Ganita,Vol. 53.no. 2,2002,117-.124 ON UNIFORMLY CONVEX FUNCTIONS V. Ravichandran We studythe classUCV of uniformly convexfunctionswhich was introduced clas by A.W. Goodman.We introducea classthatunifiesthe UCV andthe colresponding of starlikefunctionsSointroducedby RQnning.We provethat the new classis closed with prestarlikefunctionsof ordero. Someintegraloperatorson Soare underconvoluti<.rn A sufficientconditionfor functionsto be uniformlyconvexis given. discussed.

1. INTRODUCTION Let A denotethe class of all analytic functions f (z) defined on the unit disk bV/(0) - 0,,/'(0): l. Thefiurctionf eA is uniformlyconvex U: {r;lzl< l} normalized (starlike)if for every circular aray containedin U with center(eU the imagearc /(y) is convex (starlike with respectto fg). The class of all uniformly convex (starlike) functionsis denotedby UCV (UST).Note that ( r" (z)'l

(l.l)

(1.2)

/ e U C V < + R re+J@ - O # (f z ) )> 0 , z , ( e U , / r

-q)f: (z) 11, / e UST<+ Re z ) - f (6)l

17

=o, z,(etJ.

Theseclasseswereintroducedand studiedby A.W.Goodm^[?,3]. He remarked that the class UCV is preservedunder the transformatione-'" 11e'"21and no other of transformationseemsto be available.Using the following onevariablecharacterization preserve LICV. other which it is to obtain transformation easy UCV, Theorem l.l. f4,7] Let f eA. Then/eUCV if andonly if

(I .3)

' n.{r+'f'' (')I ,ltf' (')l. r . u. I

f'(z) J I f'(z) I

SincetheAlexandertyperesult/eUCV if andonly if zf eUST fails [9], the class (l .4)

Sp: {f ; f : zF'oFeUCV}

introducedby F. R{nning [7] becameinteresting.

tt7

Let f and F be analytic in U. Then f is subordinateto F (written /< F or : f(z) < F(z)if /(0) F(0)and/(U) g F(U).If f (z),g(z)eA and & n z,n g ( z ) - r * t

f ( z )- t * i

b n z,n n=2

n=2

thentheir convolutionis the function(/*g) (z)eA givenby U*g@)=z+i

anbnzn'

of starlike We introducebelow . ,r^, of functionsthat unifies severalsubclasses andconvexfunctions. Definition 1.2.Let h(z) be a convexunivalentfunctionwith h(0) - l, Re h(z) ) o ) 0. Let g(z)eA be a fixed function. Denote by Se(h)the class of all analytic functions f eA satisffing (1.5)

,\Ii e)'.(?) U . g@)_*o and < h(z). r\r * (f

z

g(z)

lf g(z) - zl (l-z)2 and h(z) : P(z), where P(z) is the Riemann mapping of U onto the parabolicregionC): {w; Rew> lw- ll}, then Se(h)= UCV. Similarly,if g(z)= zl | z and h(z) : P(z), then Se(h)= Sp. If (1.5) holds only for lzl ( r ( I then we say that f eSe(h)for lzl < r. Note that the classR, of pre-starlike functions of order a < I consistsof functions/eA satisfying 'f *

z-^.,eS*(o) (l - z)z-ze

*"/(z),I z 2

foracl

fora=l

= where S*(a) denote the class of starlike functions of order a . If g(z) zl (l-z)2-2o *6 h(z): (l + (l-2o)) zJ(l-z),then Seo): Rrl [l l] . Theorem2.

?,\.

,y /*(Hg),

If/eR'anrd-q.S*(cr),thenforanyanalyticfunctionH(z)inU,tr(U)eCo(H(U))' theclosedconvexhull of H(U). denotes where eo 111111)) In this paper, we prove that the class Se(h) is closed under convolution with prestarlike functions of order cr. We discusssome integral operatorson So.Also we give

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sufficientconditionfor a functionto be uniformlyconvex. 2. MAIN RESULTS Theorem2.l.lf feSr(h)and{eRo, then0*/eSe(h). Proof. Since/eSr(h), it followsthat z ( g *f ) , ( z )

G *f i @ ) .

< h '( z ) .

Clearly(g * f)(z)eS*(o).Let (?) H(z)='@:- f)' ' @* f)(z) 1.3,it followsthat UsingTheorem [ 0 * H ( e * / ) ] ( z< ) h(z).

[0*(e*f)](z)

Since we have

z(s*i* f)' (z) - [0* z(g* /)'](z) - [0* H(g* /)](z) )

(e*0*D@)

t0*(e*f))(z) t0*(e*f)l(z)

z(g*i* I)'@) < h(z).

( e * 0 *f i @ )

which showsthat Q*/eSr(h). and h(z):(l+(l -2u)z)l(l-z),we It g(z)is equalto zJ(l-z),2/ (l-z)2 respectively see that the classes of convex and starlike functions of order cr are closed under convolution with prestarlikefunctions of order a. If g(z) : zl (l-z) and

2l' -(l*ftr' h(z)=l+

#L'"{#))

thenwe havethe following theoremof R$nning: Corollary 2.!. tSl If f(z) is on Soand g(z) is starlikeof order Yz,then(f*g(z) is in So again. Similarly,it follows that UCV is closedunderconvolutionwith starlikefunctions of orderl/2. Sinces*rn : Rt/2,we havethefollowing

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Corollary 2.2. The classSe(P)is closedunder convolutionwith starlikefunctionsof orderll2. Corollary 2.3.Let f r0(z)) : zI'(z) I

f ,(f (z))= i|f @)+ zf'(z)J

=+f (*-'f G)de, Rek>o f,(f(z)) rou@))f (u-'ffior',

Inl
= Thenf;(/(z))eSe(P)in lzl < r; wheneverf(z)eSe(P),whererr:l 13,12:(,117-3)12 0 . 5 6 1 5 5r r, : t 4 : l . Proof.For eachi : I ,2,3,4 we havef i:ffu)*hi(z)where z h ', \( 'z ) = (l-z)' h- t.\,-(/ z = ) '-

ttL

(l _ z),

h , ( z ) = pr : l ' " l-nn ,n. h',\h' ) = i ?" (l-q)n Since the radius of starlike of order l/2 for the functionshr and h2 are ll3 and (./tZ-:)12 respectively, the resultsfor i: 1,2 follow. Sinceh: and h4 are convexand hencestarlikeof orderl12 , theresultfollows. Corollary 2.4.UCV c Sp. proof. Let f e UCV. Then,by Corollary2.3,g(z): f, POr, :

f (z) - zg'(z) e Sp.

Theorem 2.2.If f and g are in So,then the function I

,r'-tdr]L z ' 120

is in UCV. Therefore,

y > 0, cr,> 0 is alsoin So. Proof. Note that the classPAR definedby PAR- {p: I + cz*..,; lp- ll
M(z)- z' {f (z)}"e(z)- T {/(t)}"e(t)t7-'dt f

Let

N(z)= f Utr))"g(t)tr-'dt. Thenstraightforwardcomputationshowsthat zH'(z) _ M(z) H(z) (o + l)N(z) and

M'(z) = I lorf(r)*r€@)1. (".DN()-".1L* f@ e @I e PAR.ByCorollaryI in [6], we Sincef ,geSo andPARisconve*,-% (o + l)N'(z) seethat

. (cr+Y!1lt l)N(z)

e PAR. Thiscompletes theproof.

'

i

2.s.rrfe A andlWTheorem il. ], tr,"n/ e So(P). l U * g ) ( z ) I 1 Proof.Letp(z) = I +

ir.By

hypotheses,/ = So@).Sincep(z)< P(z),wehave/ e Sr(P).

Corollary2.5.Letf eA. (i)

"l#l .;, then/eucV

(ii)

.l#-tl '1, th,n./'So'

It shouldbe notedthat the aboveconditionsare also necessaryfor a function f to

t2l

be in UCV (or So)whenf(z) is a functionwith negativecoeffrcients.Sincethe classesSo and UCV are rotationally invariant,from Corollary 2.4 andtheseremarks,w€ get the followingcorollaries. Corollar! 2.6.I31f (z): z - A* is in UCV if andonly if lAl < 116. Corollar;u2.7.U\ I@):z-

is (i) in Soif if andonlyif lA"l sll(2n-l).

^f

(ii) in UCV if andonly if lA"' F ' +. n(2n-l) Corollaty 2.8. t3l f (z): zJ(l - At)'if andonly if lAl < ll3. 2 . 4 . L e t .f e A $ n = l , o ) l . L e t I , @ ) = | X ' Theorem

t - uf ( e ' z ) . T h e n

( '

(i) rrRet*,-Qf'=(?)-l=0, _f^(6)J thenf neusr. ( ,z-S)f,,_(z)l=0, then (ii) If Relr * g f neUCv. (z) /'n

L

)

(iii) ' " {'t'gl--,} = *'{?'j:]}'

L/"(") J

'so' then.rn

tf"@))

proof. We prove (i). Replacing z by ekz and e bV e*6 in the given condition and then summingover k : 0o1,2,3,..., n-1, we get the desiredresultupon simplification.The proof of (ii) and(iii) aresimilar. Let Do,n,0< cr< I denotetheclassof analyticfunctionsq(z) I + cnzn+... and

I

r l

I

letz)-;ls.,^. Note that for cr: 0, this classof firnctionswith positiverealpart Uzl. Thenwe havethe following ( Theorem2.5.Let|(z)eD*n, then/eUCV in lzl f0,wherereis givenby

r22

Proof. Let q(z) : f'(z). Since q(z)eD",n, ws have [12]

lrq'@)l- (l+c)nlzl"

lt(4 l=(t-El\l+clzl)' By Corollary2.5,f eUCV if

.t. l'f"(')l f'(z) 2 I 49 since

q(z)

I

-zf='.(?),wehave / e ucV if f'(z\

(l+c)nlzl" .l ( l - l z l "( l + c l z l " ) 2 or

c/n + [(2n+ l) + (2n- l) c]rn- I < 0, r: lzl

which is satisfiedif 0 < r ( rs. The resultis sharpfor the function

f(z)=f ffioe REFERENCES problemsin tU. Barnard,R.W.andKellogg,C., Applicationsof convolutionoperatorsto univalentfunctiontheory,MichiganMath.J.,27 (1980),8l-94.

.

155 (1991)' IZl. Goodman,A.W., On uniformlyconvexfunctions.J. Math. Anal. Appl., 364-370. 57 (1991), t3l. Goodman,A.W., On wriformly convex functions,Ann.Polon.Math. 87-92. 57 (1992), t4l. Mq W. and Minda D., Uniformly convexfunctions,Ann.Polon.Math. 165-175. M. Subclassof Uniformly starlikefunctions.Internat. t5l. Merkes,E.D. and Salasmassi, J. Math.Sci.15 (3),(1992),449454-

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Glasnik S., Someapplicationsof first orderdifferentialsubordinations t6]. Ponnusamy, Mathematicki,(5945),I 990, 287-296 t7l. Ronning, F., Uniformly convex functions and a conespondingclass of starlike functions,Proc.Amer.Math.Soc.118(1993),189-196. tSl. Ronning, F.; On starlike functions associatedwith parabolic regions. Ann. sectA, XLV; 14 ( 1991),ll7 -122. Univ.M.Curie-Sklodowska, t9]. Ronning,F., Uniform starlikeandrelatedpropertiesof univalentfunctions,J. Compl. Variable,TheoryAppl. 24 (1994),233-239. tl0].Ronning, F., Someradiusresultsfor univalentfunctions,J, Math.Anal.Appl. lg4 (1995), 3t9-327. U l].Ruscheweyh, S., Convolutions in geometric function theory, Seminaire de de I'Uniersitede Montreal,(1982). 83, Les Presses " Superieures, Mathematiques Il2).Shaffer, D.B. Distortiontheoremsfor a specialclass of analytic functions,Proc. . Amer.Math.Soc.,39(2),(1973),281-287

Departmentof Mathematics& ComputerApplications Collegeof Engineering Sri Venkateswara - 602105(India) Pennalur,Sripermubudur

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