Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 451-456
451
FUNCTIONS STARLIKE WITH RESPECT TO OTHER POINTS
S. ABDUL HALIM
DeDt.
of Mathematics
University of Malaya
59100 Kuala Lumpur
Malays i a
(Received September 26, 1990 and in revised form March 5, 1991)
In [7], Sakaguchi introduce the class of functions starlike with respect
We extend this class. For 0 .< 8
i, let S*(8) be the
s
ABSTRACT.
to symmetric points.
class of normalised analytic functions
f
defined in the open unit disc
Re zf’(z)/[f(z)-f(-z)) > 8, for some
that
S*(8),
c
other similar classes
part of some function for
f e
S
sc
(8)
z e D.
In this
D
such
paper, we introduce 2
as well as give sharp results for the real
S(8)
S(8)
s
and S
sc
(8)
The behaviour of certain
integral operators are also considered.
KEY WORDS AND PHRASES. Starlike functions of order 8, functions starlike with
respect to 8ynnetrical points, close-to-convex, integral operators.
AMS Subject Classification (1985 Revision): Primary 30 C 45.
i.
INTRODUCTION.
Let
D
S
be the class of analytic functions
Izl
lz
i},
with
f(z)
E a z
z +
n=2
For
Then
f e
0
.< 8
S*(8)
f, univalent in the unit disc
n
(i.i)
n
S*(8),
i, denote by
the class of starlike functions of order
z e D,
if, and only if, for
!zf’
Re
%),]
f(z)
8.
> 8.
In [7], Sakaguchi introduced the class S* of analytic functions f, normalised
s
by (1.1) which are starlike with respect to symmetrical points. We begin by defining
the class
S*
s’
which is contained in
K, the class of close-to-convex functions.
DEFINITION 1.
A function
f e S
s
if, and only if, for
Re
f(Z)2f(iz)
> O.
We now extend this definition as follows:
z e D,
452
S.A. HALIM
DEFINITION 2.
A function
respect
f
with normalisations
.]J
is said to be starlike of order 8, with
to symmetric points if, and only if, for
S*(8)
s
We denote this class by
z e D and 0
< l,
S*(0).
S*
and note that
.< 8
s
s
In the same manner, we define the following new classes of close-to-convex
functions, which are generalisations of the classes in E1-Ashwah and Thomas
[2].
DEFINITION 3.
A function
f
normalised by
(1.1)
is said to be starlike of order 8, with
.
respect to conjugate points if, and only if, for
2zf’(z)
Re
z e D and 0
8 < l,
S*(8).
We denote this class by
c
DEFINITION 4.
A function
f
normalised by
(1.1)
is said to be starlike of order 8, with
respect to symmetric conjugate points if, and only if, for
.
/. ?_’.i._
Re
z e D and 0
.< 8
l,
S* (8).
We denote this class by
sc
RE,%.RK.
S*
The class
For
f c
s
S*( 8), Owa et
s
al
[4]
8 <2’
;)roved that for
f(z) f(-z)
Re
2.
(eg. Wu [9]
has been studied by several authors,
z e D.
3-48’
RESULTS.
THEOREM i.
Let
f e
Re
S*(8),
s
then for
-[f(z)-f(-z)]
z
The result is sharp for
f0(z)
To prove Theorem i.,
z
re
II(I-8)
">"
i8
e. D,
211(1-8)
l+r
2
>
e 8/(1-8)
fo given by
fo(-Z) 2z(l+z2) 8-1
we first require the following lemma.
LEMMA i.
Let
g e
S*(8)
Re
and be odd.
z
Then for
>"
l+r
2"
z
re
ie
D,
and Stankiewicz
[8]).
453
FUNCTIONS STARLIKE WITH RESPECT TO OTHER POINTS
PROOF OF LEMM
Pinchuk
[5]
z
re
ie
D,
F(z 2), so
[g(z)]2
z
(2.1)
.<r.
-i
is odd, we may write
g
then for
ll2(1-s)
z
Since
S*(8),
F e
showed that if
(2.1) gives
that
.<r
-i
where on squaring both sides, gives
r
z
2 Re
Ig(z)i
+ 1
h
.< r
Thus
2Re
l-rh
>"
z
l_r
>
h
+ 1
(i+r2)2
[6]
where we have used the inequality
ig(z)l >.
+ i
r
(l+r2) I-8
for odd starlike functions of order 8.
The Lemma now follows at once.
PROOF OF THEORY4 i.
S*(8),
s
f e
Since
it follows that we may write
g(z)
for
(z)-f(-z)
2
an odd starlike function of order 8.
g
An application of Lemma 1
proves
the Theorem.
Results analogous to Theorem
i
can also be found for the classes
s(6)
a
S*s(S).
THEOREM 2.
Let
f
S*(8).
Then for
C
z=re
iS
e D,
l/2(1-6)
f(z)+f()
Re
l+r
f(z) + f()
The result is sharp for
2(28-1)/2(1-8)
2z(l+z) 2(8-I)
PROOF
Since
f e
S*(8),
C
it is easy to see that, if
F(z)
then
from
F a S*(8).
(2.1)
that
f(z)+f()
Using the same techniques as in the proof of Lemma i, it follows
S.A. HALIM
454
Re
z
The result now follows immediately.
Similarly, we have the following result, which we state without proof.
THEOREM 3.
Let
S* (8).
f
z
Then for
sc
112(1-8)
f(z)-f(-)
Re
2112tl_S
>.
z
> 2
l+r
f(-)
f(z)
The resul is sharp for
D,
re
(28-1) 12(1-8)
2z(l+z)
We now consider the results of some integral operators.
obtained analogous results of the Libera integral
f e
S*(0),
s
They proved that for
given by
h
the function
In [i] Das and Singh,
operator.
t-l[ f( t)-f(-t) ]dt
h(zJ
70
S*(0).
also belongs to
s
The result below generalises that of Das and Singh.
THEOREM 4.
Let
f e
S*(8).
Then the function
s
H(z)
2z
also belongs to
z
a+__l
a
H
defined by
ta-l[f(t)_f(_t) ]dt,
(2.2)
JO
S*(6)
s
D and a + 6
z
.for
O.
We first require the following Lemma due to Miller and Mocanu [5].
LEMMA 2.
Let M and N
be analytic in
D with M(O)
N(O)
0
and let
6
be any real
number. If N(z) maps D onto a (possibly many sheeted) region which is starlike
with respect to the origin, then for z e D,
’(z)
M(z)
Re
N--(z
6
----->Re
N-- >
Re
M’ z) <
6
N’ z)
---->Re
N(--
>
6,
and
PROOF OF THEOREM
M(z)
<
h.
(2.2) ives,
za[ f(z)-f(-z) ]-a
Iz ta-l[
f( t )-f(-t ]dt
JO
2zH’ (z)
H(z)-H(-z)
z
ta-l[ f( t)-f(-t
dt
0
(z)
Note that
M(O)
6.
N-V’
say.
N(O)
0
and for
f
S*(6),
s
FUNCTIONS STARLIKE WITH RESPECT TO OTHER POINTS
i: + zN"(z)
N’ ()
N(z)
Thus
[- z[ i
+
is starlike if, and only if
455
(z)+f’ (-z) ]
a > -8.
Furthermore, since
Re
Re
N’ (z)
iI
(z)-f(-)
>
"
H e S*(8).
Lemma 2 shows that
s
Finally, we give the following analogous results for the classes
THEOREM 5.
Let f
S*(8).
c
a+--!l
H(z)
2z
also belongs to
Then
a
H
defined by
[z ta_l[f(t
+
Jo
S*(8)
c
f(----]dt,
(2.3)
z e D and a +
for
> 0.
PROOF.
Since
(2.3)
S*(8),
c
f e
ta-l[f(t)
gives
+ f(:)]dt
t a-I
2
0
t +
z
Y. Re
0
n=2
0
ZRea
n
n=2
=2
ta_l If(t)
+
dt
f(:)]dt
0
Thus
za[f(z)
2z
+
f()]
H’(z)
a
fz ta_l
[f(t) + f()]dt
0
z
H(z) + H(:)
ta_l[f(t
+ f(:)]dt
0
where
M(0)
N(0)
0 and N e S *
On using Lemma 2 it follows that
for a + 8 > 0.
H e S*(8)
c
THEOREM 6.
Let
f e S* (8).
sc
H(z)
a+--!l
2z
also belongs to
a
H
defined by
[z ta_l[f{t
f(-:)]dt,
0
S* (8)
sc
Then
for
S(8) and S (8).
sc
c
z
D and a + 8 > 0.
S.A. HALM
456
PROOF.
For
f e S*
SC
(8), (2.4)
Hi-i)
gives
a+l
a
f-z
a + i
2(-z) a+l
(-z)
z
a
a-l[f(t)
a + i
z)a
-(a+l)
t
f(-)]dt
0
z ta_l[f(t
(-z) n+a
n= 2
n + a
(-n +
(-l)n+lan)}
f(-)]dt.
0
As before, writing
2zH’(z)
H(z)
one can show that
i.
2.
N e S*
M(z)
N(z)’
H(-I)
and hence using Lemma 2 the result follows.
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J. Ramanujan Math. Soc. .2.(i), (1987) 85-100.
5.
P.T.Mocanu, Second order differential inequalities in the complex
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S.Owa, Z.Wu and F.Ren, A note on certain subclass of Sakaguchi functions,
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B.Pinchuk, On starlike and convex functions of order 8, Duke Math. Journal, 35.,
6.
M.S.Robertson, On the theory of univalent functions, Ann. Math., 3.7, (1936),
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K.Sakaguchi, On a certain univalent mapping, J.
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Math...Soc. Jpa.n,
ii,
(1959),
8.
J.Stankiewicz, Some remarks on functions starlike w.r.t, symmetric points,
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