On Almost Increasing Sequences For Generalized Absolute Summability
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1 Joul of Applied Mthetic & Bioifotic, ol., o., 0, ISSN: (pit), (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute ubility of ifiite eie by qui-powe iceig equece i poed. Ou eult gie coectio d ipoeet to the eult of S d Seli []. Mthetic Subject Clifictio: 40F05, 40D5. Keywod : Abolute ubility, Qui-powe iceig equece. Itoductio Let be ifiite eie with ptil u ), A deote lowe tigul tix. he eie i id to be bolutely A-uble of ode, if ( whee <,. 0 Deptet of Copute Egieeig College of Egieeig Uieity of Moul, Iq, eil: wduli@hotil.co Aticle Ifo: Reied : Mch 9, 0. Publihed olie : My 3, 0.
2 44 O Alot Iceig Sequece Fo Geelized he eie i uble A,,, 0, if <. () A poitie equece γ γ ) i id to be qui- -powe iceig equece if ( thee exit cott K K(, γ ) uch tht K γ γ () Hold fo ll. It y be etioed tht eey lot iceig equece i qui - -powe iceig equece fo y oegtie, but the coee eed ot be tue. wo lowe tigul tice A d  e ocited with A follow,, 0,,..., (3),,,...,, S d Seli [] poed the followig eult. heoe.. Let A be lowe tigul tix with oegtie etie tifyig,, fo, 0, 0,,..., Ο( ), / Ο(),, Ο( ), (4), Ο( ), (5) Ο( ), (6), d let ( ) d ( ) be equece uch tht
3 W.. Suli 45, (7) If ( ) i qui - -iceig equece tifyig 0, (8) Ο( ),, (9) <, (0) Ο(), () the the eie i uble A,,, 0 < /. We e the followig coditio Ο( ),. () Re. It y be etioed tht i the poof of theoe., icoect tep though the etitio of I. he utho coide ( ) i bouded egdig thi follow fo the fct Ο(). hi ot tue, fo i qui, we y te, which iplie i Ο() tht ) ( i ot bouded. heefoe the poof of theoe. i ot lid. Le Le.. Coditio () i wee th (9) whe Poof. If (9) hold, the we he i o-deceig. while if () i tified the, Ο Ο ( ),
4 46 O Alot Iceig Sequece Fo Geelized () Ο ( ) Ο ( ) ( ) Ο ( ) Ο Ο ( )( ) Ο ( ) Ο ( ). heefoe (9) iplie () but ot coeely. Re.. Coditio (9) h bee eplced by () which i bette i the followig ee (). If i o-deceig, () i wee th (9) (ee le.) (b) he oe dtge of ou coditio i to obti the deied eult without y lo of powe though etitio. A exple the poof i coditio (9) ipoe to del with ( ) Ο, looig coideed to be Ο(). We he o uch ce i coditio ().. Coditio (4) i eliited. Le.. Coditio (8) d (0) iply Ο( ),, (3) Ο(). (4) Poof. A 0, d i o-deceig, we he
5 W.. Suli 47 () Ο () Ο() Ο Ο(). hi poe (3). o poe (4), we obee tht Ο() Ο() Ο Ο () () Ο Ο (), < () Ο() u du Ο() u du () Ο() Ο Ο (). Le.3 []. Let A be defied i theoe., the, fo.
6 48 O Alot Iceig Sequece Fo Geelized 3 Mi Reult heoe 3.. Suppoe ll coditio of theoe. e tified except coditio (9) i eplced by coditio (), d coditio (4) i eoed, the the eie i uble A,,, 0 < /. Poof. Let defiitio, we he x be the th te of the A-tfo of the eie. By d hece x 0 0, Applyig Abel' tfotio, 0 : x x.., 3 o coplete the poof, by Miowi' iequlity, it i ufficiet to how tht Applyig Holde' iequlity, we he j <, j,,3. ( ) ( ) Ο( ) Ο() Ο() Ο() Ο() Ο().
7 W.. Suli 49 Ο ) ( ) ( Ο () Ο ) ( ( ) () Ο Ο ) ( Ο(), i the ce of., 3 ( ), ( ) Ο, () ( ),, () Ο ( ), () Ο (by le.3) ( ), () Ο
8 50 O Alot Iceig Sequece Fo Geelized Ο(), Ο( ) Ο( ) Ο ( ) Ο() ( ) Ο() Ο() Ο() Ο() Ο() ACKNOWLEDGEMEN. he utho i o gteful to the efeee who guided e to ipoe witig of the ppe. Refeece [] B. E. Rhode, Icluio theoe fo bolute tix ubility ethod, J. Mt. Al. Appl., 38 (999), [] E. S d H. Seli, A ecet ote o qui powe iceig equece fo geelized bolute ubility, Joul of Iequlitie d Applictio, 009, Aticle ID675403, 0 pge.
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