Southeast Asian Bulletin of Mathematics (2003) 27:
6e7703 ffi Bulletin of Mathematics @ SEAMS.2003
Some Sufficient Conditions for Starlike Functions Associated with Parabolic Regions V. Ravichandran Department of N{athematics and Computer Applications, Sri Venkateswara College of Engineering, Pennalur 602 105, India E-mail:
[email protected]
AN,IS N{athematical Subject Classification (2000): 30C45 Abstract. An analytic functions f (z) defined on lzl ( 1 and normalized by /(0) : "f'(o) : 1 is in the class So(o) if
0,
\ I |1 . *( -"r 't( i^ # ' -"i li; lrf'(r)
This class is related to the class of uniformly convex functions introduced by Goodman. We find conditions for the inclusion So(o) C H^(a) where ff^(r) is the class of all )-spirallike functions of order a. Also we study the operator gQ) -- z(f (z)lz)^t. Keywords: Uniformly convex functions, Starlike functions, Parabolic regions
1. Introduction Let A denote the class of all analytic functions defined in the unit disk A - 1 Let S*(a),C(o) and { r t l t l < 1 } a n d n o r m a l i z e db y / ( 0 ) : 0 : f ' ( 0 ) fi (") derrote the usual classesof starlike, convex and ciose to convex functions of orcler cv, 0 ( a { 1, respectively. The function f (z) is subordinate to F(z) in A if there exists an analytic function w(z) with ur(O) : 0 and l*(r)l < 1 F(w(z)) for lzl < 1. This is written as f(z) < F(r). Let suchtlrat f(r): - 1 < B < A ( 1 . T h e n f € , 4 i s i n S . [ A ,B ] i f z f ' ( z ) l f ( z ) < ( 1 * A z ) l l r + B z ) . Similarll./ is in the classfI^(r) of .\-spirallikefunctions of order o if
R"{ r,^3!!!} [
t'cos). z € a.
/(z) )
AIso let Si (p) be the subclassof A of functions satisfying u- { tt rl -, \ ' u r ' -| 1 1 <1-t. J\z) I
698
V. Ravichandran
The function / € "4 is uniformly convex (starlike) if for every circular arc ^l contained in A with center ( e A the image arc /(7) is convex (starlike with respect to /(()). The class of all unifornily convex functions denoted by UCV was introduced by Goodman [2] in 1991. Ronning l ] and N'Ia and \,{inda [3] independently proved that / € UCV if and only if
R"{'+'i8} 'l+3
.z ? 1).
Furtlrer Ronningf4] defined the class So of functions f e A for which
R"{+&}, +3-'l
holds for all z e A. It can be observedthat f e UCV if and only if zf ' e So. Let Q:{tu; lr-ll
l z f " 1 ' tz t en j > a * l r ; l / \4
n"{r+t!t)
r
Si m i l a rl y f e S o (o ) i f
n "{ 3 l : 2 } ' o * l ' { ' , ( ?- .1) l
l. /(z) ) | J(z) | -1 ( o < 1. Then the functions in these classesare univalent. Let We have proved earlier the following
Theorem1..1-. f e A'is suchtltatl+& 16lSuppose
- 1l <
then
: [' ,f lc),zalc stz) Jo C belongsto UCV. In this paper this result is extended. We need the following: Lemma I.2. If 3-a 2
(1)
then lrft(r\ t " J
|
\ - /
^
I rtz)
| -01
|
|
z
n
afla
()\
Some Sufficient Conditions
699
imph,es f e Sr(a).
- tl < +
In particularwehavef e Sr(a)if l+&
SinceEo is the minimum distancebetweenthe point (r,0) and the points on the parabolaRe(tr - a) : l* -11, the lemma1.2followseasily.
2. Main Results It is known[1]that Si(d) c H^(a) for 0 < 13( l and l^l < cos-11(r-{3)lG-a)]. Theorem2.L 1e . t_I < B < A< 1,0 < +P < 1 _ a ( 1 a n d , ( I _ a ) ( 1_ B')> B(A- Bh. If f e ,S.[,4, B], thenthefunct,ion g d,efi,ned,by
s ( z ) :' ( ! 1 4 \ " \
z
/
is r,n the classn^@) for
t r ts c o s - , (
)
The function
0 "+ rt \t.-) \- -\ rl r" o satisfies
forBlo
B)#
forB:0
+ 9 : : :"^:" !'
andhenc e f < S*[A,B].Also.r{rt l:t,
;:
A-B ;e -z7f '6( z: ) l - A B r-gz+r-oze"' (d e [0,2"D. Now (
-^rr^\)
Re r,^:!*i t
: , t - 1 ) c o\s,+) r l,fIt _c Ao sB
,
-
A_8.
1_ d
: ctcos)
when 0 : r - ). This shows that the bound on ,\ is sharp. We have the following corollary
C o r o l l a r y 2I.f2f . e S i ( p ) , 0 ( 1 3 < L , 0 < + + bas(z): z (9)' functions d,efined
< I a n d ,1 ) 0 , t h e n t h e
i, in the class H^(*) for
- p)) l ^ l < c o s - r( t ( 1t - o \ /
V. Ravichandran
700
Theorem2.1 followsfrom the followingtwo Lemmas2.3 and 2'4. L e m m a 2 . 3L.e t - r < B g definedby .functi,on
1,0< IFP
If f e S*|A,B),thenthe
/ t(,\\_l
g(z):'l:)
/
z sz n t h e c l a s sS . h ( A - B ) + B . B l . Proof. Sinceg(z) : , (_1?)t. *. have
1 4 4 - 1 - . ry '* t' : I J 4 . z € a g(z)
f(r)'
If f e,9. [A, B], then we have zf'(z) ., 1* Az 11+Bz f@ and hence
1 +lB + (A- B)llz zg'(z) * Az <\ r1 _ ^ yI '_ r _' ^L v+l 8 " -_ . l+Bz g(z) < l i m p l i e st h a t - l S i n c et h e c o n d , i t i o n s- 1 < B < A < 1 , 0 . + P 1 @ - B ) + B < I , w e h a v es € S * b @ - B ) + B , B ) .
< B < r
The following Lemma give us the conditions for the inclusion S.IA. B) C H ^ (.' ). Lemma2.4. Let-l
< D < C < 1, 1 - a ) D(C - Da) and C-D
o{p<1. If h e S* [C , D ), th e n h e H^(a) for
l^l
lzh'(z\ I-CDl l
\
I h(r)
/
t
.
/
_
C-D
1 - Dz I - D2 |
and hence
1",^ u4!*_a\r#lr=
70r
Some Sufficient Conditions This sliows that
Now
R "{ " ' ^! : 1 }
h ( z )) -
I
if
t *cos)
(I-cD)cos)-(9 2)>acos) L-D,
or equivalently
c
cos)>
,?
I-a-D(C-D*)
Solvingfbr ). we get the result. Theorem2.5. If -1 < a,13
s ( z:)' ( 1 9 ) ' \
z
/
is in the classSo@) for 0 < 1 S =. Proof. Since
:l:1I\z)
i - ! r z g ' ( z-) 1 ) . :€ A 1
e\z)
and / € ,So(a) we have l z n ' ( z -)
l;;
zq'(2,),zeA. I tl * 1- ?(i- a)< -g\,)
Hence
ze'G) l u : ? - r l +B-. ll ga(!z] ? ) - r l +r - r ( 1 -o1 a g t z ), z€ a lsG)
I
|
which showsthat g € So(0. Bytakinga:
1 3i n t h e T h e o r e m w e h a v e :F o r - 1 { a <
l andQ(1(
f ("\
feSo@)+r(:;re,So(a) A l s o i f- 1 < o ( 1 a n d 0
<*l
(1,then
f e s o ( '+) r J J ? ) " es o ( r - ( 1- a ) r ) .
1,
V. Ravichandran
702
Si n c ef (z ) i s i n u C V(a ) i f and onl y i f zf' (z) i s i s,S o(a)w e havethe fol l ow i ng r e s u l t :I f - 1 1 a , 0 ( 1 a n d f e S r ( o ) , t h e n
I,' (+)'0, is in the class UCV(|J) for 0 < 1 < #. Using this result with Lemma 1.2 we have the following T h e o re m 2 .6 . L e t -I { a ,l 3 < I and a > (1 + o)12 and R o be defi ,nedas' i n Equat'ion (1). If the Inequality (2) is sat'isfi,edthen
l,'(f)'0, i,s i,n the classUCV(13) for 0 S t < #. In particular if a:
\, we have the result:
J, ('i) dc rz
/
tt,-t,'1
is in the classUCV(0 for 0 ( ? ( ]j *h"tteverlzf'(r)lf Q) - 1l < (t-cr)lz. If a: If2 andC:0 in the aboveresultwe havethe following Corollary2.7. Suppose f e A i,ssuchthat l+&
- 1l < tnen f,
e \ z:) J1o',' f (l c )y ' a c b e l o n gtso U C V f o r 0 < 1 < 2 . This resultis obtainedin [6]with i :2. Theorem 2 . 8 .I f - 1 < a , g 1 1 a n d , O < 7 < #
I f f e , 9 r ( a, )t h e n
g(z):'(fJ4\' \ r,szn the classfr^ (d) i,f tan2 ^ < W
z
/
, ),1 < r 12.
The proof of the Theorem follows directly from the following: L e mma 2 .9 . If f e So (r), -1 ( a 1I,
W,),1
then f e H ^(B ) proui ,dedtan2),{
Some Sufficient Conditions
703
Proof. Let 0: {tu: R,etu to trr - zf'(z)lf Q) e Q. Wc want to determine the condition on ) such that Re{e'Aur} ) p cos,\. Since Re{et^tu} : cos)R,cur- sin )Imur the above conclition is ecluivalentto cos)(R,ezr- 0) >- sin )ImT.o. Since?r'€ O this condition is satisfied rl , , 2- 2 ( 1 - . ) t a n ) r r + ( 1 - a ) ( 1 * a - 2 0 ) > 0 where u : Imtu. This is satisfied if ( 1- . ) ' t a n 2 ) ! ( t - c r ) ( 1 which gives the tlesired result.
+a-2P) I
If we take ) : 0 in the above lernnta wc get the order of starlikcness for functions in the class So(r); irr other words Sr(.) c ,S*(li) for 13< (I + a) 12.
References [1] {huja, O.P.: Some results concerning spiral-likenessof a class of arralytic functions, D emonstrat'io m,athematics26 (2-3), 687-7 02 ( 1gg3). l2l Goodman, A.W.: On Uniformly Convex Functions, Anrt. Polon. Math. 56, 87-92
(1 e s)1.
t3] \Ia, W. and Niinda, D.: Uniformly convex functions, Ann. Polon. Math.57,165-77I-t ( 1 9 9 2) . [4] Ronning, l.: Uniformly convex functions and a corresponcling class of starlike functions, Proc. Amer. X,lath.Soc. 118, 189 - 196 (1993).15] R0nning, F.: A survey on uniformly convex function and uniformly starlike functions, Ann. Uni,u.M. Curie-Sklodowska47,123-134 (1993). 16] Shanmugary, T.N. and Ravichandran, V.: Certain properties of uniformlv convex functiols, Computgti.onal methods ctnd functi,on thiory, (Ali, R. N,I, Rusiheweyh, St. and Saff, tr.B (Eds)), world Scientific Publishing co., t11-1to (1995)
