.lournal oflvluthematical und.Phl,siculScience:li,Ittl.-1(). Ntt.6.De.cemhu 199(t Printed in India
3A7
On Starlike Functionswith Respectto N-Ply Conjugateand SymmetricConjugatePoints M. S. KASI Department of lvlathernatis, l,oyala College, Madras, INDIA
V. RAVICHANDRAN Deparlrnent of Mathenntils, Anna lhrivenity, Madra-s-600025, INI)IA.
Abstract The classof starlikefunclionswith respectto n-ply corrjugate and syrunetriccorrjugatepoirrlsare of functionsarealsocon^sidered. introducedandstudied.Sorrrerelatedclasses
l. Introduction Let
A denotethe class of all analytic fuuctions "f (z) defined on the unit disk
ry =lz,lzl< l) with the normalizations "f(O)=0,/'(0)= l.
Let 5' denote the classof all
in A. Let .S*(a),((a) and (:(a) denotethe usual classes of starlikc. univalentfunctions convexand closeto convexfunctionsof order a 0 < q, < I respectivelydefinedby
.5'*(a)= lf el; Re #
> u,z e(-il
K(a)= lf el; Re{l.4r-l!_.', > u.z e(J} l'(z)
Cl(c,)= Lf e"4;Re19> o, for sorneg € K. z eLl\ g'(z)
M. S. KASI and v. RAVICIIANDRAN
A function .f e I is starlikewith respectto symrnetricpointsin [/ if
,l'' tr) x"[ ], o.z e(r. (z).f(-z)) lf (4). A functiolrf e A is starlikcwith rcspect This classis introducedandstudiedby Sakagrrchi to conjugatepointsin t/ is / satisfiesthe condition
n " { . ' f '-.f (2= }tn, z eu. lf (z) (z)) A function ./ e I is surlike with respectto symmetricconjugatepointsin ( I if it sntishcs I zf'tz\ Re{4'
I -}>0' z e(!'
l.f (z)- .f(-z)J
Denotetheclasses consisting of thesefunctions Thcscclasscs UVSi and5l respectively. are introducedby El-Ashwahand Thomas(l). Thesefunctionsare close lo convex. Sokol(fi) introduced two niore pararrreter in this classand obtainedstructural fornrurla,tlrc coclliciorl estirnate,the radiusof convexityand resultsaboutthe neighbourhoods of functions. Slrannrugarrr and Ravichandran(5) also studied the classof starlikefunctionswith respectlo syrnnrctric points.
In this paperwe introducethe classof functionsstarlikewith respectto n -ply conjugate points and certain relatedclassesof functionsand obtain structuralfomrula, distortion theorcnr and convolutiontheorems.
2. IDefinitionsand Preliminaries
Inthefollowing definitio\rs, n > l,wn = I and 0
lcl
On Starlike Functions with Respect to N-Ply (.lonjug
.109
n-k (^k,). J',Q)= 1'2t r f ll k=O
Definition2.1 : A function f e A is starlikewith respectto n -ply conjugatepointsof order u if / satisfies
zrf,(r\ I > u , , zeU.
(
Re{+'=
(l)
f
lf,(z)+f,(z)J Denote the classof all suchfunctionsby,S';'''" The functiottsstarlike with respectto r - fold qymmetricpoints having real coclTicicnrsarc obviouslyin this class. Definition2.2 : A frrnction.f e,4 is convexwith respectto n -ply conjugatepointsof order cx,if ./ satisfies
r ) I )ef'e)\' L, ne{*t>u.. l
-
r
-
-
:
-
l
(2)
ze(t.
l,/,(z)+ ./ nQ)J Dertote the class of all such fuuctions by Klo The firnctions which are convex with respect to n - fold symnretric points having rcal coefficients are in this class.
Da./inition 2.3 : A function f e A is closeto convex rvith respectto n -ply cortjugatcpoittts of order u, correspondingto the function h e ('n'() if ./ satisfies
I z/''(r) I _ : _ l > ,r.
R e { -;-::-
z e(i .
(3)
lh,(z)+hr(z)J Derrotc the classof all such functionsby C:'" .
.l o u r.A I q I h. P h t .Sc i.. Vol. 3 (), N o. (t. L)ec cttnh (x' I 996
M. S. KASI and V. RAVICHANDRAI.f
310.
rvith respectto n - fold symntetricpoints having real coefficients The frnctions close-to-convex are in tltis class.
It .f (z)=z+ZLz enz'
and g(z)=z+2I:2 hrz', are analyticin f i. then tltcir
convolutiondenotedby (.f * S\Q) is defined by (f*g)(z)=z+l
arhnzn
n=2
we needthe following lernrnasto proveour nrainresults. : t2l If g(z)eA isstarlikeint/ and.f e,I . thenfor ft(z) convc\in f/ rvith LEIv,fit,L42.1 h(0) = l' we ltave ' f (z\ < h(zl implies19- < h(z\. z e( I . ' ' gQ) g'(z) A rrroregeneralresultis in [2] and the aboveresultis sufficientfor our purpose.Lct /t,. denote the classof all prestarlikefilnctionsf (z) definedby theconditiort . .f .'R.- if andonlYif -f *eS* (l
(a).
z\z-u
*(a'). F e A- tlten LEI'llv[A 2.2 : [3] For q. ( l, let f € Ro,g eS
( r*..n\ l+lucco(F({/)). .N \ "r
/
3. Main Results
THEORE|vI3.1 : If f(z)..s';.'.o.
then the frurction ./(z) is close-to-collvexof ordcr (r atrd
ttnivalent.
.l our.Al ath.Phy.Sci., Vol.3 0. N o. 6. [ )ecemher I 996
( )n Starlike Fhnctronsv,tlh Respeclto N-Ply ('orlugtttc un.l S.l',l,nrctric('onjugcttePoints
[ ' r o o j ' : l t . / ' . S ' ; ' ' ' o . t h e n( l ) h o l d s .R e p l a c ez b y i : k z . k = ( 1 . 1 . 2. .,. , n - t
3| |
i n ( l ) : r r r da d d a l l t h c
incqualiticsto get
I zr/';ra "' I} > r r . z e ( t . I t, ct*7,
Rc{
= .fr(z) and rn-t S6r-1rz)=Tttl. wlrereu'ehaveusedfhefactthat ot-k.f'n(altz)
Tlrissltorvs
1./'r(z)+7r (x. Rc "/ h'(zl /'tlz
This showsthat f is close to convexand hcncefunctionsstarlikervith respccllo coldugatc pointsof order u areunivalent.
T'HEOREII3.2: (i\ .f .,(:'"
andif z1''esi'n'" .
(ii) ff;'' c,\'''''o Proo.f".Part(i) followseasilyfrom thedefinitions.To provc(ii). let .f e Kf,'o and LI (z) = ,.f '(z) a1d N (z) ,rif.^.. t nen.
J''(z) +-7'(i) . 2
tvl'(z) _l+(l-2u)z
N'(z)
l- z
and n- is convex.Henceby Lentnta(2). it follorvsthat z.f'(2 = _AI(z) -l+(l-2q.\z r- z + . f, ( z ) f n Q ) N ( z ) This cornpletes tlte proof. It canbe provedsimililrly that the frurctionsin ('f,'o areclose to couvexattdltettccuttivalettl. .l our -lrt qt h.Phl'..\'cr.. l'o l. 3 0. N o. 6. I.)c.ctrtnhc r I 99(t
M. S. KASI and V. RAVIC:I{AI\I)RAN
312
THE7REIvI3.3: lf .f .,S':'''o, then 2u)"
l-(l(l+rn7
@41 n
l+(l- Zur" 2(l-n)*,
'
(l- r'1
"
2(l-a)
'
(l-rn1
The results are sltarp.
Proof:Let f -5':'''". Definep(z) trndg(z) by
p(z)--W '
.f,(z)+ .f ,(z)
T hen p( z ) - l + c z n+ .... w i th
f ,(z)*7 ,Gl
and g(z) =
Re p(z) > a
and g(z) = z + oz'*l +.. .
Therefore. we have
?!:e) (l+rn1
n
2(t-lc)
(l-rn1
n
and This showsthat p(z) lies in the disk whosedianteterendpointsat l+(l-
2u)rn
l-(l-2u)rn l+rn
l-rn
Sincezf'(z) - p(z)g(z) we havethe results. The resultsare sharpfor the function/ obtainedfrorn zf ' (z) - l+ (l-2s.)zn
f (z)
l- zn
J ottrM ath.P hy. Sci., Vol.3 0, N o. 6, December I 996
is starlikc o[ ordcr rr'
On StarlikeFunctions with Respectto N-Ply (lonjug,otean.l S)mmetric (lonlugate Pttintt
Sirrcefr(z)=f(z)
and f(z\
is starlikeof order u
.r l.l
rvith rcal coeffrcieuts. *'c scc that thc
frrnctionf (z) isin sj'"'o.
Sokol[6] obtainedthe distortiontlrcorernfor thecasen = l. hrt his resultis ftrlse. Tlris is duc to l?rct tlut l./(z) *7 ttllt 2 is notan oddfunction.In factthefrrnctio n I.f Q) - .f (-z)U 2 uscdlor sirnil:rr pointsis indeedan oclclfunctiou. rvithrespect theoremin thecaseof starlikefrurctions to syrrunetric
THEORfuI 3.4 : A function J' eS'*'''ct if and only if thereexistsa fiuictiorr p(z') iuralytic in U . with p(0) = l, Re p(z) > 11satisffilg
z I "n' ( q ) * V ' " ,' "( i ) - z ttn)dq. .f(z)=Ip!)exp(:t ts n o Proof: Suppose that / =.S';'''o Let p(z) bedefined by
=-'1ry: P(z)
.fne)+f nQ)
z4!it? +J''!in rlren. p,(z)*V,
and hence
f ,ei+7,G)=.*Jli P,(q)+ I nG)-z 'Lz; 2z
,']
:
This showsthat
re)=p(,,-{iiryrrl From this, the resultfollows. The otherpart of the Theoremfollows similarly. ' For n = | and q, = 0. we havethe resultof El-AshwahandThomasil1. JourA{ath.Pfu,.Sci..Vol.3(),lr1io.6. I )e.;ctnher 1996
M. S. KASI ard V. RAVICHANDRAN
THEOREIv|3'.5:Let f .S*'n'o and g eR,, hasreal coeffrcierrls, then /*g
e,s,l'''t'
Proo^/' . Sincef . Si,'''". thefunctionG(z) defiiredby
(i(z)- J'G) +7 '(i) 2
is analyticand starlikeof order u,. Define F(z) by Zzf'(z)
f , ( z ) +f rn( ez)) ralytirc al -(1. Tlten F (z) is analytic artd Re F(z\ > a. Sinceg e- -R,,
$
'e have by lenrnm L1. L.!.
(g * ( i['')(z\ -
(g * { i)(z)
liesin the closedcollv rivalent ival llt to convexhull of F(( t\. But this is equ
z'(f *il'e) 2z
I
--
Re{
1
.,
}>u.ze(t. * l(./'*s\,G) +(f x't,Q)|
This cornpletesthe proof. Since ,( c Ro. we have the following. o n v o lution u l i n'ith conl'e.rf u r t c l i o r t s l t i t v i n g ('OROLLARY 3.1 : The class ,s:'"'o is closedunder' ccol real coefficlents. ;oel :nts rs nccessar!'.T l r i s u i l l l r c c l c i t r coefficicnt It should be noted that the assuntption(l(z) ltas real <
fronr the followittg exiuuple. Let J'Q) = z l(l- z)2 aud 0(z)= z /(l-iz).
Then /
'(z)
e ,\* ' l ' o and 0(z )
is convc)i brrt
( . / * 6 Xz ) i s n o t i n s j .r' O.
It can be shown that this class is closedunder certain integral opcrators. Also llrc class K:'"
is closedunder convolutionsrvith ftlnctions in R,, having real coefficieltls. Tlris follous
easily frour the above theorenr and the fact that "f conve.\ rvith respect to rr - trrlv corliugalc .lottr.lluth.Phy.Sci..
lbl.30. No.6, Decanher
| 996
On Starlike Functions with Resryct to N-P11,(.'onjugate and Synnetic ('onlugate Points
3|5
if and only lf zf is starlikervith respectto ,?- ply corliugalcpointsof
points of order u orderu,.
THEORE,t 3.6: The class( f'o is closedunder convolutions with frrnctionsin /(,, havingreal coefficients.
THEORE4 3.7 : lf f .s:'n,o, then
F @ = : i .f@+7(a) ds is alsoin ,\j'''o . Proof : The function (f n(z)*7 ,till l2 is starlikeof order alpha and has real coeffrcietrts. Also
t|
R'i*
'tt,7fl
l
>u,. 43-l=*"{'u'c,!<2-nl
l F n G \+ F , G l l
By lenu
r
t f n ? ) + f, ( z \ J
2.1. rveltavethe resu
The classesof ftlnctions cousistingof starlike, convex rvith respectto n - fold sytttntetric conjugatepointsciurbe introducedandthe similartheoremsfor thesenewclassesarc iltdccdtntc. Sincethe definitiousand the theoremsabout the nervclassesare similar to thoscof this paper. '
tve onrit the details.
Acknowledgments
The researchof V.R. is supported by the Council of Scientitrc and Industrial Research,New l)clhi.
IIc is
also thanlitirl to T.N. Sharunugarntbr his guidarrceduring the preparation ol the paper.
tSt11, J ou r. L t u l h. P h.y.Sc i., I/o1.3 (), N o. (t. L) ec e tn lt.!v 1
316.
M. S. KASI ard V. RAVICHAI.IDRAN
References l. 2. 3. 4. 5. 6.
El-Ashwah, R.M. and D.K. Tlroluas, Some subclassr:sof close to convc\ litttclirttts, J. RamanujanMath..Soc.,2( I ), 85-l00 ( 1987). Porurusamy,S., SomeApplicationsof tirst order differential subordinations,(ilnsrtih ltlutlrcntalic'ki, 25(45),287-296( l e90). St. Ruscheweyh,Covolutions in Geonretricl'turctionstheory, Sen. lvlatlt .Sap.,83, I''rcssercdc Montreal,Montreal,( I 982). Sakaguchi, K., On CertainUnivalentmapping,J. Math. Soc.ofJayttr, ll(l), 12-15(1959). C.hlthe radiusof univalencyof certainclasscsol'analytio Sharunugam, T.N. and V. Ravichandran, fuirctions,J.lv{ath.& PhysicalScf.,28(1994),(to appear). Sokol, J., Functions starlike with respect to corrjugatepoints, T,eszylyNaukos'c ltrlitcchniki 12,5344 ( l99l ). Rzeszowskiej,
.l our.A4at h.P hy.Sci., lbl. 3 U,N o. 6, Decenh er I 996
