Neighborhoods of Certain Class of Analytic Functions of Complex Order with Negative Coefficients

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1 Ge Mth Notes Vol 2 No Jury 20 pp ISSN ; Copyriht ICSRS Publitio 20 wwwi-srsor Avilble free olie t Neihborhoods of Certi Clss of Alyti Futios of Complex Order with Netive Coeffiiets Abedshor S Teim Deprtmet of Mthemtis Alhr Uiversity-G POBox 277 G Plestie E-mil: bedshor@yhooom Reeived Aepted 0200 Abstrt I this pper we prove severl ilusio reltios ssoited with δ - eihborhood of erti sublsses of lyti futios of omplex order with etive oeffiiets by mi use of the fmilir oept of eihborhoods of lyti futios Speil ses of some of these ilusio reltios re show to yield ow results Keywords: uivlet futios strlie futios ovex futios δ - eihborhood d lier opertor 2000 MSC No: 30 C 45 26A33

2 87 Neihborhoods of erti lss of lyti Itrodutio Let A deote the lss of futios f of the form : f 0; N :{ 23} whih re lyti i the uit dis U { :: C d < } Followi the wors of Goodm [9 d Rusheweyh [ we defie the δ -eihborhood of futio f A by see lso [2 [3[4 d [3 2 N δ f A : b d b δ I prtiulr for the idetity futio e we immeditely hve 4 N δ e A : b d b δ The bove oept of δ -eihborhoods ws exteded d pplied reetly to fmilies of lytilly multivlet futios by Altits et l [6The mi objet of the preset pper is to ivestite the δ -eihborhoods of severl sublsses A of ormlied lyti futios i U with etive d missi oeffiiets whih re itrodued below by mi use of the Rusheweyh derivtives A futio f A is sid be strlie of omplex order C \ {0} tht is * f S if it stisfies the iequlity: f 5 Re > 0 U; C {0} f Furthermore futio f A is sid be ovex of omplex order C \ {0} tht is f if it lso stisfies the followi C f iequlity 6 Re > 0 f U; C \ {0} * The lsses S d C stem essetilly from the lsses of strlie d ovex futios of omplex order whih were osidered erlier by Nsr d Aou see lso[5 6 Next for the futios f j j 2 ive by 7 f j 2 j

3 Abedshor S Teim 88 Let f f 2 deote the Hdmrd produt or ovolutio of f d f 2 defied by 8 f f 2 : 2 2 : f f Now we defie the futio φ ; by 9 φ ; For 0 2 ; where λ is the Pohhmmer symbol defied by Γ λ ; 0 0 λ Γ λ λ λ 2 λ N { 2 } Crso d Shffer [ itrodued lier opertor L by L f φ ; f Where * stds for the Hdmrd produt or ovolutio produt of two power series Defied by ϕ ϕ d ψ ψ ϕ ψ ϕ ψ ϕ ψ We ote tht L f f L2 f f Filly i term of the Crlso d Shffer[defied by let S λ deote the sublss of A osisti of futios f whih stisfy the followi iequlity : L f 2 < L f U; C \{0} ; 0 < Also let R λ ; µ deote the sublss of A osisti of futios f whih stisfy the followi iequlity : L f 3 µ µ L f < U; C \ {0} ; 0 < ; 0 µ Vrious further sublsses S d R ; µ with were studied i my erlier wors f e[7 ; see lso the referees ited i these

4 89 Neihborhoods of erti lss of lyti erlier wors Clerly i the se of for exmple the lss S we hve 4 S S d S C N; C \ {0} 2 Ilusio Reltios Ivolvi N δ I our ivestitio of the ilusio reltios ivolvi we shll require Lemm d Lemm 2 below e N δ e Lemm Let the futio f A be defied by The f is i the lss if d oly if S 2 Proof We first suppose tht f S the usi oditio 3 we et 22 L f Re > L f U or equivletly m 23 Re > U Where we hve mde use of 0 d the defiitio We ow hoose vlues of o the rel xis d let throuh the rel vlues Th the iequlity 23 immeditely yields the desired oditio 2 Coversely by pplyi the hypothesis 2 d letti we fid tht L f L f

5 Abedshor S Teim Hee by the mximum modulus theorem we hve f S whih ompletes the proof of Lemm Similrly we prove the followi Lemm Lemm 2 Let the futio f A be defied by The f is i the lss R ; µ if d oly if 25 [ µ Remr A speil se of Lemm whe d 0 < ws ive erlier by Ahuj [ Our first ilusio reltio ivolvi N δ is ive by Theorem below e Theorem If 26 δ < the 27 S N δ e Proof For futio f S of the form Lemm immeditely yields so tht

6 9 Neihborhoods of erti lss of lyti 28 O the other hd we lso fid from 2 d 28 tht < tht is 29 : δ < Whih i view of defiitio 4 proves Theorem Similrly pply Lemm 2 isted of Lemm we prove the followi Theorem Theorem 2 If 20 δ : µ the 2 R ; µ N δ Proof Suppose tht futio f R ; µ is of the form The we fid from the ssertio 25 of lemm 2 tht µ Whih yield the followi oeffiiet iequlity ; e 22 µ Mi use of 25 i ojutio with 22 we lso hve

7 Abedshor S Teim µ µ µ µ tht is δ µ : whih i liht of the defiitio 4 ompletes the proof of Theorem 2 Remr 2 By suitbly speilii the vrious prmeters ivolved i Theorem d Theorem 2 we derive the orrespodi ilusio reltios for my reltively more fmilir futio lsses see lso Equtio 5 d Remr bove 3 Neihborhoods for the lss S d R I this setio we determie the Neihborhoods for eh of the lss S d R whih we defie s follows A futio A f is sid to be i the lss S if there exist futio S suh tht 3 ;0 < < U f Aloously futio A f is sid to be i the lss ; µ R if there exist futio R suh tht the iequlity 3holds true Theorem 3 If S d 32 [ δ the

8 93 Neihborhoods of erti lss of lyti 33 N δ S Proof Suppose tht f N We the fid from the defiitio 2 tht δ 34 b δ whih redily implies the oeffiiet iequlity: 35 δ b N Next sie S we hve by [fequtio28 36 So tht b f < λ b b δ 37 provided tht is ive by 32 thus by defiitio f S for ive by 32 This evidetly ompletes our proof of Theorem 3 Theorem 4 If R d 38 δ µ ρ [ µ the 39 N δ R Remr 5 By suitble speilii the vrious prmeters i Theorem 3 d Theorem 4 we derive the orrespodi eihborhood results for my reltive more fmilir futio lsses s i [8

9 Abedshor S Teim 94 3 Suborditio Theorem Before stti d provi our suborditio Theorem for the lss S we shll me use of the followi defiitio results Defiitio 4 For two futios f d lyti futios i U we sy tht the futio f is suborditio to i U deoted by f p if there exists futio W lyti i U with w 0 0 d w < U suh tht f w Defiitio 42 A sequee { b } of omplex umbers is lled suborditio ftor sequee if wheever f is lyti uivlet d ovex i U the 4 b p f U i b is suborditio ftor sequee if d oly if 42 Re 2 b > 0 U Lemm 4 fwilf [8 the sequee { } Theorem 4 let f of the form stisfy the oeffiiet iequlity 25 the 43 f p 2[ < < > 0 U N \ {} for every futio i the lss of ovex futios I prtiulr 44 Re{ f } the ostt ftor 45 [ 2[ U i the suborditio result 43 ot be reple by y lrer oe Proof let f be defied by the oeffiiet iequlity 43 of our Theorem will hold true if the sequee

10 95 Neihborhoods of erti lss of lyti 46 2[ Is suborditio ftor sequee whih by virtue Lemm 4 is equivlet to the iequlity 47 U > 0 2[ 2 Re Now [ Re } [ [ Re r r } [ [ Re 0 [ 0 [ Re > Hee 47 hold true i U whih proves the ssertio The prove of { } [ Re U f for S f follows by ti To prove the shrpess of 2[ Cosider the futio q whih is member of the lss S d

11 Abedshor S Teim 96 Thus from reltio 4 we obti q p 2[ It be esily shows tht mi Re q 2[ 2 This show tht the ostt is best possible [ Referees [ O P Ahuj Hdmrd produts of lyti futios defied by Rusheweyh derivtives i Curret Topis i Alyti Futio Theory HM Srivstv d S Ow eds pp 3-28 World Sietifi Publishi Compy Sipore New Jersey Lodo d Ho Ko 992 [2 OP Ahuj d M Nuow Neihborhoods of lyti futios defied by Rusheweyh derivtive Mth Jpo [3O Altits d S Ow Neihborhood of erti lyti futios with etive oeffiiets Itert J Mth Ad Si [4 O Altits O O d HM Srivstv Neihborhoods of lss of lyti futios with etive oeffiiets Appl Mth Lett [5 OAltits O O d HM Srivstv Mjoritio by strlie futios of omplex order Complex Vribles Theory Appl [6 OAltits d HM Srivstv Some mjoritio problems Assoited with p-vletly d ovex futios of omplex order Est Asi Mth J [7 PLDure Uivlet futios A Series of omprehesive studies i mthemtis Vol259Sprier-Verl New Yor Berli Heidelber dtoyo983 [8 AW Goodm Uivlet futios d olyti urves Pro Amer Mth so

12 97 Neihborhoods of erti lss of lyti [9 SRusheweyh New riteri for uivlet futios Pro Amer Mth So [0 S Rusheweyh Neihborhoods of uivlet futios Pro Amer Mth So [ H Silverm Uivlet futio with etive oeffiiets Pro Amer Mth So [2 MA Nsr d MK AOuf Strlie futio of omplex order J Ntur Si Mth [3 H S Wilf Suborditi ftor sequee for ovex mps of the uit irle Pro Amer Mth So [4 H Silverm Neihborhoods of lsses of lyti futios Fr Est J Mth Si