Extension of Hardy Inequality on Weighted Sequence Spaces

Transcription

1 Jourl of Scieces Islic Reublic of Ir 20(2): (2009) Uiversiy of ehr ISS h://sciecesucir Eesio of Hrdy Iequliy o Weighed Sequece Sces R Lshriour d D Foroui 2 Dere of Mheics Fculy of Mheics Uiversiy of Sis d Bluches Zhed Islic Reublic of Ir 2 Dere of Mheics Fculy of Scieces Vli-E-Asr Uiversiy o f Rfs Rfs Islic Reublic of Ir Received: 25 Jue 2007 / Revised: 8 Aril 2009 / Acceed: 9 My 2009 Absrc Le d ( ) be sequece ih o-egive eries If ( ) 0 deoe by he ifiu of hose U sisfyig he folloig iequliy: U heever ( ) l ( ) he urose of his er is o give uer boud for he or of oeror o eighed sequece sces d() d l () d lso e() We cosidered his roble for ceri ri oerors such s orlud Weighed e esro d oso rices his roble is cosidered by soe uhors lie Bee Jso d he firs uhor o sequece sces l d eighed sequece sces for soe id of ri oerors Also his sudy is eesio of er by hg-po he Dh-hi Luor d Zog-Yi Ou Keyords: Hrdy iequliy; orlud ri; Weighed e ri Iroducio I his sudy e cosider he or of ceri ri oerors o eighed sequece sces l ( ) e ( ) d Lorez sequece sces d ( ) hich is cosidered i [] d [2] o l sces d i [5-8] d [0] o l ( ) d d ( ) for soe ri oerors such s esro oso Husdorff d Hilber oerors Assue h l is he ored lier sce of ll sequeces ( ) ih fiie or here Suose h ( ) is sequece ih oegive eries For e defie he eighed sequece sce l ( ) s follos: 2000 Mheics Subec lssificio 26D5 5A60 orresodig uhor el: +98(54) F: +98(54) E-il: lshri@hoousbcir 59

2 Vol 20 o 2 Srig 2009 Lshriour d Foroui J Sci I R Ir l( ) : < ih or here / Also if ( ) is decresig o-egive sequece such h li 0 d he he Lorez sequece sce d ( ) is defied s follos: d ( ) : < here ( ) is he decresig rerrgee of ( ) I fc d ( ) is he sce of ull sequeces for hich is i () ih or l + + d ( ) Le A L d W + L + e defie he eighed sequece sce e ( ) s follos: A e ( ) ( ):su < W ih or hich is defied s follos: A su W Our obecive i secio is o give geerlizio of soe resuls obied by [] d [2] I secio 2 e ry o solve he roble of fidig he or of ceri ri oerors o d d e ( ) d e deduce he eisece of uer boud for ceri ri oerors such s esro d oso oerors he roble of fidig he loer boud of ri oerors o eighed sequece sces is cosidered i [9] Resuls Mri Oerors o d d l () o cosider he oeror ( i ) defied by b here b i i i We rie for he or of s oeror fro l ( ) io iself d for he or of s oeror fro l io iself d for he or of s oeror o d ( ) d ( ) he folloig codiios is h e eed o cover sees for l ( ) o oes for d ( ) We ssue hroughou h: () For ll i i 0 (2) For ll i li i 0 (3) Eiher i decreses ih for ech i or i decreses ih i for ech d c i i decreses ih for ech odiio () ilies h d hece he o-egive sequeces re sufficie o deerie or of Proosiio ([5] Le 2) Le d ( i ) be oeror ih codiios () (2) d (3) he d ( ) d ( ) for ll o-egive elees i d ( ) Hece decresig o-egive elees re sufficie o deerie or of ri oeror I he folloig e se soe les hich re eeded for i resul We se ξ + ( ξ 0) d ξ i( ξ0) d lso Le ([2] Le 2) Assue h re o-egive sequeces he for ll + ( + )( ) + Le 2 ([2] Le 22) Le d be o-egive sequeces ih + L 0 d 0 for < he for ll

3 Eesio of Hrdy Iequliy o Weighed Sequece Sces Le 3 Suose h u v re o-egive ubers such h u is diverge d li v 0 he uv u 0 s Proof If e e ε > 0 sice li v 0 he here eiss ieger > 0 such h for ll > uv uv + ε u uv + ε u + Sice u is diverge here eiss ieger > such h for ll > e hve uv ε u herefore uv ε u 2 If ε 0 e hve he see Proosiio 2 ([5] Proosiio 5) Le > d ( ) be decresig sequece ih o-egive eries d le he ri ( ) be ih he folloig eries: for 0 for < he Le 4 Le > d ( ) be decresig sequece ih o-egive eries d lso be diverge Le d he ri ( c ) hve he folloig eries: for c + 0 for < he Proof is he esro ri d 0 c c for ll Sice ( ) is decresig sequece lyig Proosiio 2 e deduce h Fi such h d le + ( ) for 0 for > he + Also for A s + ds + here A + L + So h b A + ( + ) + Sice ( s ) s for 0< s < e hve b d hece ( ) + + b + ( + ) + Sice ( ) is decresig sequece + d so herefore is diverge seig + + y d ly Le 3 e ( + ) hve he see I he folloig e recll heore 8 of [3] hich is eeded for i resul heore ([3] heore 8) If > d is o-egive sequece he 6

4 Vol 20 o 2 Srig 2009 Lshriour d Foroui J Sci I R Ir i + i i Le 5 If > d re o-egive sequeces d lso is decresig he ( ) i i + i Proof Alyig heore e hve i i + i i i + i ( ) We se 0 0 for d + M su ( + )( ) suif + ( + )( ) + + We sy h ( ) is loer rigulr if 0 for < We o iroduce he firs i resul heore 2 Suose > d ( ) is decresig sequece ih o-egive eries Le ( ) be loer rigulr ri ih o-egive eries () i M Moreover if M < he is bouded o l ( ) ( ii ) If is diverge d + is decresig he herefore if ( ) is decresig sequece ih o-egive eries d is decresig d lso he M + Proof () e deduce h i Le be y sequece By Le + + ( ) M + Alyig Le 5 d he il heore of Hrdy d Lileood e hve M + his ilies h M ( M ) ii We hve su β here β if + ( + )( ) + + Le so h β 0 Le ( b ) be decresig sequece ih o-egive eries d b We se L 0 d + b + for ll We hve b d Le 2 follos h β β

5 Eesio of Hrdy Iequliy o Weighed Sequece Sces β + b + + β b Alyig Proosiio e coclude h β d so his esblishes he roof of he heore I he folloig e give soe corollries of heore 2 We ssue ( ) is decresig sequece ih o-egive eries d + is decresig d lso orollry Suose > d ( ) is loer rigulr ri ih 0 for < he su if su Moreover if he righ hd side of he bove l iequliy is fiie he is bouded o Proof We hve M su d su if his colees he roof of he see orollry 2 Assue h > d ( ) is loer rigulr ri ih 0 for < d lso ( ) is icresig sequece for ech he { su } I riculr here is he geerlized esro ri defied i Le 4 We ly he bove corollry o he folloig o secil cses Le ( ) be o-egive sequece ih >0 d + L + he orlud ri ( ) is defied s follos: + for 0 for > orollry 3 Suose > d ( ) is he orlud ri d ( ) is sequece decresig ih d > 0 he Le ( ) be o-egive sequece ih >0 he Weighed e ri M ( ) is defied s follos: for 0 for > orollry 4 Assue h > d M ( ) is he Weighed e ri d lso ( ) is icresig sequece ih d < he M orollry 5 Suose > d ( ) is loer rigulr ri ih 0 for < he if su { } Moreover if he righ hd side of he bove iequliy is fiie he is bouded o l ( ) Proof We hve M su d if his esblishes he roof We ly he bove corollry o he folloig o secil cses orollry 6 Assue h > d ( ) is he orlud ri d ( ) is icresig sequece he su 63

6 Vol 20 o 2 Srig 2009 Lshriour d Foroui J Sci I R Ir orollry 7 Suose > d M ( ) is he Weighed e ri d lso ( ) is decresig sequece ih d > 0 he M Ele Le γ here 0< γ ( log( + ) ) d + be decresig d lso herefore if ( ) is decresig sequece ih d > 0 he Also if ( ) is icresig sequece ih d < he M 2 Mri Oeror o d d e() I his r of sudy e cosider he roble of fidig he or of ri oeror d o d d e ( ) here d d e ( ) re defied s before If d( ) e deoe or of ih d if e( ) e deoe or of ih We rie for he or of s oeror fro d io iself d for he or of s oeror fro e ( ) io iself Suose is bouded ri oeror o e ( ) he he rsose ri of is bouded ri oeror o d d Le d be defied s i Le 4 d lso le be he ri rsose of he ri ( ) is defied s follos: for + 0 for > If d re esro d oso rices resecively d re geerlized esro d oso rices he roble of fidig he or of ri oerors o d d e ( ) is cosidered i [8] Also i he folloig e cosider such robles for soe rices o eighed sequece sces d d e ( ) heore 2 Suose ( ) is ri oeror sisfyig codiios () (2) d (3) If S su W < here S s + L + s s d W + L + he is bouded oeror fro d io iself d lso S su W Proof Alyig Proosiio i is sufficie o cosider decresig o-egive sequeces Le be S i d such h 2 L 0 d M su W he s S + ( ) M W Also e hve + ( + ) W herefore M d hece M Furher e e L d 0 for ll + he W S 64

7 Eesio of Hrdy Iequliy o Weighed Sequece Sces hus M his colees he roof of he heore I he folloig sees e cosider he or of esro d oso rices I is eough o cosider s S he sequece ised of becuse of he ello fcs lised i he folloig W le Le 2 () i If S W M for ll s M for ll he s ( ii ) If is icresig (or decresig) he so is S W s ( iii ) If M s he W Proof I is eleery S M s Le 22 Le 0< < X () i If d X he ( + ) is icresig d eds o ( ii ) If X + he X ( ) ( ) Proof I is eleery heore 22 If ( + ) + is decresig here 0< < he is bouded oeror o d d lso is bouded oeror o e ( ) Moreover + + I riculr ξ ( ) + here ξ is Rie s Ze fucio Proof Alyig heore 2 e hve Sice S su W s ( + ) + + ( + ) + + Le 22 ( ii ) shos h s is decresig S herefore lyig Le 2 ( ii ) e deduce h W is decresig d lso Proosiio 2 If r he ( ) W su < ( + ) s d io iself Also e hve r Proof Sice for ll s W r ( ) + heore 2 d Le 2 () i follo h r ( ) d his colees he roof Proosiio 22 If he W su < W + is bouded oeror o e ( ) d W su W + Proof Alyig heore 2 e hve S su W W Sice s d + e hve he see heore 23 Suose h 0< < he e hve here ( + ) s d io iself d lso 65

8 Vol 20 o 2 Srig 2009 Lshriour d Foroui J Sci I R Ir I riculr Proof We hve S W W + + Our W is he X of Le 22 () i hich ells us W h is icresig d eds o Le ( + ) 2 ( ii ) d ( iii ) follo he see (Of course his lso shos h r ( ) ) Refereces Bee G Iequliies coliery o Hrdy Qur J Mh Oford (2) 49: (998) 2 hg-po he Dh-hi Luor d Zog-Yi Ou Eesio of Hrdy iequliy J Mh Al 237: 60-7 (2002) 3 Hrdy GH d Lileood JE A il heore ih fucio-heoreic Ac Mh 54: 8-6 (930) 4 Hrdy GH Lileood JE d Poly G Iequliies 2d ediio bridge Uiversiy ress bridge (200) 5 Jeso GJO d Lshriour R ors of ceri oerors o eighed l sces d Lorez sequece sces J Iequliies i Pure d Alied Mheics Volue 3 Issue Aricle 6 (2002) 6 Lshriour R Weighed Me Mri o Weighed Sequece Sces WSEAS rscio o Mheics Issue 4 Volue 3: (2004) 7 Lshriour R rsose of he Weighed Me Oerors o Weighed Sequece Sces WSEAS rscio o Mheics Issue 4 Volue 4: (2005) 8 Lshriour R d Foroui D Iequliies ivolvig uer bouds for ceri ri oerors Proc Idi Acd Sci(Mh Sci) Volue 6: Agus (2006) 9 Lshriour R d Foroui D Loer bouds for rices o eighed sequece sces Jourl of Scieces Islic Reublic of Ir 8(): (2007) 0 Lshriour R d Foroui D Soe iequliies ivolvig uer bouds for soe ri oerors I zech Mh J 57(32): (2007) 66