rghts Ths work s lcesed uder a Creatve Coos Attrbuto- NoCoercal-NoDervatves 4.0 Iteratoal Lcese.
Hdaw Publshg Cororato Joural of Iequaltes ad Alcatos Volue 2008, Artcle ID 693248, 6 ages do:0.55/2008/693248 Research Artcle O Iverse Hlbert-Tye Iequaltes Zhao Chagja ad Wg-Su Cheug 2 Deartet of Iforato ad Matheatcs Sceces, College of Scece, Cha Jlag Uversty, Hagzhou 3008, Cha 2 Deartet of Matheatcs, The Uversty of Hog Kog, Pokfula Road, Hog Kog Corresodece should be addressed to Zhao Chagja, chjzhao@63.co Receved 4 Noveber 2007; Revsed Deceber 2007; Acceted 4 Deceber 2007 Recoeded by Mart J. Boher Ths aer deals wth ew verse-tye Hlbert equaltes. Our results secal cases yeld soe of the recet results ad rovde soe ew estates o such tyes of equaltes. Coyrght q 2008 Z. Chagja ad W.-S. Cheug. Ths s a oe access artcle dstrbuted uder the Creatve Coos Attrbuto Lcese, whch erts urestrcted use, dstrbuto, ad reroducto ay edu, rovded the orgal work s roerly cted.. Itroducto Cosderable atteto has bee gve to Hlbert equaltes ad Hlbert-tye equaltes ad ther varous geeralzatos by several authors cludg Hadley et al., Mzhead Bcheg 2, Mzhe 3, Hu 4, Jchag 5, Bcheg 6, adzhao 7, 8. I 998, Pachatte 9 gave soe ew tegral equaltes slar to Hlbert equalty see 0, age 226. I 2000, Zhao ad Debath establshed soe verse-tye equaltes of the above tegral equaltes. Ths aer deals wth soe ew verse-tye Hlbert equaltes whch rovde soe ew estates o such tyes of equaltes. 2. Ma results Theore 2.. Let 0 <,..., ad r 0. Let{a, } be ostve sequeces of real ubers defed for, 2,...,k,wherek,..., are atural ubers, defe A, s a,s, ad defe A,0 0.Thefor q, <0 or 0 <<, oe has k A, / r k / k k. a, A q, 2.
2 Joural of Iequaltes ad Alcatos Proof. By usg the followg equalty see 0, age 39 : h a h, a, b, a h, b h, h b h, a, b,, where a, > 0, b, > 0, ad 0 h, 2,...,, weobtathat A, A, A, A, A, a, A thus k A, A 0, A k,k 0 A,, a, A, k a, A,, 2.2 2.3, a,s A,s. 2.4 s Fro equalty 2.4 ad vew of the followg ea equalty ad verse Hölder s equalty 0, age 24,wehave /r / r, 2.5 A, /q. / a,s A q r,s 2.6 Takg the su of both sdes of 2.6 over fro to k, 2,..., frst ad the usg aga verse Hölder s equalty, we obta that k A, k /q / a,s A r s s,s q Ths coletes the roof. k / k / k / k s k a,s A q,s k s a,s A q,s s k k. a, A q, 2.7 Reark 2.2. Takg 2, q 2, r to 2., 2. becoes k 2 2 A, A 2 2, 2 2 3/2 k 3 8 2 k k 2 k /2 a, A 2, k2 k2 2 /2. a 2,2 A 2 2 2, 2 2 2.8
Z. Chagja ad W.-S. Cheug 3 Ths s just a verse for of the followg equalty whch was rove by Pachatte 9 : k r A B q k 2 q kr /2 k /2 a A 2 r r /2 b B q 2. 2.9 Theore 2.3. Let {a, }, A,, k,,adq be as defed Theore 2.. Let{, } be ostve sequeces for, 2,...,k, 2,...,. Set P, s,s, 2,...,. Letφ, 2,..., be real-valued oegatve, cocave, ad suerultlcatve fuctos defed o R 0,. The, k φ A, / M k k,k 2,...,k k q a,, φ, r, 2.0 where M k / φ P, k,k 2,...,k. 2. Proof. Fro the hyotheses ad by Jese s equalty, the eas equalty, ad verse Hölder s equalty, we obta that P, s φ A, φ,s a,s /,s s s φ,s a,s /,s P, φ,s s,s φ P, φ P, q P, r s,s φ a,s,s φ P, P, s P, / P,,s φ a,s,s s q.,s φ a,s,s 2.2 Dvdg both sdes of 2.2 by / r / r ad the takg the su over, 2,..., fro to k ad vew of verse Hölder s equalty, wehave k φ A, k φ P, q a,s /,s φ r P,,s k P, s / k φ P, M k,k 2,...,k k s,s φ a,s,s q a,s,s φ,s s q The roof s colete. M k,k 2,...,k k k q a,, φ., 2.3
4 Joural of Iequaltes ad Alcatos Reark 2.4. Takg 2, q 2, r to 2.0, 2.0 becoes k 2 2 φ A, φ2 A2,2 3 M k,k 2 k 2 /2 a,, φ 2, k2 k2 2 2 /2 a2,2 2,2 φ 2, 2 2,2 2.4 where M k 2/3 3/2 φ P, k2 2/3 3/2 φ2 P2,2 k,k 2 8. 2.5 P, 2 P 2,2 Ths s just a verse of the followg equalty whch was rove by Pachatte 9 : k r φ A ψ B k 2 /2 a M k, r k φ r 2 /2 b r q ψ, q 2.6 where k φ 2 /2 P r ψ 2 /2 Q. 2.7 M k, r 2 P Q Slarly, the followg theore also ca be establshed. Theore 2.5. Let P,, {a, }, {, }, k,,adq be as Theore 2.3 ad defe A, /P, s,s a,s, for, 2,...,k.Letφ, 2,..., be real-valued, oegatve, ad cocave fuctos defed o R.The, k P, φ A, / r k / k k q, φ a,. 2.8 The roof of Theore 2.5 ca be coleted by followg the sae stes as the roof of Theore 2.3 wth sutable chages. Here, we ot the detals. Reark 2.6. Takg 2, q 2, r to 2.8, 2.8 becoes k 2 2 P, P 2,2 φ A, φ2 A2,2 3 2 8 3/2 k k k 2 k /2 2 k2, φ a, k2 2 /2 2 2,2 φ 2 a2,2. 2 2.9
Z. Chagja ad W.-S. Cheug 5 Ths s just a verse of the followg equalty whch was rove by Pachatte 9 : k r P Q φ A ψ B k 2 kr /2 k φ /2 2 r a r q ψ /2. 2 b 2.20 Reark 2.7. I vew of L Hôtal law, we have the followg fact: l r 0 / r r ex l r 0 l / r r ex l r l /. r 0 2 r 2.2 Accordgly, the secal case whe 2, 0., ad,, let r 0, the the equalty 2.8 reduces to the followg equalty: k 2 2 φ A, φ2 A2,2 2 2 k k k 2 k 2 /2 k2 φ a, k2 2 2 /2 φ 2 a2,2. 2 2.22 Ths s just a dscrete for of the followg equalty whch was rove by Zhao ad Debath : x y 0 0 φ F s ψ G t st 2 Ackowledgets [ x ds dt xy s 0 x { φ f s } y /2 ds]2[ y t { φ g t } ] 2 /2 dt. 0 2.23 The authors cordally thak the aoyous referee for hs/her valuable coets whch lead to the roveet of ths aer. Research s suorted by Zhejag Provcal Natural Scece Foudato of Cha, Grat o. Y605065, Foudato of the Educato Deartet of Zhejag Provce of Cha, Grat o. 20050392, artally suorted by the Research Grats Coucl of the Hog Kog SAR, Cha, Project o. HKU706/07P. Refereces G. D. Hadley, J. J. Kolha, ad J. E. Pečarć, New Hlbert-Pachatte tye tegral equaltes, Joural of Matheatcal Aalyss ad Alcatos, vol. 257, o.,. 238 250, 200. 2 G. Mzhe ad Y. Bcheg, O the exteded Hlbert s equalty, Proceedgs of the Aerca Matheatcal Socety, vol. 26, o. 3,. 75 759, 998.
6 Joural of Iequaltes ad Alcatos 3 G. Mzhe, O Hlbert s equalty ad ts alcatos, Joural of Matheatcal Aalyss ad Alcatos, vol. 22, o.,. 36 323, 997. 4 K. Hu, O Hlbert equalty ad ts alcato, Advaces Matheatcs, vol. 22, o. 2,. 60 63, 993. 5 K. Jchag, O ew extesos of Hlbert s tegral equalty, Joural of Matheatcal Aalyss ad Alcatos, vol. 235, o. 2,. 608 64, 999. 6 Y. Bcheg, O ew geeralzatos of Hlbert s equalty, Joural of Matheatcal Aalyss ad Alcatos, vol. 248, o.,. 29 40, 2000. 7 C.-J. Zhao, Iverses of dserse ad cotuous Pachatte s equaltes, Acta Matheatca Sca, vol. 46, o. 6,. 6, 2003. 8 C.-J. Zhao, Geeralzato o two ew Hlbert tye equaltes, Joural of Matheatcs,vol.20,o.4,. 43 46, 2000. 9 B. G. Pachatte, O soe ew equaltes slar to Hlbert s equalty, Joural of Matheatcal Aalyss ad Alcatos, vol. 226, o.,. 66 79, 998. 0 G. H. Hardy, J. E. Lttlewood, ad G. Pólya, Iequaltes, Cabrdge Uversty Press, Cabrdge, UK, 2d edto, 934. C.-J. Zhao ad L. Debath, Soe ew verse tye Hlbert tegral equaltes, Joural of Matheatcal Aalyss ad Alcatos, vol. 262, o.,. 4 48, 200.