189
Internat. J. Math. & Math. Sci.
I1990) 189-192
VOL. 13 NO.
NOTE ON CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS OF ORDER
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577, Japan
M.K. AOUF
Department of Mathematics
Faculty of Science
University of Qatar
P.O. BOX 2713
Doha-Qata r
Department of Mathematics
Faculty of Science
University of Mansoura
Mansoura, Egypt
(Received April 25, 1988)
ABSTRACT.
The object of the present paper is to show a result for functions belonging
to the class
R(e) which is the subclass of close-to-convex functions in the unit disk
U.
Close-to-convex of order e, class P(e), class
KEY WORDS AND PHRASES.
R(e),
starlikeness bound.
1980 AMS SUBJECT CLASSIFICATION CODE. 30C45.
I.
INTRODUCTION.
Let A denote the class of functions of the form
f(z)
+
z
n=2
(I.I)
a z
n
which are analytic in the unit disk U
zl
[z:
<
the class A is said to be close-to-convex of order
I}.
A function f(z) belonging to
if and only if it satisfies the
condition
Re{f’(z)}
for
some
e (0
<
>
(1.2)
e
I) and
for
z e U.
We
denote
by
consisting of functions which are close-to-convex of order
P(a) the
subclass
in the unit disk U.
of
A
S. OWA, M.K. AOUF AND M.A. NASR
190
R(a) be the subclass of A consisting of all functions which satisfy the
Further, let
condition
[f’(z)- 1[ <
(0
for soe
<
4
(1.3)
-a
z e U.
1) and for all
R(a) c P(a) for
It is clear that
0
(
a
<
1.
Nunokawa, Fukul, Owa, Saltoh and
R(a). Also,
Seklng [I] have showed the starlikeness bound of functions in the class
the starllkeness bound of functions belonging to the class
P(a) was given by Fukul,
owa, Ogawa and Nunokawa [2].
2. MAIN RESULT.
In order to prove our main result, we have to recall here the following lemma due
to Lewandowski, Miller and Ziotkiewlcz
LEMMA.
Let
:
and irt
u
181
8 be real and
D
+
u
/
C, D
iu2,
+ iv
2.
(u,v) is continuous in D;
(li)
(ei8,0)
(lit)
Re{#(lu2,vl)}
Re{#(eI8,0)} >
0 when
Vl
Let
p(z)
(p(z),zp’(z))
2
i +
plz + p2 z +
e D for all z e U. If
e
Re{#(p(z),zp’ (z) )}
then
Re{p(z)}
>
0 (z
a complex valued function
Suppose that the function
(i)
D and
#(u,v) be
(C is the complex plane),
c C x C
v
v
<
[3].
/2. Let
>
0;
(lu2,v I)
-
0
(u,v) satisfies
e D and
2u2sln8
+ u2
2
2cos8
be regular in the unit disk U such that
(z e U),
U).
Applying the above lemma, we derive
THEOREM.
Let the function f(z) defined by (I.I) be in the class
R(e).
Then
Re {e iB
f(z)
z
>
0,
(2.1)
SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
191
where
Re{e
for
IB] <
can
see
>
f(z)}
Sin-l(1
/2
e
we
f(z) e R(=) that
It follows from
PROOF.
).
iB f(z)
z
that
(z
0
E
2.3)
U)
Defining the function p(z) by
(2.4)
p(z),
p(z)
e
iB
+
PlZ
+
P2 z
2
+
Is
regular
in
U.
Taking
the
differentiations of boh sides in (2.4), we have
e
p(z) + zp’(z).
f’(z)
(2.5)
It follows from (2.3) that
Re{eilSf’(z)}
Re[p(z) + zp’(z)}
>
(2.6)
O.
Setting
#(u,v)
u
+ v
p(z) and v
(note that u
zp’(z)),
(2.7)
we see that
(i)
(il)
#(u,v) is continuous
(ei8,O)
Re[#(iu2,v
2‘v t)
C;
C
Re{#(eiS,0}
e D and
(Ill) for all (iu
in D
cos8
>
O;
D such that
Vl
-
)}=
v
2u2sln8
2
+ u
2
2cos8
2u2sing
+ u 22
2cos
Therefore, the function
#(u,v) defined by (2.7) satisfies the conditions
in
Lemma.
S. OWA, M.K. AOUF AND M.A. NASR
192
Using Lemma, we have
Re[p(z)}
I f(z)
Re {e
>
z
0
which completes the proof of Theorem.
Letting
COROLLARY.
e
0 in Theorem, we have
Let the function f(z) defined by (I.I) be in the class R(O).
Then
Re[.
f(z------D}
>
z
0
(z
g
U).
REFERENCES
I.
2.
3.
NUNOKAWA, M., FUKUI, S., OWA, S., SAITOH, H. and SEKINE, T. On the
starlikeness bound of univalent functions, Math. Japon., to appear.
FUKUI, S., OWA, S., OGAWA, S. and NUNOKAWA, M. A note on a class of analytic
functions satisfying Re{f’(z)} > e, Bull. Fac. Edu. Waka[ama Univ. Nat. Sci.
36 (1987), 13-17.
LEWANDOWSKI, Z., MILLER, S.$. and ZLOTKIEWICZ, E. Generating functions for some
classes of univalent functions, Proc. Amer. Math. Soc. 56 (1976), 111-117.