NEIGHBOURHOODS OF A CERTAIN SUBCLASS OF STARLIKE FUNCTIONS. P. Thirupathi Reddy. E. mail:

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1 NEIGHOURHOOD OF CERTIN UCL OF TRLIKE FUNCTION P Tirupi Reddy E mil: reddyp@yooom sr: Te im o is pper is o rodue e lss ( sulss o ( sisyig e odio wi is ( ) p < 0< E We sudy eigouroods o is lss d lso prove eessry d suiie odio i erms o ovoluios or uio o e ( 000 Muje Clssiiio: 30 C 45 Keywords: Neigourood uordio Hdmrd Produ Iroduio: Le deoe e lss o uios o e orm () JGRM 03 ll Rigs Reserved

2 P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 - wi re lyi i e u dis E{: <} Furer le e e sulss o osisig o ose uios re uivle i E Le CV d T deoe e sulsses o osisig o ovex d srlie uios respeively I d g re y wo uios i su d g e e ovoluio or Hdmrd produ o d g deoed y g is deied y ( g) Clerly d ( ) Te lss o srogly srlie uios ws rodued idepedely y r d Kirw [] d iewi [4] Deiio : uio is sid o e srogly srlie o order α ( 0<α ) deoed y T( α ) i rg απ E () JGRM 03 ll Rigs Reserved

3 P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 - I erms o suordio T( α ) e reried s e lss o uios sisyig e relio α p Reely Nlisi e l [] geerlie e lss T( α ) y lss ( i sisies Z p < 0< E (3) Z ilvi [3] Te oio o T eigourood ws rodued y eil mll d Deiio : For 0 T d { } eigourood o lyi i E is deied y T sequee o o-egive rels T TN g : T (4) We se lemm wi we eed o eslis our resuls Lemm []: Le ( Te se o vlues lie i ellipse ( X ) Y were Te mjor d mior xes JGRM 03 ll Rigs Reserved 3

4 4 o e ellipse re give y 0 θ d 0 θ I is pper we rodue ew lss o uios d sudy e properies o eigouroods o uios i is lss wi geerlies e ree resuls o Nlisi e l[] Now we deie e lss s ollows Deiio : Le e e lss o ll uios () deied o E s ; R (5) Were d re s deied i Lemm Now we give rerio o e lss y mes o ovoluio Teorem : i d oly i 0 E d or some () JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

5 5 Proo: Le us irs ssume 0 d or some () d E ee we ve 0 Equivlely s vries desries ellipse 0 ee lies iside e ellipse or Coversely le e JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

6 P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 - ( ) R Equivlely 0 ( ) Normliig e uio w i e res we ge 0 i E were () is e uio deied i (5) To ivesige e T- eigouroods o uios elogig o e lss ( we eed e ollowig Lemms Lemm : Le () is i ( e σ were σ d σ 0i is eve i is odd Proo: Le () ( e or R JGRM 03 ll Rigs Reserved 6

7 7 3 Te omprig e oeiies o eer side we ge is odd we is eve we Hee we is eve σ were σ we is odd JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

8 8 is σ σ s we ve we is eve d JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

9 9 we is odd Lemm : I or every ε ε < < we ve ε ε ε F e or some E Proo: Le ε F e y Teorem 0 F ε d E Equivlely 0 ε ε or ε ee JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

10 0 Teorem : I or every ε ε < < we ve ε F e TN were /γ d γ Proo: Le g is i d TN e g g > γ γ γ 0 or JGRM 03 ll Rigs Reserved P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 -

11 P Tirupi Reddy Jourl o Glol Reser i Memil rives () Novemer 03 - Reerees: D r d WE Kirw O some lsses o ouded uivle uios J Lodo m o() (969) L Nlisi J Tgmi d R Prvm O sulss o uiormly srlie uios oues si ull M (997) T eil-mll d EM ilvi Neigouroods o lyi uios Jourl d lysis memique 5(989) J iewi ome emrs oerig srlie uios ull d Polo i er i m 8(970) Deprme o Memis Gpy Egieerig CollegeHuer RodRgsipe Wrgl (P) INDI JGRM 03 ll Rigs Reserved