On Some Integral Inequalities of Hardy-Type Operators

Transcription

1 Advces i Pue Mhemics, 3, 3, h://d.doi.og/.436/m Pulished Olie Ocoe 3 (h:// O Some Iegl Ieuliies of Hdy-Tye Oeos Ruf Kmilu, Omolehi Joseh Olouju, Susi Oloye Akeem Deme of Mhemics, Uivesiy of Iloi, Iloi, Nigei Emil: kuf@uiloi.edu.g, omolehi_joseh@yhoo.com, susikm@gmil.com Received Ail, 3; evised My 5, 3; cceed Jue 8, 3 Coyigh 3 Ruf Kmilu e l. This is oe ccess icle disiued ude he Ceive Commos Aiuio Licese, which emis uesiced use, disiuio, d eoducio i y medium, ovided he oigil wok is oely cied. ABSTRACT I ece ime, hdy iegl ieuliies hve eceived eios of my eseches. The im of his e is o oi ew iegl ieuliies of hdy-ye which comleme some ece esuls. Keywods: Hdy s Ieuliy; Mesule; Weigh Fucios & Hdy-Tye Oeos. Ioducio The clssicl hdy iegl ieuliy eds: Theoem Le f e o-egive -iegle fucio defied o,, d. The, f is iegle ove he ievl, fo ech d he followig ieuliy: f ydy d f d () holds, whee []). is he es ossile cos (see This ieuliy c e foud i my sdd ooks (see [-7]). Ieuliy () hs foud much iees fom ume of eseches d hee e umeous ew oofs, s well s, eesios, efiemes d vis which is efe o s Hdy ye ieuliies. I he ece e [8], he uho oved he followig geelizio which is eesio of [9]. Theoem Le f L X, g L X d fg L X e fiie, o-egive mesule fucios o,, d wih such h. The, he followig ieuliy holds: whee, d d d T fg C f g R k C l k l k () d k R k k. [] lso oved he followig iegl ieuliy of Hdy-ye mily y Jese s Ieuliy: Theoem 3 Le g e coiuous d odecesig o [, ],, wih g fo. Le d f e oegive d Leesgue- Sieljes iegle wih esec o g o,. Suose is el ume such h, he d d,,,, d g f g g C g f g (3) Coyigh 3 SciRes.

2 6 R. KAMILU ET AL. whee,,,,, C g g g. Ohe ece develomes of he Hdy-ye ieuliies c e see i he es [-6]. I his icle, we oi ou some ohe Hdy-ye ieuliies which will comleme he ove esuls () d (3).. Mi Resuls The followig lemm is of icul iees (see lso [8]). Lemm. Le,,, d le f e o-egive mesule fucio such h f d. The he followig ieuliy holds: d f l f d (4) Poof Le he, I f d, I f d y Holde s ieuliy, we hve, l I f d d f d We eed o show h hee eiss, such h fo y,, euliy i (4) does o hold. If ohewise, hee eis decesig seuece i N,, such h fo N he ieuliy (4), wie, ecomes euliy. The, o evey N hee coesod el coss c d d o oh zeo, such h eveywhee i., c f d lmos Thee eiss osiive iege N such h fo > N, f lmos eveywhee i (,). Hece, c c d d d fo N, d lso f d lim f d c This codics he fcs h f d. The lemm is oved. Theoem 4 Le f L X, g L X e fiie o-egive mesule fucios o,, d wih such h, he he followig ieuliy holds: whee C fg dd C fg d (5) l R k l k d Poof R k k. k fg d d f d g d d l fg d d l d fg d l fg d d C fg d Coyigh 3 SciRes.

3 R. KAMILU ET AL. 6 whee C is s sed i he seme of he heoem d his oves he heoem. The e esuls e o cove fucios s i lies o Hdy-ye ieuliies. Lemm. locl miimum of fucio f is glol miimum if d oly if f is sicly cove. Poof The ecessy follows fom he fc h if oi is locl oimum of cove fucio f. The f z f fo y z i some eighohood U of. Fo y y, z y elogs o U d sufficiely close o imlies h is glol oimum. Fo he sufficie, we le f e sicly cove fucio wih cove domi. Suose f hs loc l miimum d suc h h d ssumig f f. By sic coveiy d fo y,, we hve, f f f f f f. Sice y eighohood of cois ois of he fom wih,, hus he eighohood of cois ois fo which f f. Hece, f does o hve locl miimum, codicio. I mus e h, his shows h f hs mos oe locl miimum. Lemm. Le d c. If is osiive cove fucio o (,c), he Poof h dd h d hd d h dd h d d h d h d Hece he oof. Lemm. Le h, e o-egive fo,, o decesig d. he, d h d, d h Poof Le e coiuous d cove, If hs coiuous ivese which is eccessily cocve, he y (6) (7) Jese s ieuliy we hve, d, d d Tkig u u h h d,, we oi h, d h, d d d fo, we hve which we wie s h, d h, d d d, d d, h h d This comlee he oof. Theoem 5 If d c, le f, g e defied o (,) such h f, g c, he Poof e l fg d d e fg d e l fg d d e l fgl dd e l ( fg)de l d d Sice f e is cove fucio, lyig Jese s ieuliy o he ove gives e l fg d d fgd e l d e fgdd e fg d d e fg d The esul follows. (8) Coyigh 3 SciRes.

4 6 R. KAMILU ET AL. Theoem 6 Le g e coiuous d odecesig o,,, wih g fo d. Le d f e oe- give d Leesgue-Sieljes iegle wih esec o g o,. Suose is el ume such h he, whee d d,,,, g f g g C g f dg (9) C g g g g,,,, Poof d I he ieuliy (.5), we le, d g dg h g g f The, he lef hd side of (.5) ecomes d he igh hd side educes o d d g g f g g g f g g f dg d d d g g g g f g g g g g g f dg g g g f dg g g g g f dg Hece, ieuliy (.5) ecomes d d g f g g g g g f g fo, we hve g f dg g g g g f dg Iegig oh sides wih esec o g d he isig oh sides o owe yields d g f dg g g g g g f dg dg Coyigh 3 SciRes.

5 R. KAMILU ET AL. 63 Alyig Mikowski iegl ieuliy o he igh hd side imlies g f g g g g g d d g g g f g dg dg Sice Hece, we hve,,,, d g g g f g g g C g f g d d d,,,, d g f g g C g f g Which comlee he oof of he Theoem. 3. Coclusio This wok oied cosidele imoveme o Adego- Sheikh d Imou esuls d licios fo mesule d cove fucios e lso give. REFERENCES [] G. H. Hdy, N oes o Theoem of HILBERT, Mhe- Ieuli- mische Zelschif, Vol. 6, 9, [] R. A. Adms, Soolev Sces, Acdemic Pess, New Yok-Lodo, 975. [3] G. H. Hdy, J. E. Lilewood d G. Poly, ies, Cmidge Uivesiy Pess, Cmidge, 95, MR46395( 3:77e), Reied 99. [4] A. Kufe d L.-E. Pesso, Weighed Ieuliies of Hdy Tye, The Ameic Mhemicl Mohly, Wold Scieific, New Jesey, Lodo, Sigoe, Hog Kog, 3. h://d.doi.og/.4/59 [5] A. Kufe, L. Mligd d L.-E. Pesso, The Hdy Ieuliy Aou Is Hisoy d Some Reled Resuls, Vydvelsky Sevis Pulishig House, Pilse, 7. [6] C. P. Niculescu d L.-E. Pesso, Cove Fucios d Thei Alicios. A Coemoy Aoch, Sige, Beli, Heideleg, New Yok, Hog Kog, Lodo, Mil, Pis, Tokyo, 5. [7] B. Oic d A. Kufe, Hdy Tye Ieuliies, Logm, Hlow, 99. [8] K. Ruf, J. O. Omolehi d J. A. Gdey, O Some Refieme of Resuls o Hdy s Iegl Ieuliy, Ieiol Joul of Scieific Comuig, Vol., No., 7,. 5-. [9] Y. Bicheg, Z. Zhuohu d L. Deh, Noe o New Geelizios of Hdy s Iegl Ieuliy, Joul of Mhemicl Alysis d Alicios, Vol. 7, No., 998, h://d.doi.og/.6/jm [] A. G. Adego-Sheikh d C. O. Imou, A Iegl Ieuliy of he Hdy s Tye, Kgujevc Joul of Mhemics, Vol. 9, 6, [] S. S. Dgomi d N. M. Ioescu, Some Covese of Jese s Ieuliy d Alicios, Revue d'alyse Numéiue e de Théoie de l Aoimio, Vol. 3, No., 994, [] C. O. Imou d A. G. Adego-Sheikh, O Some Weighed Mied Nom Hdy-Tye Iegl Ieuliies, Joul of Ieuliies i Pue d Alied Mhmics, Vol. 8, No. 4, 7,. -. [3] S. Kijse, L. Nikolov, L.-E. Pesso d A. Wedesig, Hdy-Tye Ieuliies vi Coveiy, Mhemicl Ieuliies & Alicios, Vol. 8, No. 3, 5, [4] K. Ruf d J. O. Omolehi, Some Noes o Iegl Ieuliy Reled o G.H. Hdy s Iegl Ieuliy, Puj Uivesiy Joul of Mhemics, Vol. 38, 6, [5] M. Z. Sky d H. Yildiim, Some Hdy Tye Iegl Ieuliies, Joul of Ieuliies i Pue d Alied Mhemics, Vol. 7, No. 5, 6,. -5. [6] L. Zhogue, G. Migzhe d L. Deh, O New Coyigh 3 SciRes.

6 64 R. KAMILU ET AL. Geelizios of he Hile Iegl Ieuliy, Joul of Mhemicl Alysis d Alicios, Vol. 36, No., 7, h://d.doi.og/.6/j.jm Coyigh 3 SciRes.