On Quasi - f -Power Increasing Sequences

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1 Ieaioal Maheaical Fou Vol o Quasi - f -owe Iceasig Sequeces Maheda Misa G Deae of Maheaics NC College (Auooous) Jaju disha Mahedaisa2007@gailco B adhy Rolad Isiue of echoy Golahaa disha Idia iaady@gailco Daaa Bisoyi Deae of Maheaics LN Mahavidyalaya Kodala Gaja disha Idia dbisoyi2@gailco U K Misa Deae of Maheaics Naioal Isiue of Sciece ad echoy allu Hills Golahaa disha Idia uaaa_isa@yahooco Absac A esul coceig absolue idexed Suabiliy faco of a ifiie seies usig Quasi - f - owe iceasig sequeces has bee esablished Maheaics Subjec Classificaio: 40A05 40D5 40F05

2 378 M Misa B adhy D Bisoyi ad U K Misa Keywods: Quasi-iceasig Quasi - - owe iceasig Quasi - f - owe iceasig idex absolue Suabiliy suabiliy faco Ioducio ( ) 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 he sequece ( a ) is said o be quasi- -owe iceasig if hee exiss a cosa K deedig uo wih K such ha (2) K a a fo all I aicula if 0 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 o-egaive 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 ( ) a quasi- iceasig sequece Le sequece of osiive ubes such ha α is a quasi- f -owe iceasig sequece he he ( ) a be a ifiie seies wih sequece of aial sus { s } Le ( ) as 0 he he sequece-o-sequece asfoaio (4) s 0 0 be a α is f

3 quasi - f -owe iceasig sequeces 379 defies he ( ) he seies N - ea of he sequece ( s ) geeaed by he sequece of coefficies { } a is said o be suable N ( [ ] ) if (5) < he seies a is said o be suable N ; 0 if (6) < 2 Kow heoes Dealig wih quasi- -owe iceasig sequece Bo ad Debah [2] have esablished he followig heoe: 2 heoe: Le ( ) be a quasi- -owe iceasig sequece fo 0 < sequece If he codiios (22) () (2) ( ) (23) ( ) (24) ( ) ad < ad ( ) be a eal

4 380 M Misa B adhy D Bisoyi ad U K Misa (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 heoe: Le he sequece ( ) he eal sequece ( ) be a quasi- -owe iceasig sequece fo 0 < < ad saisfies he codiios (22) ( ) ad (222) ( ) Fuhee suose he codiios (23) (24) ad hold whee ( ) ax( ) (223) ( ) < N he he seies a is suable Recely exedig he above esuls o quasi- f -owe iceasig sequece Sulaia [5] have esablished he followig heoe: 23 heoe: f ( f < be a sequece Le ( ) be a quasi- f - Le ) ( )0 0 owe sequece ad ( ) a sequece of cosas saisfyig he codiios (23) as 0

5 quasi - f -owe iceasig sequeces 38 (232) < (233) ( ) (234) ( ) ad (235) ( ) whee is he ( C) ea of he sequece ( ) N a he he seies a is suable I wha follows i his ae we ove he followig heoe 3 Mai heoe Le ( ) Le f ( f ) ( ) be a sequece ad ( ) a sequece of cosas such ha (3) 0 as (32) < (33) () (34) (35) ( ) (36) ( ) be a quasi- f -owe sequece

6 382 M Misa B adhy D Bisoyi ad U K Misa he he seies a is suable N ; 0 I ode o ove he heoe we equie he followig lea 4 Lea: Le f ( f ) ( ) 0 < 0 be a sequece ad ( ) be a quasi - f - owe iceasig sequece Le ( ) be a sequece of cosas saisfyig (3) ad (32) he (4) 0( ) ad (42) < 4 oof of Lea: As 0 ad is o-deceasig we have ( ) () () ( ) () his esablishes (4) Nex () ()

7 quasi - f -owe iceasig sequeces 383 ( ) () () < ( ) () () du u du u () () () () () his esablishes (42) 5 oof of heoe: Le ( ) be he sequece of ( ) N ea of he seies a he 0 0 a ( ) a 0 Hece fo a a ) (

8 384 M Misa B adhy D Bisoyi ad U K Misa (say) I ode o ove he heoe usig Miowsi s iequaliy i is eough o show ha 234 < j j Alyig Holdes iequaliy we have ( ) () ) ( () ) ( ) 0( () () Nex 2 () () () () as i he case of Nex 3

9 quasi - f -owe iceasig sequeces 385 () () ( ) ) ( ( ) ( ) 0() () ) ( 0 ( ) ( ) 0() () 0() 0() () () Fially 4 () () ( ) () ) ( ) ( ()

10 386 M Misa B adhy D Bisoyi ad U K Misa () () his colees he oof of he heoe ( ) Refeeces [] 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(2004) [3] LLeide A ece oe o absolue Riesz suabiliy facos J Ieq ue ad Al MahVol-7 Issue-2aicle-44(2006) [4] WSulaia Exesio o absolue suabiliy facos of ifiie seies J Mah Aal Al 322(2006) [5] WSulaia A ece oe o absolue absolue Riesz suabiliy facos of a ifiie seies JAl Fucioal Aalysis Vol-7 o Received: cobe 202

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