Coefficient Inequalities for Certain Subclasses. of Analytic Functions

Transcription

1 I. Jourl o Mh. Alysis, Vol., 00, o. 6, Coeiie Iequliies or Ceri Sulsses o Alyi Fuios T. Rm Reddy d * R.. Shrm Deprme o Mhemis, Kkiy Uiversiy Wrgl , Adhr Prdesh, Idi reddyr@yhoo.om, *rshrm_005@yhoo.o.i Asr I his pper, e irodue eri su lsses o lyi uios d sudy he Fekee-Sego iequliy or he uios i hese lsses. Ceri ppliios o hese resuls or he uios deied hrough ovoluio re lso oied. Mhemis Suje Clssiiio: 0C5 Keyords: Alyi uio, suordiio, Fekee-Sego iequliy, ovoluio. *Correspodig Auhor. I. Iroduio Le A e he lss o uios ormlied y (.) hih re lyi i he ope ui disk { C : <} D. Le S e he sulss o A osisig o uios hih re uivle i D.

2 77 T. Rm Reddy d R.. Shrm For uios d g, lyi i D, e sy h he uio is suordie o g i here exiss Shr uio () lyi i D ih (0) 0 d () < ( D) suh h () g(()) ( D) e deoe his suordiio y p g ( D) I priulr, i he uio g is uivle i D, he ove suordiio is equivle o (0) g (0) d (D) g (D) d Re ϕ > hih mps he ope ui disk D oo regio srlike ih respe o d is symmeri ih respe o he rel xis. Le ϕ e lyi uio i D ih ϕ( 0), ϕ ( 0) > 0, { } 0 * Le S ( ϕ) d ( ϕ) A sisyig p ϕ C e he sulsses o A osisig o uios d p ϕ ( D) (.) respeively. These lsses ere irodued d sudied y M d Mid[]. These lsses ere urher geerlied y Rvihdr, Pologlu, oll * d Se []. They hve irodued he lsses S ( ϕ) d C ( ϕ) o e he lsses o uios A or hih p ϕ d D p ϕ respeively, (.) here is o-ero omplex umer. Reely Rmhdr, Sivsurmi, Srivsv d Smih [] irodued d sudied he lss M,, ( ϕ) o e he lss o uios A or hih ( ) p ϕ (.)

3 Coeiie iequliies 77 Here 0 <, 0, 0 d D. All hese uhors hve oied he Fekee-Sego iequliy or he uios i hese lsses. They oud eri ppliios o hese resuls or he uios deied hrough ovoluio d riol derivives. Moived esseilly y he ove meioed orks, i he prese pper e irodue eri su lsses o lyi uios d oi he Fekee-Sego iequliy or he uios i hese lsses. Ceri ppliios o hese resuls or he uios deied y ovoluio re lso oud. The iequliy or he iverse uio is lso disussed here. The resuls o his pper uiy d geerlie severl erlier resuls i his direio. Deiiio (.): Le ϕ e uivle sr like uio ih respe o hih mps he ope ui disk D oo regio i he righ hl ple d is ϕ 0 d ϕ ( 0 ) > 0. A symmeri ih respe o he rel xis, ih uio A is sid o e memer o he lss ( ϕ) M i,,, p [ ϕ ] (.5) Here 0 <, 0, 0 0 <, d is o-ero omplex umer. I is oed h i) M ( ϕ) ( ϕ),,, M deied d sudied y,, Rmhdr, Sivsurmi, Srivsv d Smih []. * ii) M ( ϕ) S ( ϕ) d M ( ϕ) ( ϕ),0,,0,0,,0 Rvihdr, Pologlu, oll d Se []. * iii) M ( ϕ) S ( ϕ) d M ( ϕ) ( ϕ),0,,0 d Mid []. iv) M ( ϕ) ( ϕ ),,,,,0,, C deied d sudied y C deied d sudied y M M deied d sudied y Rvihdr, Drus, Hussi Kh d Surmi [].

4 77 T. Rm Reddy d R.. Shrm Deiiio (.): For ixed lyi uio g g ( g > 0) e deie M,, g ( ϕ) g M ( ϕ) *,,, A, o e he lss o ll uios A or hih here * deoes he ovoluio eee o uios. I is oed h M ( ϕ) M ( ϕ),,,,,,,,. Deiiio (.): Le e lyi uivle uio i M ( ϕ). A uio d ( r ),,, iverse o i ( ) < D 0. is sid o e he Here r 0 is greerh he rdius o he Koee domi or he lss ( ϕ) To prove our resuls e require he olloig lemms. Lemm : [] I... M.,,, P is uio ih posiive rel pr i D he or y omplex umer v, e hve {, v } v Mx. d The resul is shrp or he uios P () give y P P. Lemm : [] I... P is uio ih posiive rel pr i D he or y rel umer v. v v v ( v 0) ( 0 v ) ( v )

5 Coeiie iequliies 775 he v < 0 or v > he equliy holds i d oly i P or oe o i s roios. I 0 < v < he he equliy holds rue i d oly i P or oe o is roios. I v 0 he equliy holds rue i d oly i P ( 0 ) or oe o i s roios. I v he equliy holds rue i d oly i P () is he reiprol o oe o he uios suh h he equliy holds rue i he se he v 0. Alhough he ove upper oud is shrp, i he se he 0 <v <, I e urher improved s ollos: v v 0 < v d v ( v) < v. Fekee-Sego i equliy or M ( ϕ),,, I his seio e oi he Fekee-Sego iequliy or he uios M ϕ. i he lss,,, Theorem (.): Le... re re rel. I ( ϕ) M,,, ϕ ih > 0 0 d s he or y omplex umer, e hve r mx, [ ] ( ( ) ) (.) here [ ( ) ] ( ( ) ) [ ( ) ] is shrp. d he resul

6 776 T. Rm Reddy d R.. Shrm Proo: Sie ( ϕ) rom deiiio (.) y suordiio priiple M,,, here exiss Shr s uio () i D ih (0) 0 d () < suh h ( ) [ ϕ( )], D Deie uio P() suh h P ( ) ϕ... ( ) ( ) ( )... (.) Replig (), simpliiio, e ge d ih heir equivle expressios, er ( ) ( ) ( ) [ ( ) ] [ ( ) ] ( ) ( )]... ( ) ( )... (.) Comprig he oeiies o d o oh sides e se d [ ( ) ] (.) [ ( ) ] ( ) ( ) ( ) ( ) ( )] [ ( ) ] (.5) Deie oher uio P () suh h

7 Coeiie iequliies P here P () is uio ih posiive rel pr d lyi i D ih P (0). Solvig () i erms o P () e ge ϕ As ϕ P e hve Comprig he oeiies o d o oh sides, e ge s (.6) d (.7) From equios (.), (.5), (.6) d (.7) e ge (.8) d

8 778 T. Rm Reddy d R.. Shrm (.9) here For y omplex umer osider v (.0) here v The resul (.) o ollos y ppliio o Lemm (.) o he equio (.0) d he resul is shrp i.e. i P I P. Cosiderig s rel umer d leig e prove he olloig heorem y pplyig Lemm (). Theorem (.): I ϕ,, M he or y rel umer d or

9 Coeiie iequliies 779 σ σ σ d or e hve i σ i σ σ i σ (.) Furhermore I σ σ he d i σ σ he (.) d hese resuls re shrp.

10 780 T. Rm Reddy d R.. Shrm Proo: Sie ϕ,, M he rom equio (.0) ih e hve v (.0) here v y pplyig Lemm () o he R.H.S. o he equio (.0) e ge olloig ses. Cse (): I σ he. Aer simpliiio, e ge v 0 v -v v (.) Cse (): I σ σ he Whih o simpliiio redues o 0 v v (.) Cse (): I σ he

11 Coeiie iequliies 78 Whih implies h v v v v (.5) From equios (.0), (.), (.) d (.5) e ge he resul (.) Cse (): Furhermore i σ σ he hih implies h 0 v v v Cse (5): I σ σ he hih implies h v v v

12 78 T. Rm Reddy d R.. Shrm hih proves he resul (.) Shrpess: I < σ or > σ he equliy holds i d oly i I σ < < σ he he equliy holds good or i σ he he equliy holds good i d oly i 0 I σ he equliy holds good i d oly i Theorem.: I () φ,,, M d () 0 r d < is he iverse uio o he or y omplex umer, e hve, mx d d (.6) here d r 0 is greerh he rdius o he Koee domi or he lss ϕ,,, M

13 Coeiie iequliies 78 Proo: As (.) e hve - (() d is he iverse uio o rom deiiio - ( d ) ( d d )... Comprig he oeiies o d o oh sides e ge d 0. Ad d d 0 For y omplex umer, d - d - d -. d ih - (.7) d The resul ollos rom equios (.) & (.7) d is shrp or P() d. Reerees. M W. d Mid. D: A uiied reme o some speil lsses o uivle uios: I proeedigs o he oeree o omplex lysis. Z Li.e F. Re, L-yg, S. Zhg. I. Press 99.. Rmhdrm.C Sivsurmy. S Srivsv H.M d Smih. A. Coeiie iequliies or eri sulsses o lyi uios d heir ppliios ivolvig he o Srivsv operor o riol lulus mhemil iequliies d ppliios Pre pri.

14 78 T. Rm Reddy d R.. Shrm. Rvihdr, V. Pologlu, Y olol d Se. A: Ceri sulsses o srlike d ovex uios o omplex order. Hepe Jourl o Mhmis d Sisis, (005) 9-5. Reeived: Augus, 009