Far East J. Math. Sci. (FJMS) 12(1) (2004), 4l-5I
CERTAIN APPLICATIONS OF FIRST ORDER DIFFE RENTIAL SUBORDINATION V. RAVICHANDRAN ( ReceivedJuly 31, 2003 )
Submitted bv Saburou Saitoh
Abstract For the function f(z) = z * a222 + "' defined on I zl.
l, we find h(z)
o"2S'1ry11z)+ zf'(z)lfQ) < h(z) implies
zf'(z)lfQ) < q(z)
such that
^^dk$-r#.he)
impries
1. Introduction
and Defrnitions
Let A,, be the class of all functions f(") = z * a,ralzrt'+l* "' which andlet A1 -- A. A function f(z)e A is are analyticin L -{z; lrl.t} starlike of order 0, 0 ( cr < 1 if
ou({'G)) , o \ f(z) ) for all z e A,. The class of all starlike functions of order a is denoted by S-(o). We write S-(0) simply as S*. Recently Li and Owa [3] have proved the following: 2000 Mathematics Subject Classification: 30C45. Key words and phrases: starlike function, differential subordination. O 2004 Pushpa Publishing House
42
V. RAVICHANDRAN Theorem L. If f(") . A satisfies
' ""{#("r"8.')}
c[
e A
g '
L/
for some o (o > 0), then f(z) e 5.. In fact, Lewandowski et al. [2] and Ramesha et al. [8] have proved weaker form of the above theorem. If the number - alz is replaced by -o2(I-c-)14,
(o < cr < z) in the above condition, Li and Owa [3] have
proved that f(z) rs in S-(cr/2). Later Ravichandran et al. [9] have proved the following: Theorern
2. If f (") . A,, satisfies
.[u-+], zeL, ""{#0T8.')i'"p[p.+-'] (0 . o, p < 1), then f("). S-(B). Also Nunokawaet al. [5] refined the results of Rameshaet al. [8]. Recently Padmanabhan [7] has improved his earlier results in different direction. In this paper, we obtain sufficient condition for a more general class of starlike functions
s -\"./ ( h=) { r e A t 4 9 . L'
f @)
n(')}. )
When h(z) = (t + Az)l(t + Bz), - 1 < B < A < L, the class S'(h) reduces to the class S-[A, Bl. When h(z) = (1 + (1 -zu)z)l(r-
"),0 < a < 1, the
class S- (h) reducesto the class S- (o) of starlike functions of order ct. Recently Frasin and Darus [1] have proved that if f e A and
(rf(r))
rf'(z)
, jW-'-fw
.H,
(o<.,<1),
FIRSTORDERDIFFERENTIALSUBORDINATION
43
then z 2 f' Q)
fa
1 <1-cr.
In this paper, we find h(z) such that
ktQ--r'fi,\?,)
f'(")
implies
1S
d,isleA^an"d,0an'd'$ be
onalytic hr. a domain, D cotttahtirlg q(L) with Q(ur)'. 0 when' ro e q(A). Set Q@) = zq'(z)6@GD,h(z) = e(q(r))+ QQ). Supposethat I. Q@) is starlike uniualent in" A,,antd
2.Re!9>oforzea,. Qk)
rf o(p(r)) + zp'(z)$(p(")) * o(q(r)) + zq'(z)$(q(")), then p(z) - q(") on"dq(z) is th,ebestdomhtanr't. 2. Main Results We begin with the following: Lemrna
1. If
p(z)
ond
q(z) are an'alytic irt' A,
ut iu a l e n t, a ,9 a re re a l rtu m b e r s atud
' o * "9^,r9^)\] 'Bne{tr- cx)+ zuq(z)*B[r q'lz))) x t
q(z)
is coruuuc
44
V. RAVICHANDRAN
CTTLd
(1 - ")p(z) + apz(z) + pzp'(") < (r - ")q( z) + aqz(z) + pzq'(z), then p(") . q(z) and q(z) is the bestdominant. Proof. Let e(r) = (1 - u)w + uwz and 0(r) = g. Then clearly e(r) and Q(ru)are analytic in C and 0(r) * 0. Also let .
= pzq'(z) QQ) = zq'(z)Q@QD and
h(z)- e(q(r))+QQ) = (1- a)q(z)+oqz(")+Bzq'(z). Since q(z) is convex univalent,
zq'(z) is starlike
univalent.
Therefore
QQ) is starlike univalent in A, and
- o)+ Zuq(z). R"?# = o.{,1 u(t. #&)}, o i for z e A. Therefore the result follows from Theorem A.
Notethat the conditionUo"{0 - a) + zuq(z).U(t.
OtE)i
' o is
satisfied by any convex function that maps A onto a convex region in the right-half plane when 0 < a < 1 and B > 0. Theorem
S. If q(z) is coruuexuniualent and 0 < cr ( 1,
o"{++zq(z)*('* #E)1,, antd
"2 -f6-*-7649 * o [,'(3) < (1-
then,+9
f(")
a)q(z)+ oqz(") + azq,(z),
< q@) and, q(z) is the bestd.omina,tlt. r\
45
FIRST ORDER DIFFERENTIAL SUBORDINATION Proof. Let p(z) =
Then a computation showsthat #.
-zp'(z) M _, - r -,j zf'(z) w - j 6 zf'(z) which shows that zp'(z)
zf'(z)
-/-\ , P \ z ) + 6 = r_,* i @ Therefore. we have
^."'f'(4 _ ^ "f'(") zf'(z) "-d-"-fGi-M
="l'!,9) +pQ)-rfpa) " p\z) L
J-
= azp'(z) * opz(") - up(z) and hence we have
319' * o "'!r',?) = (1- a)p(z)+ opz(") + azp'(z). f(") f(") By using Lemma 1, (with cr = P), we have the result. Let H(a,h)
be the class of functions f e A with f(")1" 40, z e L',
such that
I \ trl @ -) - 2 a ( 2 2 + 2 2 ) + L - 2 2 w t uh e ' e er e t vh I / r ,) , w Il f c H I r (\a I. . I c o r o rlrl- |a' I Jr y 1 fI e L / ,,t h, , 1 _ "(rlt
e
I
I |
i
0
eS*.
"),
t
46
V. RAVICHANDRAN Proof. Let uq take q(z) = (I + z)lQ - z). Then azq'(z)+ oq2("1+ (1 - a)q(z) = he)
and the result follows from our Theorem B. Let
-1 < B < A < l.
functions f eA
Note that the class s-[A, Bl
consists of
satisfying zf'(z) .I+Az jWt L*8",
zea"
For the class S.[A, B], we have the following: Corollary 2. Let cr > 0 and -I < B < A < L. If
@ * B(I - zu)lz+ A(1 - c)B + uAfzz , $ + Az)2
th,en / . S. [A, B]. The following special case of the above corollary improves an earlier result of Obradovic et aI. [6]: C o r o l l a r y S . L e tc t > 0 a n d O < A < I .
If
"'{,'?) < 1 + (r + zu)Az+ uAzzz, {9 * o f(") f(r) thertt9
Let us= 1+ (t + za)ez + aA2z2.Then for z = eij, we have + oAze2isl lw -1 | = l(r + Zu)Aeio = l(1+ 2u)A + aAzrisl > (r * 2u)A - aA2. Hence we have the following:
FIRST ORDER DIFF'ERENTIAL SUBORDINATION C o r o l l a r y 4 . L ect r > 0 a n d 0 < A < I .
47
If
!9 * o'2{,'(?)< 1 + f(t + zgez - uAzlz, f(") f(") t h e t t4 9 p < t + A z . llz)
I n p a r t i c u l a r , w i t hA = L l Q + c r ) , 0 < i , < 1 + o , w e h a v e
Q + z Q e - ,= , A+z? 1 " - a # We will show that L+Za ^ tv r+cr
--
-
?,"2 (t+c)"
A-------------;
Z
lv.
Note that this condition is equivalent to L+2a-=gl->1+a I+cr, or o - 'sl > o I+a or 1 + cr ) tr, which is true by our assumption. Thus we have the following: Corollarv 5 [6]. If 49
j 6 - * *- 7o'2[,'(:) 7
< 1 + ]"2 (u> o, o < r. < a + 1),
t h e n " t ^ ' ! 1 )+. tr *r1 2 ' Ilz)
In fact Obradovic et aI. [6] proved the above results by a different method. To prove our next result, we need the following lemma which can be proved by an application of TheoremA:
V. RAVICHANDRAN
48
Lemrna 2. Let q(z) be an'alytic in, A,, q(0) = | an'd zq'(z)lqQ) be starlihe uniualen't itt'L. If p(z) = 1 + cz + ... is an'alytic itt,A,an,d satisfies
#
p(z)
(zf(z))"_ u -7@ z"tr?r) < h@), f(r) then,
t!9
o(zt : \ / = 1!@. f"(")
A computation shows that
zp'!4-etzf'(z) -z"l:!",)=@t9))" -z"fi,Q,)
6=2t76-'f6--f@
fQ)
By Lemm a 2, we have the result. By takins qQ) = (1 + A")10 + Bz), we have the following: Corollary 6. If
@ @ - - z " l : , ( < ) .(,A , P ) " . , ( - 1< B < A < r )
Wi-"j6-W
FIRSTORDERDIFFERENTIALSUBORDINATION
49
then 'Q)-r+Az -zFz f@ r+Bz i
Proof. Let q(z) = (1 + A")10 + Bz). If
h(z)= zq'(z)lq@l= ffi is starlike univalent in A, then the result follows from Theorem 4. We prove that the function h(z) is indeed starlike. A computation shows that
e@),= +&-,-lh.ihl =
= wL - 'ABzz Now
ReQ(e'e)=Reffi _l+r(A+
B ) ( l - A B r z ) c o s o -A z B z r a l(t + Areit) (t * Brrio)12
_ (1 - ABr2)(t + ABr2 + r(A + B)cos0) ,'"n
16*eni\1*affi
provided L + A B r 2 + r ( A + B ) c o s 0 > 0 . S i n c e- 1 < B < A < I , + r ( A + B ) c o s 0 > ( f - A r ) ( L - B r ) > 0 w h e n( A * B ) > 0 . + r(A+ B)cos0 t (t * Ar)Q + Br)> 0 when (A + B) s 0. Corollary 7.If f e A and
-("f("\)" 7 6 - ' _,zf'(z) -,_? f(4 tzz'
I+ABr2
A l s oL + A B r 2
50
V. RAVICHANDRAN
then
o""zl(r) , ,. f"(") AIso by takine q@) = 1 + (l - u)2, we have the following: CorollaryS. If f e A and
("f(")) , "f'(4 ., (r; o)", (o < cr< 1), - = jT-"-fW J l l l - - a ! ,, then
j6
"2f'(")
1
u L (tz -\ =
( 1' o ) l
<1-cr.
Note that the function
1aTJ-SZ
maps A onto the region
l,.Jd#l.J& This is a circular disk whose diametric end points are at l-cr 2-u
. brnce
la-l I
l| * l | < = .
2 -a
a - 1 and c
t = , the image of tu(z) contains the disk 2-s '
cr
T h u si,f ("f(")) , zf'(z) -7W-"}w
t
1 - a 2 - o '
then ("f ("))" , zf'(z) . -TT-'-fW Hence we have the following:
(1 - a)z
t+-(lr;)z'
FIRST ORDER DIFFERENTIAL SUBORDINATION
51
Corollary I [1]. If f e A and
(rf(r)) , zf'(z) . H , -7W-'i6
(o<,,<1),
thert 2 2f' Q)
-_--=--I
1 <1-cr.
f'(r) References tll
B. A. Frasin and M. Darus, On certain analytic univalent functions, Internat. J. Math. Math. Sci. 25(5)(2001),30b-310.
I2l
Z. Lewandowski, S. S. Miller and E. Zlotkiewicz, Generating functions for some classesof univalent functions,Proc.Amer. Math. soc. b6 (1926),l1l-117.
t3l
J.-L. Li and S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math. 33(3)(2002),313-318.
t4]
S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, Pure and Applied Mathematics No. 225, Marcel Dekker, New York, 2000.
l5l
M. Nunokawa, s. owa, s. K. Lee, M. obradovic,M. K. Aouf, H. saitoh, A. Ikeda and N. Koike, Sufficient conditions for starlikeness, Chinese J. Math. 24(3) (1996), 265-27r.
16l M. Obradovic, S. B. Joshi and I. Jovanovic, On certain sufficient conditions for starlikenessand convexity,Indian J. Pure Appl. Math. 2g(g) (1998),27r-275. l7l
K. S. Padmanabhan,On sufficient conditionsfor starlikeness,Indian J. Pure Appl. Math. 32(4)(2001),543-550.
t8l
C. Ramesha, S. Kumar and K. S. Padmanabhan. A sufficient condition for starlikeness,ChineseJ. Math. 23(2)(199b),t67-17I.
tgl
V. Ravichandran, C. Selvaraj and R. Rajalakshmi, Sufficient conditions for functions of order cr, J. Inequal. Pure Appl. Math. 3(b) (2002), Article No. 81. http://jipam.vu.edu. au
Department of Computer Applications Sri Venkateswara Collegeof Engineering Sriperumbudur 602 105, India e-mail:
[email protected]
