Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 439-446
439
ON SOME RESULTS FOR X-SPIRAL FUNCTIONS OF ORDER
MILUTIN
OBRADOVI
Department of Mathematics
Faculty of Technology and Metallurgy
4 Karnegieva Street
Ii000 Belgrade
Yugoslavia
and
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577
Japan
(Received March I, 1986)
ABSTRACT.
of order
The results of various kinds concerning h-spiral functions
in the unit disk U are given in this paper. They represent
mainly the generalizations of the previous results of the authors.
KEY WORDS AND PHRASES. %-spiral of order
,
starlike of order
subordination, univalent.
1980 MATHEMATICS SUBJECT CLASSIFICATION.
30C45.
I. INTRODUCTION.
.
Let An denote the class of functions of the form
f(z)
z
+
k=n+l
akzk
which are regular in the unit disk U
(n
{z.
zl
N
{1, 2, 3,
})
(1.1)
< i}.
For a function f(z) belonging to the class An we say that f(z) is
-spiral of order
if and only if
Re
for some real
lei%
(lI
zf’(z)
f(z)
}
> cos%
< /2), for some
(1.2)
(0 _<_
<
I), and for all z
U.
440
OBRADOVI and
M.
S(a)
the class
n
and S%(a) instead
We denote by
we write A
of all such functions
S(a),
of A I and
has been considered by Libera [I].
S%(a)
,
S. OWA
Sn(a),
and S
Sx
S(0)
We note that
S0(a)
n
S (a)
In the case n
i
respectively. The class
S0(0)
and
S
where S
denote the classes of functions which are -spiral, starlike
of order a and type (i.i), and starlike, respectively.
-
Let f(z) and g(z) be regular in the unit disk U. Then a function
f(z) is said to be subordinate to g(z) if there exists a regular function
-
w(z) in the unit disk U satisfying w(0)
that f(z)
0 and
lw(z)
<
I (z
U) such
g(w(z)). For this relation the following symbol f(z)
is used. In particular, if
subordination f(z)
g(z)
g(z) is univalent in the unit disk U the
g(z) is equivalent to f(0)
g(0) and f(U) g(U).
2. RESULTS AND CONSEQUENCES.
First we give the following result due to Miller and Mocanu [2].
.
LEMMA I.
and let u
Let (u,v) be a complex valued function,
C, D C XC (C is the complex plane),
D
uI
+
iu 2, v
v
I + iv 2. Suppose that the function (u,v)
satisfies the following conditions:
(u,v) is continuous in D;
(i)
(I,0)
(ii)
D and Re{(l,0)} > 0;
Re{(iu2,vl)} __<
(iii)
v
I <=-n(l +
0 for all
(iu2,v I)
E
D and such that
u)/2.
1 + pn z n
Let p(z)
such that (p(z),zp’(z))
+ Pn+l z n+l +
be regular in the unit disk U
D for all z s U. If
Re{(p(z), zp’ (z))
> 0
(z
U),
then
Re{p(z)
> 0
(z e U).
By using the above lemma, we prove
THEOREM I.
Let the function f(z) defined by(l.l)be in the class
S
%(a)
n
and let
n
0 <
Then we have
__<
(2.)
2(i
) cos
(z
2B(I- a)cos
+
n
U).
RESULTS FOR k-SPIRAL FUNCTIONS
PROOF.
441
If we put
28(I
(2.2)
a) cos I + n
and
Be il
f(z)
+ B,
(i- B)p(z)
(2.3)
z
B satisfies (.2:1) then p(z)
where
p(z)
i
is regular in the unit disk U and
+ pn zn + Pn+l zn+l +
From (2.3)after taking the logarithmical
differentiation we have that
zf’ (z)
Beil
zp’ (z)
B ei
(I- B).
(2.4)
f(z)
B)p(z) + B
(I
and from there
e
iX
zf’(z)
e
acos
-(2.5)
cos X +
8{(1- B)p(z) + B}
f(z)
Since f(z) e
Re
{
e
SX()
n
il
B)zp’(z)
(i
iX
then from (2.5)we get
B)zp’(z)
(i
cos X
+
(z
> 0
g
U).
(2.6)
B)p(z) + B}
B{(I
Let consider the function (u,v) defined by
(u,v)
(it is noted u
D
B)v
(i
iX
e
cos X
+
B)u + B}
{(
zp’(z)). Then (u,v) is continuous in
p(z) and v
(I- )cos I > 0.
(- {-B/(I-B)}) C. Also, (I,0) e D and Re{(l,0)}
Furthermore, for all
D such that v I < -n(l
(iu2,v I)
+
(i- )cos
Re{(iu2,vl)}
(1
) cos
Re
{
nB(l
)cos
(1
(I
B)
{482(i
28{(1
< 0
u22)/2 we have
(i- B)v I
B)iu 2 + B}
B{(l
B(I- B)v 1
x +
8{(I
<
+
B)2u + B 2}
B)(I + u)
2{(i
2
2
c)
B)2 2
cos X
2
B)2up
+ B 2}
2 2
-n }u 2
+ B 2}
}
M.
42
OBRADOVI and
S. OWA
a)cos I
<__ 0o Therefore,
because 0 < B < i and 28(1
n
the function
(u,v) satisfies the conditions in Lemma i. This proves that Re{p(z)}
> 0
for z s U, that is, that from
8e
il
Re
B
>
z e U),
which is equivalent to the statement of Theorem I.
COROLLARY I.
I(0)
n/2 cos I in Theorem i, we have
0 and 8
Taking a
Let
te
function f(z) def;ned by (i.I)be in the class
Then
Sn
f(z)
Re{[--
z
--I
I
}
(z
>
U).
2
0 in Theorem i, then we have the former result
If
REMARK I.
(n/2 cos %)e
0 in Corollary i, then we have the
given by the authors in [3]. If
earlier result given by Golusin [4].
Let 8 be a fixed real number, 0 < 8 < I, and let the
THEOREM 2.
function f(z) be in the class
z[
g(z)
=<
where 0 <
(I
Sl(a).
Let
f(z)
z
(i
)/(i
e) and
function g(z) is in the class
(z e U),
z)2Ue-il cos I
i
8
y(l
(2.7)
). Then the
S(8).
Let 8 (0 < 8 < i) be a given real number. Then from (2.7)
PROOF.
by using the logarithmic differentiation and some simple transformations,
we have that
e
il
zg’ (z)
8 cos
g(z)
zf’ (z)
+
acos
f(z)
zf’ (z)
a cos
f(z)
l}
+ (i
l}
+
y)ei
l+z
cos I
g
i
{B
2
a
i(l
cos l
i
z
) sin I.
(2.8)
443
RESULTS FOR %-SPIRAL FUNCTIONS
From (i0) we have
Re
[ei%
zg’ (z)
g(z)
Re
{e
zf’ (z)
il
+
cos I
cos I
f(z)
i
z
> 0
(z
because of the suppositions for y, f(z) and
E
U),
(2.9)
given in the statement
of Theorem 2. Thus we complete the proof of Theorem 2.
COROLLARY 2.
,
Let 0 < 8 <
and let f(z) e S
Then the
function g(z) defined by
f(z)
g(z)
SI(B).
belongs to the class
PROOF.
__< 8
Since 0
<
,
it follows that I
__< (I
8)/(1
) and
I in Theorem 2. Also, we have
we may choose y
REMARK 2.
(2.10)
z)2(-B)e-il cos
(i
In particular, for I
if f(z) e S () and 0
__< 8 __<
,
0 in Corollary 2 we have that
then the function
f(z)
g(z)
z)2(
(I
8)
belongs to the class S (8).
This is the earlier result given by the authors [5].
Letting
0 in Theorem 2, we have
,
COROLLARY 3.
Let f(z) e S () and let 0
function g(z) defined by
g(z)
z[
f(z)
z
i
(I
z) 2
=<
8 < I. Then the
M.
444
S*(8),
belongs to the class
i-
(i
OBRADOVI
AND S. OWA
where 0 < y
=<
(i
B)/(I
) and
).
In order to derive the following
theorem, we shall apply the next
result given by Robertson [6].
LEMMA 2.
Let the function f(z) belonging to A
be univalent in
the unit disk U. For 0
t =< i, let F(z,t) be regular
in the unit disk U
with F(z,0) f(z) and F(0,t) 0. Let
p be a positive real number
=<
for
which
F(z)
F(z,t)
lim
t
+0
F(z,0)
zt p
exi, sts. Further, let F(z,t) be subordinate
to f(z) in the unit disk U
for 0 < t < i. Then
F(z)
Re
< 0
(z
f’(z)
u).
If, in addition, F(z) is also regular in the unit disk U and
Re{F(O)} # O,
then
f’(z)
Re
F(z)
}
< 0
(z e U).
(2.11)
Applying the above lemma, we prove
THEOREM 3.
Let the function f(z) be in the class A, and let
the function g(z) defined by
g(z)
f(z)
I
z
cos k
cos
=<
(2.12)
e < i. If the function
i
G(z,t)
i
s
+
be univalent in the unit disk U, where
and 0
ds
0
cos
{
is a real number with
lI
<
/2
G(z,t) defined by
-i%
(i- te
)f(z) -cos (i- t
2)
iz
0
f(s)
ds
s
}
(2 13)
-
RESULTS FOR h-SPIRAL FUNCTIONS
is subordinate to g(z), that is,
fixed
and
,
s().
G(z,t)
445
g(z), in the unit disk U for
and for each t (0 __< t __<_ I), then f(z) is
in the class
PROOF.
It is easy to show that G(z,0)g(z) and
G(0,t) 0.
We choose p
I and F(z,t) to be the function G(z,t)
defined by (2.13)in
Lemma 2. Then we have
G(z)
G(z,t)
lira
t
+0
G(z,0)
G(z,t)/t
lim
t
+0
zt
-e-i%f(z)
(I
(2.14)
cos)z
From (2.o14)we know that
G(z) is regular in the unit
disk U and
cos
Re{G(O)}
Since
I
cos A
f’(z)
g’(z)
# O.
ecos
e cos
i
z
it follows from (2.11)in Lemma 2 that
g’(z)
Re
G(z)
}
< 0
(z e U),
which is equivalent to
Re
{e
i
zf’ (z)
cos
f(z)
}
> 0
(z
U).
Consequently, we prove that f(z) is in the class S
If we put
COROLLARY 4.
0 in Theorem 3, then we have
-
Let the function f(z) belonging to the class A be
univalent in the unit disk U such that
(i-
te- )f(z)
where % is real such that
class
Sl(0)
II
f(z)
< n/2 and 0
(z
__<
U),
t < i. Then f(z) is in the
This is the former result due to Robertson [6].
446
M.
For
OBRADOVI and
S. OWA
0 in Theorem 3, we have the following result for starlike
functions of order
.
Let the function f(z) be in the class A and let
COROLLARY 5.
the function g(z) defined by
f(z)
g(z)
I
ds
0
(0
<
< i)
s
be univalent in the unit disk U. If
G(z, t)
ds
0
s
-
g(z)
in the unit disk U, then f(z) belongs to the class S (e). This is the
previous result given by Obradovi [7].
REFERENCES
i. Libera, R. J., Univalent l-spiral functions, Canad. J. Math.
449
19(1967),
456.
2. Miller, S. S. and Mocanu, P. T., Second order differential
inequalities in the complex plane, J. Math. Anal. Appl. 65(1978),
289- 305.
3.
Obradovi, Mo
and Owa, S., An application of Miller and Mocanu’s
result, to appear.
4. Goluzin, G. M., Geometric Theory_of Functions of a Co_m_plex Vari_a_le,
Moscow, 1952.
5. Owa, S. and
Obradovi, M.,
A note on a Robertson’s result, Mat. Vesnik,
to appear.
6. Robertson, M. S., Applications of the subordination principle to
univalent functions, P_acii_c_.J._M_a_t_h_.
1!_(1961), 315- 324.
7. Obradovlc, M., Two applications of one Robertson’s result, M__a_t__V_es_nih_,
_(3) (1983), 283- 287.