Internat. J. Math. & Math. Sci.
VOL. ii NO. 3 (1988) 493-496
493
A NOTE ON CERTAIN SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577, Japan
LIU LIQUAN
Department of Mathematics
Heilongjiang University
Harbin, China
WANCANG MA
Department of Mathematics
Northwest University
Xian, China
(Received June 25, 1987 and in revised form July 31, 1987)
ABSTRACT.
to class
The object of the present paper is to show a result for functions belonging
P’(I-e,0)
which is a subclass of close-to-convex functions in the unit disk U.
KEY WORDS AND PHRASES.
,
Close-to-Convex of order
Class P’(e), Class P’(I-,0),
subordination.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
30C45.
INTRODUCTION
Let A denote the class of functions of the form
f(z)
z
+
(I.i)
anZ
nffi2
which are analytic in the unit disk U
is said to be in the class
{z:
P’(=) (according
Izl
<
i}.
to Goodman
A function f(z) belonging to A
[4]) if and only if it satisfies
the condition
Re{f’(z)}
for some
(0
<
>
=
(1.2)
i) and for all z e U.
Note that P’(a) the subclass of close-toFurther, let P’(I-e,0) (according to
convex functions of order e in the unit disk U.
Goodman [4]) be the subclass of A consisting of all functions which satisfy the
condition
If’(z)
for some
(0 <
<
11
< 1
(1.3)
e
i) and for all z
U.
S. OWA, L. LIQUAN, and W. MA
494
It is clear that P’(l-a,0) is the subclass of P’(a) for 0 &
Fukui, Owe, Saitoh and Sekine [I] showed that functions in
Izl
< r
Nunokawa,
are starlike in
is the root of the equation
I, where r
(I-=)r212) 2
(21(3-=)) 2 (r-
1
I.
P’(l-e,0)
Sin-l((1-a)r)
+
log
.
2
Also, Fukui, Owe, Ogawa and Nunokawa [2] proved that functions in P’(a) are starlike
in
Izl
< r
2, where r 2 is the smallest root in
Sin
2(1-a)r
-1
[0,i) of the equation
.
1
+ log
(2a-1)r 2
r
2
For the functions f(z) and g(z) belonging to A, we say that f(z) is subordinate to
g(z) in D if there exists an analytic function w(z) in U such that lw(z)l < for
z’ U and f(z) g(w(z)). We denote by f(z)- g(z) this subordination. In particular,
g(0)
if g(z) is univalent in U the subordination f(z)- g(z) is equivalent to f(0)
and f(U)
g(U) (cf. [3]).
MAIN RESULT
In order to prove our main result, we have to recall here the following lemma due
to Miller and Mocanu [5].
2.
Let q(z) be an injective mapping of U onto Q, with q(0)
Let p(z)
LEMMA.
q(z)
be analytic in U with
and
i, such that
+ pl z + p2 z 2 +...
is regular on U except for at most one pole on U.
p(Izl
<
Iz01)
p(z)
z0p’(z 0)
Q, then
c
If there exists a point z e U such that p(z
0
e i8
and w
where m
0
I.
-
O)
e 8U
=-q-l(p(z0)).
mw0q’(w0),
Applying the above lemma, we derive
Let the function f(z) defined by (I) be in the class P’(I-a,0).
THEOREM.
f(z)
z
PROOF.
Let q(z)
holds true if
p(z)
E
Assume that p(z)
(I
a)z
I +
(1.4)
2
+ (i
i
a)z/2 and p(z)
f(z)/z.
It is clear that the result
for z e U.
for z e U and the subordination
U.
Then there exists a point z e U such that
0
Therefore, applying the lemma, we get
f’(z0)
Then
p(z 0)
e
p(z)-q(z)
does not hold in
8q(U) and p(Izl
<
Iz01)
c
q(U).
z0p’(z 0) + p(z 0)
nw0q’(w0) + q(w0)
m(l-)w 0
+ (l-)w 0
2
2
+
(m + 1) (l-a)w 0
+I
(1.5)
CERTAIN CLASS SUBCLASS OF CLOSE-TO-CONVEX FUNCTIONS
where
R
>-
[Wo[
and
I.
95
Thus
(R+ i) (i -a)
a
11
[f’(z 0)
which contradicts the hypothesis that
in U.
a,
(1.6)
2
f(z)
e
P’(1-a,0).
So we Rust have p(z)-q(z)
This coRpletes the proof of Theorem.
Finally, we have
CORALLARY 1.
Let the function f(z) defined by (1.1) be in the class P’(1-a,0),
Then
Re
ei
f(z)
>
0,
Z
where
181
Sin-l(l
/2
CORALLARY 2.
)/2.
Let the function f(z) defined by (I.I) be in the class P’(l-a,0).
Then
f(z)
Re
>0.
Z
ACKNOWLEDGEMENTS.
The authors would like to thank the referee of the paper for his
thoughtful encouragement and numerous helpful advlces.
i.
2.
REFERENCES
NUNOKAWA, M., FUKUI, S., OWA, S., SAITOH, H. and SEKINE, T., On the Starlike
Boundary of Univalent Functions, to appear.
FUKUI, S., OWA, S., OGAWA, S. and NUNOKAWA, M., A Note on a Class of Analytic
Functions Satisfying Re{f’(z)} >
Bull._ Fac. Edu. Wakayama Univ. Nat. SGi.
,
36 (1987) 13-17.
3.
4.
5.
DUREN, P.L., Univalent Functions, Grudleheren der Mathematischen Wissenschaften
9, Springer-Verlag, New York, Berlin, Heidelberg and Tokyo, 1983.
GOODMAN, A.W., U.n.ival.ent Functions. Vol. !L Mariner Publ. CoRp. Inc., 1983.
MILLER, S.S. and MOCANU, P.T., Second Order Differential Inequalitiesln the
Complex Plane, J. Math. LAnal,_ AppI.. 65 (1978) 289-305.