Internat. J. Math. & Math. Sci.
VOL. 19 NO. 3 (1996) 613-614
613
A NEW CRITERION FOR STARLIKE FUNCTIONS
LING YI
Department of Mathematics
llarbin Institute of Technology
IIarbin, P ft. China
SHUSEN DING
Department of Mathematics
The Florida State University
Tallahassee, Florida, 32306-3027
(Received November 22, 1994 and in revised form January 19, 1995)
ABSTRACT. In this paper we shall get a new criterion for starlikeness, and the hypothesis
of this criterion is much weaker than those in [1] and [2].
KEY WORDS AND PHRASES. Starlike functions, univalent functions, analytic functions.
1991 AMS SUBJECT CLASSIFICATION CODES. 30C45, 30C55.
INTRODUCTION AND PRELIMINARIES.
Let A be the class of functions f(z), which are analytic in the unit disc D {z Izl < 1}, with
0. Let S be the set of starlike functions, S
f(0) f’(0)
f(z) e. A, Re(zf’(z)/f(z)) > O, z E D}.
R. Singh and S. Singh in [1] proved that if f(z)E A and Re[f’(z)+ zf"(z)] > O, z a_. D, then
1.
f(z)
s.
-,
Recently, R. Singh and S. Singh in [2] proved that if f(z) A and Re [f’(z) + zf"(z)] >
then f(z) S.
In this paper we shall show that the asscrtion of R.. Singh and S. Singh holds under
weaker hypothesis.
LEMMA
1.
for all real z, y _<
Suppose that the function : C D C satisfies the condition Re(iz,
and all z e D. If p(z)
+ pz +... is analytic in D and
z
D,
a
much
y;
z) < 6
+-
Re(p(z), zp’(z); z) > 6, for z e D
then Re(p(z)) > 0 in D.
A general form of this lemma can be found in [3]. In [4] the authors got the following result.
and
LEMMA 2. Let a > 0, # < 1. If the function p is analytic in D, with p(0)
Re [p(z) + ozp’(z)] > fl,
then Re(p(z)) > (2#- 1)+ 2(1- #)F(1,
function. This result is sharp.
,
+ 1" -1),
z
zE
D
D, where F(a, b, c; z) is a hypergeometric
By taking a
in lemma 2, we obtain
LEMMA 3. Let # < 1. If the function p is analytic in D, with p(0)
ne [p(z)+ zp’(z)] > #,
then Re(p(z)) > (2#- 1)+ 2(1 -#)/n2,
z
z
eD
D, and the result is sharp.
and
614
2.
L. YI AND S. DING
MAIN RESULT
THEOREM. If I(z) A and
+ zf’Cz)] >
Re [/’(z)
4(1
in 2)
-0.263, z
+2
(1)
D
then f(z) e S.
PROOF. By using lemma 3, from (1) we have
3(1 -In2)
> 0,
2(I I2) +
Re(f’(z)) > I-
From (2) and lemma 3,
we
have
Re f(z) > -2 +
z
Now, we let
-2 + 2(-I.a
p(z)
(:)
z.D.
3
2(1
zf’(z)/l(z) and ,X(z)= f(z)/z, then p(z) is analytic in D and p(0)= 1, Re{,(:)} >
A simple computation shows that
f’(=) + f"()
a(z)[p(=) + =’()1
(p(=), =n’(=); z),
where (u, v; z) ,X(z)(u + v). Using(l), we have Re[(p(z), zp’(z).z)] > 1Now for all real ::, y < -1/2(1 + ::u), we have
Re [q,(/::, y; z)]
for each
z
D.
(U- x2)RelA(z)] <
-(1 + 3x)Re [,(z)] < -Re [,()]
Note
z
s
for each :
D.
(4)
from(4) weget
4(1
for all
(3)
z 6 D.
0.526,
In 2) +
in2)
+2
D. Thus by lentna 1, Re [p(z)] > 0 in D, that is, f(z) q S.
.
,
{I e .,4. Re[f’(z) + zf"(z)] > z e D}. It was proved in [4] that if
f() R(o) o -m
-0.61), then/(z) is univalent, and the constant ao can not be replaced
3
by any less one. Our present theoren yields R
C S. Thus, a natural problem which
arises is to find inf{B" R(B)c 8}.
REMARK. For < 1, let R(O)
(1 (i-.)’+)
REFERENCES
[I] R. SINGH and S. SINGH, Starlikeness and
MariaeCurie
Sklodowska, Lublin,
convexity of certain integrals. Ann. University
XXXV, 16, Sec.A(1981), 45 47.
[2] R. SINGH and S. SINGH, Convolution properties of a class of starlike functions.pr0. A.bt.S.
106 (1)(1989), 145- 152.
[3] S. S. MILLER and P. T. MOCANU, Second order differential equations in the complex plane.
J. Math. Anal. Appi. 65(1978), 289 305.
[4] LING YI, BO GEJUN and LUO SItENGZI[ENG, On an inequality for some analytic functions,
J. Math. nes. and Exp. (2)13(1993), 259- 262.