Available olie a h://sciog Egieeig Maheaics Lees 2 (23) No 56-66 ISSN 249-9337 ABSLUE INDEED SUMMABILIY FACR F AN INFINIE SERIES USING QUASI-F-WER INCREASING SEQUENCES SKAIKRAY * RKJAI 2 UKMISRA 3 NCSAH 4 Deae of Maheaics Raveshaw Uivesiy Cuac-7533 disha Idia 2 Deae of Maheaics DRIEMS agi Cuac-75422 disha Idia 3 Deae of Maheaics NIS alu Hills Behau-768 disha Idia 4 Deae of Maheaics SBWoe scollege(auo) Cuac-753 disha Idia Absac: A esul coceig absolue idexed suabiliy faco of a ifiie seies usig Quasi - f - owe iceasig sequeces has bee esablished Keywods:Quasi-iceasigQuasi - - owe iceasig Quasi - f - owe iceasig idex absolue suabiliy suabiliy faco 2 AMS Subjec Classificaio: 4A5 4D5 4F5 INRDUCIN: A osiive sequece a is said o be alos iceasig if hee exiss a osiive sequece b ad wo osiive cosas A ad B such ha () Ab a Bb fo all * Coesodig auho he sequece a is said o be quasi- -owe iceasig if hee exiss a cosa K deedig uo wih K such ha Received Decebe 6 22 56
57 ABSLUE INDEED SUMMABILIY FACR (2) K a a fo all I aicula if he a is said o be quasi-iceasig sequece I is clea ha evey alos iceasig sequece is a quasi- -owe iceasig sequece fo ay oegaive Bu he covese is o ue as is quasi- -owe iceasig bu o alos iceasig Le f f be a osiive sequece of ubes he he osiive sequece a is said o be quasi- f -owe iceasig if hee exiss a cosa K deedig uo f wih K such ha (3) K f a f a fo [4 ] Clealy if is a quasi- f -owe iceasig sequece he he quasi- iceasig sequece Le a be a ifiie seies wih sequece of aial sus s Le osiive ubes such ha is a f be a sequece of as he he sequece-o-sequece asfoaio (4) s defies he N - ea of he sequece s geeaed by he sequece of coefficies he seies a is said o be suable N [ ] if
SKAIKRAY RKJAI UKMISRA NCSAH 58 (5) he seies a is said o be suable ; N if (6) he seies a is said o be suable ; N if (7) uig ; N educes o ; N 2 RELIMINARIES Dealig wih quasi- -owe iceasig sequece Bo ad Debah[2] have esablished he followig heoe: 2 HEREM: Le be a quasi- -owe iceasig sequece fo ad be a eal sequece If he codiios (2)
59 ABSLUE INDEED SUMMABILIY FACR (22) () (23) ( ) (24) ( ) ad (25) 2 ae saisfied whee is he (C) ea of he sequece ( a ) he he seies a is suable N Subsequely Leidle[3] esablished a siila esul educig ceai codiio of Bo He esablished: 22 HEREM: Le he sequece be a quasi- -owe iceasig sequece fo he eal sequece saisfies he codiios ad (22) ( ) ad (222) ( ) Fuhe suose he codiios (23) (24) ad (223) ( )
SKAIKRAY RKJAI UKMISRA NCSAH 6 hold whee ( ) ax he he seies a is suable N Recely exedig he above esuls o quasi- f -owe iceasig sequece Sulaia[5] have esablished he followig heoe: 23 HEREM: Le ( f ) f be a sequece Le owe sequece ad a sequece of cosas saisfyig he codiios (23) as be a quasi- f - (232) (233) (234) ad (235) whee is he C ea of he sequece a a he he seies is suable N We ove he followig heoe 3 MAIN RESULS: be a sequece ad be a quasi- f -owe sequece Le Le f ( f ) sequece of cosas such ha (3) as a (32)
6 ABSLUE INDEED SUMMABILIY FACR (33) () (34) (35) (36) he he seies 4 LEMMA: a is suable N ; I ode o ove he heoe we equie he followig lea Le f f be a sequece ad be a quasi - f - owe iceasig sequece Le be a sequece of cosas saisfyig (3) ad (32) he (4) ad (42) 4 RF F HE LEMMA: As ad is o-deceasig we have () ()
SKAIKRAY RKJAI UKMISRA NCSAH 62 () ) ( his esablishes (4) Nex () () () () () () du u du u () () () () () his esablishes (42)
63 ABSLUE INDEED SUMMABILIY FACR 5 RF F HE HEREM: Le be he sequece of N ea of he seies a he a a Hece fo a a ) ( 4 3 2 (say) I ode o ove he heoe usig Miowsi s iequaliy i is eough o show ha 4 23 j j Alyig Holde s iequaliy we have
SKAIKRAY RKJAI UKMISRA NCSAH 64 () ) ( () ) ( ) ( () () Nex 2 () () ) ( () as i he case of Nex 3
65 ABSLUE INDEED SUMMABILIY FACR () () ) ( () ) ( () () () () () () Fially 4 () () ()
SKAIKRAY RKJAI UKMISRA NCSAH 66 ( ) ( ) ) ( () ) ( () his colees he oof of he heoe REFERENCES [] HBo A Noe o wo suabiliy ehods ocae Mah Soc98 (986) 8-84 [2] HBo ad LDebah Quasi - - owe iceasig sequeces Ieaioal joual of Maheaics ad Maheaical Scieces44(24)237-2376 [3] LLeide A ece oe o absolue Riesz suabiliy facos J Ieq ue ad Al MahVol-7 Issue- 2aicle-44(26) [4] WSulaia Exesio o absolue suabiliy facos of ifiie seies J Mah Aal Al 322 (26)224-23 [5] WSulaia A ece oe o absolue absolueriesz suabiliy facos of a ifiie seies J Al Fucioal Aalysis Vol-7o438-387