205
Internat. J. Math. & Math. Sci.
(1994) 205-207
VOL. 17 NO.
A CRITERION FOR P-VALENTLY STARLIKE FUNCTIONS
MAMORU NUNOKAWA
SHIGEYOSHI OWA
Department of Mathematics
SEIICHI FUKUI
Department of Mathematics
Department of Mathematics
Gunma University
Wakayama University
Wakayama 640, Japan
Aramaki, Maebashi, Gunma 371, Japan
Kinki University
Higashi-Osaka, Osaka 577, Japan
(Received December 22, 1992 and in revised form April 19, 1993)
ABSTRACT. The object of the present paper is to prove
a criterion for p-valently starlike
functions in the open unit disk.
KEY WORDS AND PHRASES. Analytic, open unit disk, p-valently starlike.
1991 AMS SUBJECT CLASSIFICATION CODE. Primary, 30C45.
1.
INTRODUCTION.
Let A(p) be the class of functions of the form
f(z)
E
zp +
an zn
(p E N
{1,2,3,--. }),
(1.1)
n=p+l
which are analytic in the open unit disk U
to be p-valently starlike in U if it satisfies
{z: Izl < 1}. A function f(z) belonging to A(p) is said
.
Rf zf’(z) > 0
e f(z)
(z U).
(1.2)
We denote by S(l) the subclass of A(I) consisting of functions f(z) which are p-valently starlike in
t
(d. [ll).
Recently, Nunokawa [4] has shown that
THEOREM A. If f(z) A(p) satisfies f(z) 0(0 <
Re
"zf’(z)
f(z)
z
I; < 1+
< 1) and
(z.U),
(1.3)
_.
then f(z)
In the present paper, we derive a new criterion for the class S(p) involving the above result
Nunokawa
by
[4].
2. A NEW CRITERION.
To derive our main result, we have to recall here the following lemma due to Jack [2] (also,
due to Miller and Mocanu [3]).
LEMMA. Let w(z) be analytic in U with w(0)= 0. If w(z)l attains its maximum value on
the circle zl r < at a point z0, then we can write
zow’(zo) W(Zo),
where k is a real number and k >
1.
(2.1)
S. OWA, M. NUNOKAWA AND S. FUKUI
206
Now, we prove
THEOREM. If f(z)
< 1) and
A(p) satisfies f(z) # 0(0 <
l+
’" t
s’<=)
)-(l
>o
(z U),
(2.2)
then f(z) S(p) and
-7- p
(2.3)
( e v).
<p
PROOF. Define the function w(z) by
p(1
f(z)
(2.4)
+ w(z)).
Then w(z) is analytic in V and w(0)= 0. It follows from (2.4) that
+-if(z)’
so
p(1 + w(z)) +
(2.5)
+ w(z’
that,
zS’(z)f(z) (1 + zf ’(z)if(z) ]’=
Suppose that there exists a point
zw’(z)
V(1 + w(,) )"
V such that
0
I(z)
Then, applying Lemma,
+
I,(z0)
((z o) #
).
we can write
ZoW’(Zo) kw(zo)
W(Zo) eiO(o # r). Thus we have
f(z)
zoft(zo)
+ zoS"(zo)’
ft(z
O) ]
=1+
(k
+
p(1
>= 1)
+ eio)- -
2p(1 + cosO)
Note that the condition (2.2) implies
s.)(
(zcU),
zf’(z) 1+ St(z) #a
where
that
a
(2.S)
> + 1/4v. Therefore, (2.7) contradicts our condition (2.2). Consequently, we conclude
_
IzS’(z)
(2.9)
that is, that f(z) S(v).
in Theorem, we have
Letting v
COROLLARY. If f(z) A(1) satisfies f(z) # 0(0 < [zl < l) and
f(z),
St:S’O)(’
S’I:)7
(z
>0
(2.10)
then f(z) S(1) and
:f’(*)
s-7-(Ty- <
(ev).
(2.11)
CRITERION FOR P-VALENTLY STARLIKE FUNCTIONS
ACKNOWLEDGEMENT. The research of the first author
207
supported in part by Japanese
Ministry of Education, Science and Culture under Grant-in-Aid for General Scientific Research
was
(No. 04640204).
REFERENCES
1.
GOODMAN, A.W., On the Schwarz-Christoffel transformation and p-valent functions,
Trans. Amer. Math. Soc. 68, (1950), 204-223.
JACK, I.S., Functions starlike and convex of order J. London Math. Soc. 3 (1971), 469-
,
474.
MILLER, S.S. z MOCANU, P.T., Second order differential inequalities in the complex
plane, J. Math. Anal. Appl. 65 (1978), 289-305.
NUNOKAWA, M., Certain class of starlike functions,
to appear.