Internat. J. Math. & Math. Sci.
VOL. 19 NO. 2 (1996) 409-410
409
TWO CRITERIA FOR UNIVALENCY
N. SAMARIS
Department of Mathematics
University of Patras
Patras 26110, Greece
(Received January ii, 1994 and in revised form February 28, 1995)
ABSTRACT. In the present paper
be univalent in
]z[ <
we
give two criteria for the functions f(z)
z
+ 2.: +
to
1
KEY WORDS AND PHRASES. Univalent functions, univalency.
1990 MATHEMATICS SUBJECT CLASSIFICATION. Primary 30C45.
Let A denote the class of functions which are analytic in the unit disk U
{z Izl < 1}
we
0.
denote
class
of
B
the
functions
which
A
convex and
are
By
E
univalent,
f’(0)
f
bo, lnded in U. In the present paper we prove the following theorems.
THEOREM 1. Let f E A, satisfy the condition
asd
%2
!_-7[
f,-
f,
z
for some q e B, where
q, < 1
1!
z
in U.
q2
sup{Iq(z)l; z U}. Then f is univalent in U.
Let f A, q B satisfy the condition
THEOREM 2.
q0
1
Z
Z
l7[f() q(z)}l < A
for some q E
(1)
B, where
,
q’(z)
znf{Iqoq(z
I;-
in
q
U,
(2)
U}
f is univalent in U.
If we put q(z) z in the Th.1 we get the Ozaki and Nunokawa theorem [2].
If we put q(z) z in the .Th.2 we get the Nunokawa,Obradovic and Owa theorem
PROOF OF THEOREM 1. If
Then
q2
then
is analytic in
1
=
f’-q-l
q
U and
f(z)
If we put q(w)
[1].
we
1
+ q(z)
1
q’ (w)(w)dw + c.
get
I
f(z)
q(z)
I
qoo
o
q(z)
(q-X())d + c
(3)
410
Fr,u tie ccmdition q
N. SAMARI S
H
,’ud tle elation
(3)
we
get
+(v-’ ()),
]o
From Schwarz’s Lemma and condition (1)
get
f(z,
f(z2
we
get O(z)] <
q(z:
q(z,
i_<
in /_;. Now from the relation
(4)
(4)
q(22)1
Iq(z,
q
If f(:l)= .f(z2) then it is obvious that q(zl)= q(z) or
PROOF OF THEOREM 2. If we put
z: f
P(z)
then we get
P’(z)=-z[f(z=
z: q
q-
f
=
q(z)
and
lP’(=)l < A
Iq’(=)l
in U.
From the relation
q<,)
we
p,(q-l())
d
get
IP(z)l _< AIq(z)l
Now, the condition (1) of Th.1
in U.
is obvious.
REFERENCES
1.
2.
NUNOKAWA, M., OBRADOVIC, M. and OWA, S. One criterion for univalency. Proc.
Amer. Math. Soc. 106 (1989) 1035-1037.
OZAKI, S. and NUNOKAWA, M. The Scharzian derivative and univalent functions, Proc.
Amer. Math. Soc. 33 (1972),392-394."
we
(5)