Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 621-623
621
AN APPLICATION OF MILLER AND MOCANU’S RESULT
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka 57 7
Ja pan
ZHWOREN WU
Department of Mathematics
Tongji University
Shanghai, People’s Republic of China
(Received April 7, 1988 and in revised form October 15, 1988)
ABSTRACT.
The object of the present paper is to give an application of Miller and
Mocanu’s result for
a certain integral operator.
Integral operator, Miller and Mocanu’s result, set H
KEY WORDS AND PHRASES.
complex
valued function.
1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.
I.
INTRODUCTION.
Let A denote the class of functions of the form
f(z)
z +
(I.I)
a z
n
n=2
which are analytic in the unit disk U
to the class
Izl
{z:
<
A, we define the integral operator J
c
c
J (f(z))
c
+
z
tc-lf(t)dt
c
I}.
by [I
For a function f(z) belonging
p. 126, Equation (2.1)]
(c
>
-I).
(1.2)
0
{1,2,3,...},
[2].
In
Bernardi
and Livingston [4].
[3]
J1
Recently, Owa and Srlvastava [I] have proved a property of the operator J (c > -I).
c
The
operator J
particular,
when c
c
the operator
N
was
was
introduced
With the above operator J
DEFINITION.
Let H
c
h(r,s,t):
by
studied earlier by Libera
we now introduce
c
be the set of complex-valued functions h(r,s,t);
?
C
(C is the complex plane)
622
S. OWA AND Z. WU
such that
(i)
h(r,s,t)
(ii) (0,0,0)
(ill)
[h(O,O,O) <
D and
g
m+c e iO
}h(e iO ci0
real 0
2.
is continuous in D(L
me
iO
C3;
I;
>
+ L
whenever
m+c e i 0 me i 0+ L) e D with
I.
and real m
Re(e-iOL)
)
m(m-l)=
c+l
for
AN APPLICATION OF MILLER AND MOCANU’S RESULT.
We begin with the statement of the following lemma due to Miller and Mocanu [5].
(0 < r < I) and
0 in U. If z 0
LEMMA. Let w(z) e A with w(z)
0
ro eis0
mw(z 0)
z0w’ (z 0)
and
w" z0)
z
0
Re {I +}
w’ z
O)
(2.2)
I.
)
where m is real and
m,
Applying the above lemma, we derive the following
THEOREM.
Let h(r,s,t)
H
A satisfy
and let f(z)
c
Dr--.. C
(Jc(f(z)),f(z),zf’(z))
3
and
{h(J
for c
>
U.
-1 and
f(z)zf’(z)){ <
(f(z))
(2.3)
Then we have
IJ c (f(z)){
<
(2.4)
(z e U)
where J (f(z)) is defined by (1.2).
c
w(z) for f(z) e A, we have w(z) E A
PROOF. Letting J (f(z))
c
Since
0 (z e U).
w(z)
(c + l)f(z)
Z(Jc(f(z)))’
and
-CJc(f(z)),
we have
f(z)
c
=w(z)
+z w’ (z)
and
zf’(z)
z2’(z).
zw’ (z) +
Suppose that there exists a point z 0
r0 el80
(0
m+c
e i80
<
r
0
<
I) such that
max
Then, using the above lemma, we have
J
(f(z0))
e i80
f(z0)
z0f’(z 0)
me 180
+ L,
APPLICATION OF MILLER AND MOCANU’S RESULT
where L
z20w"(z0)/(c+l).
z
Re{
oW"
z
-i
L)
623
Further, applying the lemma, we see that
0)
z
Re
w,z0
w"
z
0
meI0--------
m- I,
that is,
Re(e
re(m-l)
(2 7)
c+l
Therefore, the condition h(r,s,t) Hc implies that
i00 m+c
c+
c
i0
i00
(2.)
This contradicts our condition
w(z)[
Jc(f(z))
<
(2.3).
nsequently, we conclude that
U.
for all z
us
we complete the proof of the assertion
of the theorem.
Taking c
COROLLARY.
0 in the theorem, we have the following
Let h(r,s,t)
(F(z),zF’(z),z(zF’(z))’)
g
H0,
and let f(z)
DC C and
lh(F(z),zF’(z),z(zF’(z))’)l <
Then
IF(z)l <
(z
F(z)
g
A satisfy
3
(z
u).
(2.9)
U), where
J0(f(z))--
dr.
0
ACKN OWLE D CMENT.
The authors of the paper would llke to thank the referee for his comments.
REFERENCES
I.
OWA,
2.
BERNARDI, S.D., Convex and starlike univalent functions, Trans. Amer. Math. Soc.
135 (1969), 429-446.
3.
Some classes of regular univalent functions, Proc. Amer. Math.
LIBERA, R.J.
Soc.16 (196 5), 755-758.
4.
LIVINGSTON,
5.
MILLER, S.S. and MOCANU,
S. and SRIVASTAVA, H.M., Some applications of the generalized Libera
integral operator, Proc. Japan. Acad. set. A Math. Scl. 62 (1986), 125-128.
A.E., On the radius of unlvalence of certain analytic functions,
Proc. Amer. Math. Soc. 17 (1966), 352-357.
P.T., Second order differential inequalities
complex plane, J. Math. Anal. Appl. 65 (1978), 289-305.
in
the