On A Subclass of Harmonic Univalent Functions Defined By Generalized Derivative Operator

Transcription

1 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: N. D. Sgle Dertmet of Mthemtics, Asheb Dge College of Egieerig, Asht, Sgli, (M.S) Idi Y. P. Ydv Dertmet of Mthemtics, S. B. Ptil Polytechic, Idur, Pue, (M.S) Idi 4306 ABSTRACT I the reset er, subclss of hrmoic uivlet fuctios is defied usig geerlied derivtive oertor d we hve obtied mog others results like, coefficiet iequlities, distortio theorem d covex combitio. 000 AMS subject clssifictio: Primry 30C45, Secodry 30C50. Keywords: Uivlet fuctios, rmoic fuctios, Derivtive oertor, Covex Combitios, Distortio Bouds. INTRODUCTION A cotiuous fuctio f() is sid to be comlex-vlued hrmoic fuctio i simly coected domi D i comlex le C if both Re( f ) d Im( f ) re rel hrmoic i D. Such fuctios c be exressed s h( ) g( ) (.) where h () d g () re lytic i D. We cll h () s lytic rt d g () s co-lytic rt of f(). A ecessry d sufficiet coditio for f() to be loclly uivlet d sese-reservig i D is tht [] O A Subclss of rmoic Uivlet Fuctios Defied By Geerlied Derivtive Oertor Let S ' ' h ( ) g ( ) for ll i D. be the fmily of fuctios of the form (.) tht re hrmoic, uivlet d oriettio reservig i the oe uit disk U { : }, so tht h( ) g( ) is ormlied by f (0) h(0) f (0) 0. Further h( ) g( ) c be uiquely determied by the coefficiets of ower series exsios. where h( ), g( ) b, U, b, (.) C for,3,4,... d b C for,,3,... We ote tht this fmily S ws ivestigted d studied by Cluie d Sheil-Smll [ ] d it reduces to the well-kow fmily S the clss of ll ormlied lytic uivlet fuctios h () give i (.), wheever the co-lytic rt g () of f() is ideticlly ero. Let S deote the subfmily of S cosistig of hrmoic fuctios of the form h( ) g ( ) Where b U b = = h()=+, g ()=(-),,. (.3) For h( ) g( ) give by(.), we defie the derivtive oertor itroduced by Shqsi d Drus [8] of f() s, 56 Pge

2 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: D D h( ) ( ) D g( ), (.4) m, m, m, where m, D h( ) ( ) C( m, ) m Dm, g( ) ( ) C( m, ) b, b, C( m, ) C. m Defiitio: The fuctio h( ) g( ) defied by (.) is i the clss S (, m, k,, ) if D m, Dm, Re k (.5) D m, Dm, where 0 k, 0. Also let S (, m, k,, ) S (, m, k,, ) S (.6) We ote tht by seciliig the rmeter, esecilly whe 0, S k (, m, k,, ) reduces to well-kow fmily of strlike hrmoic fuctios of order. I recet yers my reserchers hve studied vrious subclsses of exmle [],[3],[4],[6]d [8]. S for I the reset er we im t systemtic study of bsic roerties, i rticulr coefficiet boud, distortio theorem d extreme oits of foremetioed subclss of hrmoic fuctios.. MAIN RESULTS Theorem: Let h( ) g( ) be give by (.). If coditio k k k k C m, C m, b where (.), 0, 0 k, N 0, the f() is sese-reservig hrmoic uivlet i U d f S (, m, k,, ). Proof: If the iequlity (.) holds for coefficiets of h( ) g( ) the by (.), f() is oriettio reservig d hrmoic uivlet i U. Now it remis to show tht f S (, m, k,, ). Accordig to (.4) d (.5) we hve D m, Dm, Re k D m, Dm, which is equivlet to A () Re B () 563 Pge

3 where Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: A( ) k D kd m, m, d m, Usig the fct tht, Re( w) B( ) D if w w it suffices to show tht A( ) B( ) A( ) B( ) substitutig vlues of A() d B() with simle clcultios we led to (, ) k k C m k k C( m, ) b (, ) k k C m k k C( m, ) b k k C( m, ) k k C( m, ) b k k C( m, ) k k C( m, ) b 0. By ssumtio. ece roof is comleted. The fuctios x y k k k k where x y (.3) shows tht the coefficiet boud give (.) is shr. Theorem : Let h( ) g ( ) be so tht h () d g ( ) give by (.6). The f S (, m, k,, ) oly if if d k k k k C( m, ) C( m, ) b, (.4) where,0,0 k. 564 Pge

4 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: S m k S m k For oly if rt, we Proof: The if rt follows form Theorem with the fct the (,,,, ) (,,,, ). show tht f S (, m, k,, ) if the coditio (.4) is ot stisfied. Note tht ecessry d sufficiet coditio for Let f h g give by (.6) to be i S (, m, k,, ) is tht which is equivlet to D m, Dm, Re k D m, Dm, Re k Dm, k Dm, Dm, k k C( m, ) k k k C( m, ) b Re 0. C( m, ) k C( m, ) b The bove coditios must hold for ll vlues of, r. Choosig o ositive xis where 0 r. we hve k k C( m, ) r k k k C( m, ) b r k (, ) (, ) C m r C m b r 0. (.5) or equivletly if the coditio (.4) dose ot hold the the umertor i (.5) is egtive for r sufficietly close to. Thus there exists 0 r0 i (0,) for which the quotiet i (.5) is egtive.this cotrdicts tht required coditio for f (,,,, ) S m k d hece roof is comleted. Theorem 3: Let f be give by (.6). The f S k, ; ( ) ( ) f x h y g if d oly if 565 Pge

5 where, ( ), Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: h h ( ),,3,4,... k k g ( ),,,3,... k k x 0, y 0, x x y 0. S m k re I rticulr, the extreme oits of (,,,, ) Proof: Let The ( ) ( ) f x h y g h d g. k k x y x k k y k k k k b x y x d so f S (, m, k,, ). Coversely, suose tht f S (, m, k,, ). d Settig 566 Pge

6 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: k k x,,3,... k k y b,,,3,... where x y we obti f xh( ) yg ( ) s required. Theorem 4: Let f S (, m, k,, ) the for r we hve k k f( ) b r b r k k k k d k k f( ) b r b r k k k k Proof. Let f S (, m, k,, ).Tkig bsolute vlue of f we obti b r b r b r b r k k k b r b r k k k k k k b r b r k k k k b r b r k k k k k k k b r br. 567 Pge

7 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: The forthcomig result follows from left hd iequlity i Theorem.4. Theorem 5:The clss of S (, m, k,, ) is closed uder covex combitio. Proof: For,,3,... i suose f S (, m, k,, ) f b i i i the by Theorem i where k k k k b i i. (.6) For ti, 0t i i, the covex combitio of f my be writte s i t f t t b ( ) i i i i i i i i i hece by (.6) k k ti i i k k ti bi i k k k k t i i i i i ii t i d therefore t f S (, m, k,, ) i i i This comletes the roof. 568 Pge

8 Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue ISSN: REFERENCES. O. P. Ahuj, R. Aghlry d S. B. Joshi, rmoic uivlet fuctios ssocited with k- uiformly strlike fuctios, Mth Sci. Res. J.,9 () (005), J. Cluie, T. Sheil-Smll, rmoic uivlet fuctios, A. Acd. Sci. Fe A. I. Mth., 9 (984), J. M. Jhgiri, rmoic fuctios strlike i the uit disc, J. Mth. Al. Al., 35 (999), J. M. Jhgiri, Y. C. Kim d. M. Srivstv, Costructio of certi clss of hrmoic close-to covex fuctios ssocited with Alexder itegrl trsform, Itegrl Trs d Sec Fuct., 4 (003), J. M. Jhgiri, G. Murgusudrmoorthy d K. Vijy, Slge tye hrmoic uivlet fuctios, South J. Pure d Al. Mth., (00), Al-Oboudi, F. M. O uivlet fuctios defied by geerlied Slge oertor, IJMMS, 7(004), G. S. Slge, Subclss of uivlet fuctios, Lect Notes i Mth. Sriger-Verlg, 03 (983), K. Al-Shqsi d M. Drus, Differetil Sdwich theorems with geerlied derivtive ortor, Iter. J. Com. Mth. Sci.,, () (008), Silverm, rmoic uivlet fuctios with egtive coefficiets, J. Mth Al Al., 0(998), S. Ylci, O certi hrmoic uivlet fuctios defied by Slge derivtive, Soochow J.Mth., 3 (3) (005), Pge