Convex

Certain Properties of Uniformly Functions T. N. Shanmugam and V. ltavichandran

We investigate the classes of normalized function" IQ) analytic in the Abstract. ", tl,t$ * I ranger over the open unit disk'Q - {, ,l"l < l} for which # parabolic region O = {tr : lur - 1l < Reur}. We also discuss some intagral operators and radius problems.

' 1

Introduction

'

Let A denote the class of all functions f (z) analytic in the open unit disk D= tz ,lzl< 1) with /(0) = /'(0)-1=0. LetS'and Kdenotethesubclasses of .,4 consisting of starlike and convex univalent functions respectively. In [21 Goodman introduced the class UCV of uniformly convex functions. Ma and Minda [3] and Ronning [6] independently proved that / € UCV if and only if

""#+1>l#l

(1.1)

for all z € D. Further Ronning [6] defined the class So of functions t e A for which

k+8,1#-l

(1.2)

holdsfor all z € D. It can be observedthat / e UCV if and only if zt' e Sr. Let Q = {w: lur- ll < Reu}. It followsthat (1.1)and (1.2)areequirnalent ""d {t} are in O respectively.Note that O is a to saying that 1 + {i:} parabolicregionsymmetricwith respectto the real a:cisand (f ,0) is its vertex. The function,t(z) with k(0) = k'(0) - I = 0 and.

W*1-?,[^'(#)]'*'

(1.3)

is an example of function in UCV [3].

2

Integral Operators

We discuss some integtal operators under which the class UCV is invariant.

rnd Frrnctiol Thcorv (CMFT'91) Mathork Cornprrirtlonrl enrl E. B. Setf (Edr.). PP 3l9-t2{ R. M. Ali. St. Rrrrehcrcyh Co. Ptc. Ltd. CopyrlSht @fO06 by Wotld Scientific PtblirhinS in eny form tcrctvcd All riShtr of reptorltrrtion tsBN 0tl-o2-21t0-o

319

T. N. Shonmlnotn

and V. Rouichottdran

Tlreorem I Let /, e UCV and let o, b! real nnnlser.s srrclt that a; ) 0 and DL,0, ( l. Then the function tz

n

sQ)- JoflUirclt"'ac also belongs to UCV. Proof: SinceieUCV,

^ " 2gg, ,(,z\+'))"t > i ^ l r f i ' k ) l =l r g " ( r ) l ?",1IMI'l$l' Henceg€UCV. I The upper bound DL, or S I is the best possible.For the functions f ;(z) = lc(z) where lc(z) is given by (1.3), we have

r

il!i

f;

n

4!'Q),, r j:T+L=r'

forz = -1. If Il=, o, ) l, thenf /UCv. Corollary t Let At,Az,...,An be complex numbers such that l,a,l S I for d = 1 , 2 , . . . , n . L e t 0 S a , 1 l a n d D L , d , 1 1 . T I r c nt h e f u n c t i o n

f'+ Id( g(z)-1il ro ,=l(l A,q'Yn''--

f,^l 1 t l'l tl f4 lrt i . t

:i

r g _ i r * )=*l t_ : i r , 2 s,v)

alsobelongsto UCV. l o r i = L , 2 , . . . , n . S i n c el A , l S i , / , e U C V f o r e a c h P r o o f :I ' e I f ; ( z ) - t r (see[2]). Also t!Q) = 6_-+;t". An applicationof TheorenrI with thesef (z) yieldsthe corollary.f

1l

; ;

Theorem2 Suppose f e Ais such thatl#-

'l

g(z)-l'(f) (K

: ,, ,,

ll < i

belongsto UCV.

Unifonnly

Proof:

Conucr

g?,l

Functions

Since

z g " ( z_ ) rlzf'(z)

-_ 1r 1j 76,.="Lfe) ' wrr h:rve

' ; '| l i €1 t2'0\" ( ' \ l I ' i ; l l

n " z s : . ' ,+( 2l >, ) t - l Y " ( ) 1

tt'tz) ,

3

Radius Problems

we find under uoriiu, contlitions on / e ,4 the largest Il such that (1.2) is satisfied for lzl < I?. We call /? the .9n radius. Let P(cr) denote the class of all analytic functionsp(z) definedon D, rvith p(0) = 1, satisfying.Rep(z)) c for all z € [0. Denote P(0) bV P. Let a> l. Let lio : min{lro -ol : lro - ll = Reu}. A sirnple computation gives -l I7 '""

( a- I 2

ir

t vt ;=2

if c ) f

i."
Now D(o, /?") = {ut : lw - ol < /?"} is the largest disk centerc
l ' f ( ' )- a|l< /"r " |n for all z € D, then f e Sn.

Lerrrrrra2 (t5l) lt QQ) is analytic for z € lD arrrl lOtr,lt 1I

l z 6 ' k ) + t h ?.- ).1l I

t+ zQQ) |

l-r

br z €D, tlten

f t r r l z=l r ( l '

Lentrrra3 ([4]) Srtpposcthat 7\z) = 1 * {:,,2"* ... is nnalytir:iuxl Rcp(z) > 0 for z € lD. ?fien

ll d P l , , , ' , , ' ' , u[ r , r l=z rl ( l . p(r)l-l-':

322

T. N. Slnnmrgom and, V. Roatchondron

Theorem 3 ([U)

If p(z) is in P(o), then

lP(')' r

| + (I _ 2a)r2 . ,(t _ o)r |

r-r'

fr^o rr.l z=l r ( 1 "

ljt:;t-

A function t e A is called close-to-starlikeol order o e [0, lJ if there exists a function g starlike of order a with

n"/!"1r o. 9\z)

The class of all such functions is denoted by CS(o). g(z) = z is the only function starlike of order l.

See [1]. When o = l,

Theorem a The.9o radius R for the class CS(o) is

n - lzlz- o) + vtGL rza+ r4-l Proof: Lel g(z) be a starlikefunctionof ordera with l{z) = IE , n. :4+ b in P(a) and therefore,by Theorem3, we have -o)r lrg'(") l+(l -2a)r2l-z(t

lM and

=l-,,-

Tl

I ' H( 4 1

zr

lh(,)lsr-r

Thereforewe have ltl'(r)

I + ( l - 2 o ) r 2 l- z ( z - o ) r

lT6ffi-l:",-l

= r-",'

This representsa circular disk intersectingthe real axis in

l-2(2-a)r+(l-Za)rz l-12

and

r, =

| + 2(2 - c)r + (1 - 2a)r2 l-12

Forr=RrvetrauesFd_}andforrlesstharrthisva|tte,the disk lies completely inside the parabolic region Q. Note that the fitnctiott The result is sharp for the function f (r) - f#-. g(z) with I of Theorem the hypothesis satisfies p=fr=2.t. f /(z) lf we take g(z) - z in Theorem l, the So radius can be inrproved. Lct .9,, > O denote the classof all analytic function f (z) = z * a,.zn*... with nc f forz€D.

.:ii t{

:

323

Unitormly Conuex Ftnctions

t

h t)

Theorem 5 Tle So radius for the class.9,.is

l'/4,1.r-^f"". Tfiis result is sfiarp.

Proof: Since ,U4"

e P, from Lemma 3 we have

$

lrf (r)

l1d

tt;(z)l . 2rlrn - ', 1l :_ l |tz 1 . ' d=|i 7 .

satisfiesthe Tlre result followsfrom Lemma1. The functionf(z): # of Theorem5. Sincen" {1,$ = l++l - lI tor z = exP(f )r, the Sp lrypothesis I is sharp. radius

- ll < I for z ED, Theorem6 It f € A andg € .5'(o),rr e [0,lJ rvitl l#\ then the.9oradiusR of f (z) is -l

-'

n=[r-2o*zJr+(l-rr)2J Proof: Using Theorem 3 iurd Leltttna 2 it' is scclr t'lrat lrl'(r\

It@

-

1 2+ ( 3 - - 2 r r ) r . 1 * (l - 2rr)r2 | .

r-r2

l-

r-rz

(2) tt t*:=+l3)I the parabola This circularcliskrouches

= |. Thisgives

parabola the value of .R. For r less than this value the circular disk is inside the the = which satisfies function for the sharp is IQ) result O. Tlre {a\*, = I with lrypothesis 9(z) 6-,i'r=m.

Acknowledgements. The authors arc thankful to the refereefor his su88estions and comments. The secondauthor wishesto thank the Corrncilof Scientific and In
.n, ,-. - \ l

:

'

J

2

4

'l'.

N. Shanmugorn ond, V. Rouiclmndran

References [tJ Goodman, A. w., (Jnivalent Flrnctions, Yol. 2, polygonal publ. House, Washington, New Jersey,1983,97. [2f Goodman, A. !v., on uniformly convex furrctions, Ann. polon. Math.56 (1991), 87-92. [3f Ma, w. and D. Minda, uniformly convex functions, Ann. polon. Math 5T (1992), 165-175. [al MacGregor, T. H., The radius of univalenceof certain analytic functions, Proc. Amer. Math. Soc. 14 (1963),514-b20. [5] MacGregor, T. H., The radius of univalenceof certain analytic functions, II., Proc. Amer. Math. Soc. 14 (196J), SZI-L}4. [6] Ronning, F., Uniformly convexfunctions and a correspondingclassof starlike functions, Proc. Anter. Math. Soc. lf 8 (lgg3), 189-196.

T. N. Shonmugom Department of Mathematics Anna University Madras 600 025, India [email protected] V. Rauichandron Department of Mathematics Anna University Madras 600 025, India