5 Proc Pist Acd Sci Nilh 43: A 5-6 Al-ih 006 PATIAL SUMS OF CETAIN CLASSES OF MEOMOPHIC FUNCTIONS Nilh A Al-ih Girls Collg o Eductio iydh Sudi Arbi civd Jury 006 cctd Fbrury 006 Couictd by Pro r M Iqbl Choudhry Abstrct: Usig Al-Oboudi dirtil ortor study so clsss o roorhic uctios A cssry d suicit coditio or blogig to ths clsss coicits xtrl uctios d rtil sus r studid Kyords: Mroorhic covx strli covolutio dirtil ortor xtrl oits rtil sus 000 Mthtics Subct Clssiictio 30C45 Itroductio d diitios Lt dot th clss o uctios o th or 0 N 3 ( hich r rgulr d ultivlt i ucturd dis U{:0< <} Th uctio ƒ blogig to is sid to b roorhic strli * uctio i d oly i it stisis > ( Also uctio ƒ blogig to is sid to b roorhic covx uctio i d oly i it stisis > (3 ctly Al-Oboudi [] did dirtil ortor o lytic uctio ƒ o th or ( E-il: _ldih@hotilco (4 s ollos Lt 0 d ( ( ( ( ( or 0 d Ν 0 {03} Wh hv Sãlãg dirtil ortor [4] It c sily b s tht ƒ is giv by (4 Th [ ( ] [ ( ] Alyig this dirtil ortor to -vlt roorhic uctio i hv ( (
Crti clsss o roorhic uctios 5 d ( ( ( ( (5 or ll d < N {3} d Ν 0 {03} iitio A uctio o th or ( (6 Iclusio d suicit coditios I th xt thor rov tht ll uctios i th clss M( r -vlt roorhic strli o th ordr W d th olloig l L [] Lt b o costt d rgulr uctio i U (0 0 I ttis its xiu vlu o th circl < r < t 0 hv 0 ' ( 0 hr is rl ubr d Thor Lt b i M( hr Th or ll Ν 0 d 0 < d 0 blogs to M( i d oly i hr d N 0 ( < ( U (7 ( 0 < > 0 d M ( M ( Proo Lt M ( Th ( ( ( ( W c s tht or dirt slctios o d hv y dirt clsss o - vlt roorhic uctios iitio I ( 0 (8 th Τ ( i d oly i ( < ( U W hv to sho tht ( ilis th iqulity ( Lt us di i U by < ( (3 hr ( 0 d Ν 0 0 < > ( ( (4
53 Nilh A Al-ih irtitig (4 logrithiclly obti ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (5 W cli tht < i U For othris [by L ] thr xists 0 i U such tht 0 ' ( 0 (6 hr ( 0 d Fro (5 d 6 obti ( Thus ( ( ( ( ( ( ( ( (7 (8 hich cotrdicts ( Hc < d ro (3 M( Corollry All M( blog to (β or ll hr β I th xt thor driv so rortis o th ortor d giv lictio o th olloig Millr-Mocu l [3] L [3] Lt Φ(uv b colx vlud uctio Φ : C C x C (C is colx l d lt uu iu d vv iv Suos tht th uctio Φ(uv stisis th coditios (i Φ(uv is cotiuous i ; (ii (0 {Φ(uv}>0; (iii {(Φ(iu v }0; or ll (iu v Thor such tht ( u v I h c is lytic uctio i U d Φ(h h'; or ll U th h>0 I M( th M( β or β hr Proo β ( 4( ( Lt us di by 7 4 ith ( 4( ( (9 7 β 4 Th h is lytic i th o dis U irtitig (9 logrithiclly obti ( Thus ( ( β h ( ( β ( β h( ( β h ( ( β ( β h(
Crti clsss o roorhic uctios 54 d ( β ( β ( β h ( [( β ( β h( ] ( β h ( ( β ( β h( ( β ( h( > 0 Lt us di uctio Φ(uv by Φ ( u v ( β ( ( β v [( β ( β u] u Thor 3 I i U d did by (6 d [ ( ][ ( ( ( ( ] ( ( ( (0 th M( β Proo Suos (0 holds or ll dissibl vlus o d Usig th hyothss sil clcultio shos tht or ll U hv Th st (i β Φ(uv is cotiuous i C (ii (0 {Φ(0} > 0 ( β (iii For ll (iu v such tht ( u { Φ( iu v } ( β ( iu v ( β v [( β ( β iu ] ( β ( β ( β ( u { } 0 β β u Th uctio Φ(uv stisis th coditios o L Cosqutly obti ( 4( ( < 4 Hc th thor 7 W o driv suicit coditio or uctio to b i M( by hyothsis Accordig to th xiu odulus thor gt M( β Wh T( th th bov suicit coditio is lso cssry d hv th olloig Thor 4 Lt b did by (8 Th T( i d oly i [ ] [ ( ( ( ( ] ( ( ( (
55 Nilh A Al-ih Proo Hr oly d to rov th i rt Lt T( Th ( ( Usig th ct tht {} or ll hv th olloig tht ( [ ( ( ( ( ] ( ( ( ( ( [ ] ( < Lt us choos vlus o o th rl xis so is rl Uo clrig th doitor i ( d lttig through th rl vlus obti th rquird coditio Th uctio ( ( ( [ ( ] [ ( ( ( ( ] is xtrl uctio or th thor Corollry ( I T( th (3 ( ( ( [ ( ] [ ( ( ( ( ] or ch 3 Th qulity holds or th uctio giv by (3 Corollry (3 T( T( or ll N 3 Closur thors Thor 3 Th clss T( is closd udr covx cobitios Proo Lt g T( d lt is giv by (8 d g b b 0 For 0 δ it is suicit tht th uctio h did by h ( δ δ g ( U blogs to T( Sic h [( δ δ b ] lyig th Thor 4 gt ( δ [ ( ] [ ( ( ( ( ] δ [ ( ] [ ( ( ( ( ] [( δ ( δ( δ ]( ( ( δ ( ( ilis h T( b Fro Thor 3 it ollos tht th closd covx hull o T( is th s s T( No dtri th xtr oits o T(
Crti clsss o roorhic uctios 56 Thor 3 Lt hr 3 U d N 0 Th T( i d oly i it c b xrssd s σ hr σ 0 d σ Proo Suos tht σ σ Sic σ σ σ it ollos ro th Thor 4 tht T ( Covrsly suos tht T d sic st or σ d σ σ Fro Thor 4 hv 0 d σ σ It ollos tht σ Corollry (4 Th xtr oits o T( r th uctio 3 hr is giv by (3 4 Prtil sus o ( d t( Folloig th rlir ors by Silvr [5] o th rtil sus o lytic uctios study th rtio o uctio o th or (8 to its squc o rtil sus o th or (4 hr th coicits o stisy th coditio ( W lso dtri shr lor bouds or d
57 Nilh A Al-ih It is s tht this study ot oly givs s rticulr cs th rsults o Silvr [6] but lso givs ris to svrl rsults Thor 4 I is o th or (8 d stisis th coditio ( d 3 th or ( U (4 (43 Th rsults (4 d (43 r shr or uctio giv by (3 Proo Lt us di th uctio by (44 It is suicit to sho tht Fro (44 c rit d No i or quivltly (45 Fro th coditio o (0 it is suicit to sho tht
Crti clsss o roorhic uctios 58 hich is quivlt to (46 To s tht th uctio giv by (3 shos tht this rsult is shr lt h r To rov th scod rt o this thor rit hr This lst iqulity is quivlt to Mig us o ( gt (43 Filly qulity holds i (43 or th xtrl uctio giv by (3 r 4 irt choics o d giv th bov rsult or y ll o clsss Corollry (5 Lt 0 i Thor * is giv by (6 d Th or U
59 Nilh A Al-ih d This rsult is shr or 3 (47 Corollry (6 Lt d i it stisis th or U d Th rsult is shr or th uctio giv by (48 Thor 4 I o th or (6 stisis th coditio ( d 3 th (49 d (40 hr U d Proo W rit hr
Crti clsss o roorhic uctios 60 No i No it is suicit to sho tht tht is 0 To rov (40 lt us di by hr No i It is suicit to sho tht th lt sid is boudd by hich is quivlt to 0 r 4 irt choics o giv th bov rsult or y ll o clsss o roorhic uctios For xl Corollry (7 Lt b giv by (6 d i it stisis th coditio
6 Nilh A Al-ih th d or U d [ ( ] ( ( ( ( ( ( ( ( ( ( 34 3 I both css th xtrl uctio is giv by (47 Corollry (8 Lt b o th or (6 d stisy th coditio ( ( ( ( 3 Th d or U d ( ( ( ( ( 3 I both css th xtrl uctio is giv by (48 rcs Al-Oboudi FM 004 O uivlt uctios did by grlid Sãlãg ortor Itr J Mth & Mth Sci 7:49-439 Jc IS 97 Fuctios strli d covx o ordr J Lodo Mth Soc :469-474 3 Millr SS d Mocu PT 987 Scod ordr dirtil iqulitis i colx l J Mth Al Al 65:89-305 4 Sãlãg GS 983 Subclsss o uivlt uctios I: Colx Alysis- Fith oi Fiish Sir Prt (Buchrst 98 Lctur Nots i Mthtics Vol 03 36-37 Srigr Vrlg Brli 5 Silvr H 997 Prtil sus o strli uctios J Mth Al Al 09:-7 ( ( ( ( ( 3