On a class of analytic functions defined by Ruscheweyh derivative

Transcription

1 Lfe Scece Jourl ;9( O clss of lytc fuctos defed by Ruscheweyh dervtve S N Ml M Arf K I Noor 3 d M Rz Deprtmet of Mthemtcs GC Uversty Fslbd Pujb Pst Deprtmet of Mthemtcs Abdul Wl Kh Uversty Mrd KPK Pst 3 Deprtmet of Mthemtcs COMSATS Isttute of Iformto Techology Islmbd Pst sml@yhoocom (S N Ml mrfmths@hotmlcom (M Arf hldoor@hotmlcom (K I Noor mohs976@yhoocom (M Rz Abstrct: The m of ths pper s to troduce clss of lytc fuctos defed by usg geerlzed jows fuctos d Ruscheweyh dervtve The coeffcet boud cluso result d rdus problem hs bee dscussed ths pper Severl ow results hve bee deducted from our m results s specl cses by ssgg prtculr vlues to the dfferet prmeters [Ml SN Arf M Noor KI d Rz M O clss of lytc fuctos defed by Ruscheweyh dervtve Lfe Sc J ;9(: (ISSN: Key words: Alytc fuctos Jows fuctos Ruscheweyh dervtve bouded boudry rotto bouded rdus rotto Mthemtcs Subject Clssfcto: 3C5 3C Itroducto Let A be the clss of fuctos of the form f z z z ( whch re lytc the ope ut ds E { z : z < } If f d g re lytc E we sy tht f s subordte to g wrtte f g or f ( z g( z f there exsts Schwrz fucto wz E such tht f ( z g( w( z Let P[ A B be the clss of fuctos h lytc E wth h ( d Az h( z B < A Bz Ths clss ws troduced by Jows [ The clss P[ A B s coected wth the clss P of fuctos wth postve rel prts by the relto [ ( B h( A ( B h ( A h P A B P ( Lter Poltoğlu [ defed the clss P[ A B α s: Let P[ A B α be the clss of fuctos p lytc E wth p ( d p ( z {( α A α } B z (3 Bz B < A α < From (3 t c esly be see tht p P[ A B α f d oly f there exsts h P[ A B such tht It s lso oted tht P[ p z α h z α α < z E ( P the well-ow clss of lytc fuctos E wth postve rel prt Noor [3 cosdered the geerlzed clss P[ A B α of Jows fuctos whch s defed s follows A fucto p s sd to be the clss P[ A B α f d oly f p( z p ( z p ( z (5 p p P A B α B < A d [ α P[ A B α P A B α d P[ P the well-ow clss gve d studed by Pchu [ < It s cler tht [ For y two lytc fuctos f ( z z d f ( z b z z E the covoluto (Hdmrd product of f d f s defed by lfescecej@gmlcom

2 Lfe Scece Jourl ;9( f f z b z (6 Usg Hdmrd product Ruscheweyh [5 troduced ler opertor D : A A It s defed s z D f ( z f ( z ( z (7 z ϕ z > wth ϕ ( (! ( ρ s Pochhmmer symbol gve s ( ρ ρ( ρ ( ρ ( ρ N Moreover for { } D f ( z The fucto D f ( z N ( ( z z f z! ws the clled (8 (9 th order Ruscheweyh dervtve of f For the pplcto of Ruscheweyh dervtve see [6 8 The followg detty c esly be estblshed D f ( z D f ( z z D f ( z ( Now usg ll these cocepts we defe the followg clss Defto A fucto f ν [ α f d oly f A B b D f ( z A s the clss [ α P A B z E b b D f ( z > B < A α < d b C { } Assgg cert vlues to dfferet prmeters we hve dfferet well-ow clsses of lytc fuctos s c be see below Specl cses ( ν [ α b ν ( α b the well-ow clss defed by Lth d Njud Ro [9 ( ν [ AB α C[ AB α ν [ AB α S[ AB α the well-ow clss defed by Poltoğlu [ ( ν [ A B V [ A B ν [ AB R [ A B V [ A B d R [ A B deote the clss of jows fuctos wth bouded boudry d bouded rdus rottos respectvely gve by Noor [ For the detl o the subject of fuctos of bouded boudry rotto Jows fuctos d relted topcs we refer the wor of Noor etl [ d Arf etl [ Prelmry Results We eed the followg results to obt our m results Lemm Let p( z q z P [ A B α The for ll ( ( α A B q ( Ths equlty s shrp The proof follows from ( (5 d the coeffcet boud of h P[ A B gve by Aouf [3 Lemm [ Let u u u v v v d ( uv ψ be complex vlued fucto stsfyg the codtos: ψ uv s cotuous dom D ( ( ( D ( ( u v Reψ > Re ψ wheever ( u v D v u h z c z s fucto lytc E hz zh ( z D d d d If such tht Re ( hz zh( z ψ > for z E Re hz > E Lemm 3 Let p P [ A B the wth lfescecej@gmlcom

3 Lfe Scece Jourl ;9( The for z r < AB r ABr ( Br Re pz pz A B r ABr ( Br ( The proof s mmedte by usg (5 d the h P A B see [5 growth result of [ Lemm Let p P [ A B The for z r < r wth ( ( B ( AB r A B B A B r Re p( z zp ( z ( Br ( ( AB rabr The result follows drectly by usg Lemm 3 3 M Results Theorem 3 Let f ν [ AB α b B < A > C { } The ( (! ϕ wth b α < ( ( ( (3 b A B α d ϕ s gve by (8 Ths result s shrp Proof Set D f z p( z (3 b b D f ( z so tht p P [ A B α Let p( z The (3 c be wrtte s q z D f ( z D f ( z bd f ( z q z whch mples tht ϕ ( q ϕ q b ϕ q (3 Usg Lemm we obt b( AB( α( ϕ ϕ ( ϕ ϕ ϕ ( For ϕ ( (! ϕ ( Therefore (3 holds for Assume tht (3 s true for m d cosder m ϕ m ϕ mϕ j ( m ( m! ϕ m mϕ j j m mϕ j m ( (! Therefore the result s true for Usg mthemtcl ducto (3 holds true for ll Ths result s shrp for > α < b C { } d s c be see from the fuctos f ( z whch re gve s Az Az Bz D f z Bz α α b b D f z For dfferet vlues of A B α b d we obt the followg results [6 Corollry 3 If f ν [ α R ( α the lfescecej@gmlcom

4 Lfe Scece Jourl ;9( Ths result s shrp ( ( (! α Corollry 33 If f ν [ α V ( α ( ( α the! Ths result s shrp Theorem 3 For rel b > ν [ ABα b ν [ β b β ( β z E < s oe of the roots of b ( ( ( ( α [ λ λ λλ α b B B ( B ( b β ( b [ ( B ( A ( b( β λ Δ ( b β ( b [ ( B ( A ( b( β λ Δ d Δ ( b( ( β b( ( α Proof Suppose f ν [ AB α b p z d set D f ( z b b D f ( z (33 (3 (35 (36 p s lytc E wth p ( The by smple computtos together wth (36 d ( yeld zp ( z 3 D f ( z ( pz b b D f ( z p z (37 b b ( b ν α Sce f [ AB b 3 b t follows tht zp ( z 3 ( pz P[ AB α p z or equvletly ( α zp ( z α α p( z 3 Defe ϕ ( z [ pz P AB z ( ( z ( ( z z (38 d by usg covoluto techques gve by Noor [3 we hve zp z zp z pz p z p z p z By usg (38 we see tht ( 3 3 zp ( z p ( z p ( z 3 p zp ( z ( z 3 α p ( z P[ A B α α z E Now we wt to show tht p P[ A B β β ( β < s oe of the root of (33 Let p ( z ( β h ( z β The α ( β α ( β zh ( z β h ( z P[ A B β 3 α h ( z ( β Usg the fct llustrted ( we hve lfescecej@gmlcom

5 Lfe Scece Jourl ;9( ( λ h ( z ( h ( z ω B ωzh ( z ( A ( h ( z ω ( λ h ( z ( h ( z ω B ωzh ( z ( A ( h ( z ω P β α ( β 3 β β α ( β α u h z v zh ( z ω ω λ d We ow form the fuctol ψ ( uv by choosg d ote tht the frst two codtos of Lemm re clerly stsfed We chec codto ( s follows ( A ( u ω ( A ( u ω B λ u u ω ωv ψ ( uv B λ u u ω ωv ω ( B v ( λ ( ω ω ( B v ( λ ( ω λ u B A u λ u B A u λ ω [ λ( B ( A λ ω [ λ( B ( A Now d ( ωv u ( B ( λ ω ( ωv u ( B ( λ ω λ B A u λ B A u ψ ( u v Tg rel prt of ψ ( u v we hve ( v u ( B ( v u ( B λ ω λ ω [( λ ω ( B ( A [( λ ω ( B ( A u [ λ ( ω v u ( B [( λ ω ( B ( A u v u Re ψ ( u v As ω d fter lttle smplfcto we hve > > so pplyg A Bu Cu Re ψ ( u v (39 D A [ [ λ ω B λ ω B B ( B ω ( B ω λ ( B ( λ ω ( B ( λ ω ( AB ( A C ( B ( ω d D λ ωv u B λ [ [( λ ω ( B ( A u The rght hd sde of (39 s egtve f A d B From A we hve β to be oe of the roots of b ( ( ( ( α [ λ ( λ ( λλ α b B B ( B wth β < d lso for β < we hve B Sce ll the codtos of Lemm re stsfed t follows tht h P d cosequetly [ p P β Hece from (36 ν [ β b f By choosg the prmeters A B b d we obt the followg ow result proved [7 Corollry 35 Let f V ( α The f R ( β β s root of β ( α β wth β < whch s lfescecej@gmlcom

6 Lfe Scece Jourl ;9( β ( α α α 9 For α Corollry 35 we hve the followg well ow result [8 V ( C R S for z E Theorem 36 Let f ν [ AB b b > (rel d > b( < The D f ( z mps z < r oto covex dom r s the lest postve root of the equto r r r r r 3 wth < 3 A B B B A B ( AB AB d ( 3 ( A B Ths result s shrp Proof Sce f ν [ AB b D f ( z b( p( z the (3 D f ( z p P [ A B Usg the detty ( we hve from (3 ( z D f ( z b p z (3 D f ( z Logrthmc dfferetto of (3 yelds ( z( D f z ( D f z zp ( z p( z pz b( The we hve ( ( D f ( z z D f z Re Re p( z zp ( z pz d hece by usg Lemm 3 d Lemm ( ( D f ( z z D f z Re ( ( B r Re pz A B r ABr {( ( ( } ( ( AB r ABr ξ r A B B A B r B A B r ( ( ( AB r ABr ξ 3 r r r r 3 Re pz > provded 3 T( r r r r r ( > 3 3 A B B B A B 8 8 AB AB 3 AB d ξ A B r B A B B r We hve ( T > d T ( < Therefore D f ( z mps z < r oto covex dom r s the lest postve root of the equto T( r lyg For D f ( z such tht ( z ( D f z b p z D f p ( z ( AB z ABz B z we hve lfescecej@gmlcom

7 Lfe Scece Jourl ;9( ( ( ( D f ( z 3 3 z D f z r r r r ( ( ( AB r ABr ξ for z r Hece ths rdus r s shrp By choosg the prmeters A B b d we obt the followg ow result see [8 Corollry 37 Let f S The f mps z < r oto covex dom r s the lest postve root of the equto 3 r r 6r r wth r < whch s r 3 Ths s lso shrp Referces [ W Jows Some extreml problems for cert fmles of lytc fuctos A Polo Mth 8 ( [ Y Poltoğlu M Bolcl A Şe d E Yvuz A study o the geerlzto of Jows fucto the ut dsc Act Mthemtc Acdeme Pedgogce Nyíregyházess ( [3 K I Noor Applctos of cert opertors to the clsses relted wth geerlzed Jows fuctos Itegrl Trsform Spec Fuct (8( [ B Pchu Fucto wth bouded boudry rotto Isrel J Mth ( [5 S Ruscheweyh A ew crter for uvlet Fucto Proc Amer Mth Soc 9(( [6 A A Lupş O specl dfferetl superordtos usg geerlzed Sălăge opertor d Ruscheweyh dervtve Comput Mth Appl 6(( 8-58 [7 K I Noor S N Ml O subclss of qus-covex uvlet fuctos World Appl Sc J (( -9 [8 K I Noor M Arf O some pplcto of Ruscheweyh dervtve Comp Mth Appl 6( [9 S Lth S Njud Ro Covex combtos of lytc fuctos geerlzed Ruscheweyh clss It J Mth Educ Sc Techology 5(6 ( [ K I Noor O some tegrl opertors for cert fmles of lytc fucto Tmg J Mth ( [ K I Noor M Arf Mppg propertes of tegrl opertor Appled Mth Lett 5( [ M Arf K I Noor M Rz W Hq Some propertes of geerlzed clss of lytc fuctos relted wth Jows fuctos Abst Appl Aly vol ( rtcle ID 7983 pp [3 M K Aouf O clss of p-vlet strle fuctos of order α Iter J Mth Mth Sc ( [ S S Mller Dfferetl equltes d Crtheodory fuctos Bull Amer Mth Soc 8 ( [5 R Prvthm T N Shmug O lytc fuctos wth referece to tegrl opertor Bull Austrl Mth Soc 8 ( [6 K I Noor Hgher order close-to-covex fuctos Mth Jpoc 37((99-8 [7 K I Noor W Hq M Arf d S Mustf O bouded boudry d bouded rdus rottos J Iequ Appl vol (9 rt ID pp [8 A W Goodm Uvlet fuctos Vol I II Mrer Publshg Compy Temp Flord U S A 983 /8/ lfescecej@gmlcom