455
[nternat. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 455-467
GENERALIZED CLASSES OF STARLIKE
AND CONVEX FUNCTIONS OF ORDER
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577
Japan
and
C.Y. SHEN
Department of Mathematics and Statistics
Simon Fraser University
Burnaby, B.C., V5A IS6
Canada
(Received June I, 1985)
We have introduced, in this paper, the generalized classes of starlike
ABSTRACT.
by using the fractional calculus.
and convex functions of order
We then
proved some subordination theorems, argument theorems, and various results of
modified Hadamard product for functions belonging to these classes.
We have also
established some properties about the generalized Libera operator defined on these
classes of functions.
KEY WORDS AND PHRASES. Subordination, fractional derivative, fractional integral,
generalized Libera operator.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 26A24, 30C45.
i.
INTRODUCT ION.
Let
A
denote the class of functions of the form
f(z)
z +
7 a zn
n
n=2
which are analytic in the unit disk
the subclass of
S
A
=a<
< I}.
Furthermore, let
consisting of all univalent functions.
Re(Zf’(z)
f(z)
(0
{z:Iz
U
is said to be starlike of order
for some
(i.i)
i)
S
denote
f(z)
of
if
> a
and for all
A function
(1.2)
z
U
We use
S* ()
to denote the class
456
S. OWA AND C. Y. SHEN
to
Re(l +
K()
We use
f(z)
that
K()
zf" (z)
f’ (z)
>
(1.3)
to denote the class of all convex functions of order
K()
K(0)
K
K()
and
[3],
(0 _<-
S*()
[4],
Pinchuk
S*(d)
zf’ (z)
if and only if
[I],
were introduced by Robertson
MacGregor
belonging
if we replace (1.2) by
is said to be convex of order
S
f (z)
Similarly, a function
of all starlike functions of order
< i).
Note
S*()
and that
The class S*()
S*(0)
and
and studied subsequently by Schlld
S*,
K()
[2],
and others.
Many essentially equivalent definitions of the fractional calculus (that is,
fractional derivatives and fractional integrals) have been given in the literature
[5,
(cf., e.g.,
Chapter
13], [6], [7], [8], [9], [i0, p.
28 at
seq.],
and
[ii]).
For our discussion, it is more convenient to use the following definitions which
[12
were employed recently by Owa
-
The fractional integral of order
DEFINITION 1.1.
f (z)
functlon
by
D
I
where
> 0
z
and by Srivastava and Owa
f(z)
f(z)
f()
l-i d
J
0
Is
(1.4)
(z-.)
an analytlc function
log(Z-)
to be real when
DEFINITION 1.2.
f (z)
function
is defined, for a
z
1
a simply-connected region of the
in
Z-plane containing the origin, and the multiplicity of
requiring
%
[13].
z-
l-i
(z-)
is removed by
> 0
l
The fractional derivative of order
is defined, for a
by
z
d
F(1-1) dz
1
D f(z)
z
l
j
0
0 _<-
where
I < 1
f(z)
f()
(1.5)
(z-)
is an analytic function in a simply-connected region of
the z-plane, and the multiplicity of
(z-)
is removed as in Definition i. 1
above.
Under the hypotheses of Definition 1.2 the fractional
DEFINITION 1.3.
n+%
derivative of order
is defined by
Dn+f(z)
z
0 _<-
where
Let
< I
S*(,)
n
N
d
n
dz n
l
D f(z)
z
(1.6)
{0,1,2
O
be the class of all functions
f(z)
in
S
satlsfying the
inequality
Re(A(l’f)
f (z")
fcr
i, 0
< i, and for all
A(l,f)
Also let
A(l,f)
K(C,l)
S*(,l)
> e
F(1-l) z
(1.7)
z
U
where
f(z)
I+IDI+
z
be the class of all functlons
for
< 1
and
0 <-
< 1
(1.8)
f(z)
in
S
We note that
such that
S*(,0)
S*()
and
CLASSES OF STARLIKE AND CONVEX FUNCTIONS
K(,0)
K().
Thus
classes
S*(e)
and
S*(,l)
K()
[14].
introduced by Owa
K(,I)
and
respectively.
are the generalizations of the
with
a
_
n
C (, l)
T
Recently, Owa and Shen
be the subclass of
S
z
f(z)
n=2
0
for all
S*(,I)
The classes
[15]
were
K(,I)
and
conslsting of all functions of the form
a z
n
n
(1.9)
T* (,l)
We introduce the classes of function
n
K(,I)
and
proved some coefficient
S*(,I)
inequalities for functions belonging to the classes
Let
457
and
as follows
S*(,l) N T
T*(,I)
K(,l) q T
C(,I)
T* (e,l)
The classes
cases
T*(,0) and
T*(,l)
C(,I)
and
C(,0)
C(,I)
and
by Silverman
(i.i0)
(I.ii)
[14 ],
[16].
were studied by Owa
were studied by Silverman
and the special
Thus the classes
provide an interesting generalizatlon of the ones considered
[16].
In sections 2, 3 and 4, we shall prove several results for functions
S*(,l)
belonging to the generalized classes
K(,l)
T*(,I)
and
S*(,l;a,b) of functions in section 5.
then introduce the class
section, we shall study a certain Integral operator defined on
2.
C(,I)
We
In the last
A.
SUBORDINATION THEOREMS.
f(z)
Let
and
g(z)
be analytic in the unit disk
f(z)
is subordinate to
w(z)
analytic in the unit disk
f(z)
g(w(z)).
disk
U
U
f(z)
,< g(z),
f(z)
.<
g(z)
if there exists a function
w(0)
which satisfies
In particular, if the function
then
g(z)
f(0)
if and only if
lw(z)
0,
< i, and
is univalent In the unit
g(0)
and
g(U)
f(U)
[18]).
[17],
(cf.
g(z), written
Then we say that
U
In order to prove our first theorem, we require the following lemma due to
Miller, Mocanu, and Reade
LEMMA 2.1.
g’ (0) @ 0
[19].
Re(8) > 0, Re(y) >- 0, f(z)
Let
A,
and
g(z)
A
with
and
Re
{(8-1) zg’
(z)
rain(Re(7’)
(5
+
g(z)
z@"(z) +
g’ (z)
1
>-6
(2.1)
where
(2.2)
0
and
60
If
f(z)
-<
2Re (B) Re (y)
II/l
/
I-1
g(z), then
z
(Z’
0
z
1
f(t) tY-idt)
(7-"
1
B
i
g (t)
0
ty -idt)
(2.4)
458
S. OWA AND C. Y. SHEN
Using Lemma 2. i, we can prove
THEOREM 2.1.
S*(,l) (0 <_
Let the function
< i).
< i;
f(z)
defined by
be in the class
(i.I)
Then
z
A(l, f)
1
dt
f(t)
,<
(2-i) (i +
1
log(l-z))
z
(2.5)
0
where
A(l,f)
(1.8).
is given by
Note that the function
PROOF.
I+ (i-2) z
g(z)
onto the half domain such that
U
S*(,I)
from the definition of the class
A(l,f)
g(z)
(zg"(z)
g’ (z))>
l, y
Taking
that
(2.6)
1
analytic wlth
is
(z
0
g’ (0)
2 (l-s)
M
0
and
U)
(2.7)
Lemma 2.1, we see that the function
in
the hypotheses of Lemma 2.1.
g(z) satls fies
Thus we have
z
z
I
1
This Implies
l-z
Furthermore, the function
I + Re
Re (w) >
l+(l-2)z
g(z)
f(z)
defined by
U)
(z
l-z
maps the unit disk
g (z)
!(l’f))dt-<
f (t)
I
1
0
l+(l-2)t
dt
l-t
(2.8)
0
which implies (2.5).
COROLLARY 2.1.
S*(0,I) ( <
i).
Let the function
f(z)
defined by
(i.i)
be in the class
Then
z
1
I
A(,f)
Jf (t)
dt
0
COROLLARY 2.2.
S*()
(0 <-
< I).
Then
tf’ (t)
(--f(----)
j
dt
0
Similarly, we have
THEOREM 2.2.
log(l-z)
(2 9)
defined Dy
f(z)
(2-i)(i +
Let the function
< i; I < i).
K(,I) (0 _<-
-
Let the function
z
1
1
1
(I.ii
be in the class
1
--z log(l-z))
(2.10)
defined by (i.I)
f(z)
be in the class
Then
z
1
I
A(A(I,f))
(%(l,f)
dt
-< (2-i) (i
+
1
(2.11)
log(l-z)
0
where
A(l,f)
PROOF.
is given by (1.8).
Note that
f(z)
Consequently, on replacing
COROLLARY 2.3.
K(,I)
f(z)
by
Let the function
f and only if
A(l,f)
f(z)
A(%,f)
S*(,l).
in Theorem 2.1, we have Theorem 2.2.
defined by (I.i)
be
in
the class
CLASSES OF STARLIKE AND CONVEX FUNCTIONS
K(O,I) (I < i)
Then
?
1
459
A(A(I, f)
i
(2.12)
0
COROLLARY 2.4.
(0 & e < I).
K()
-
Let the function
Then
Z
I
1
tf" (t)
(I + -f,
(t))
at
0
3.
ARGUMENT THEOREMS.
f(z)
defined by (i.i) be in the class
1
-z log(l-z))
(2-i) (I +
(2.13)
In this section, we derive the argument theorems for functions belonging to
the classes S*(,I)
and K(e,l).
THEOREM 3. i.
S*(,l) (0 _<arg
< i).
i+I l+l
D
A(I, f)
w(z)
lw(z)
(i. i) be in the class
2(i-)
Izl
+(-2)
Izl
sin-l(
(z (U)
(3.1)
f(z)
i+ (I-2) w (z)
f(z)
and
Z
deflned by
In view of (2.6), we can write
PROOF.
where
<-
("f(z)
z
f (z)
Then
A0 (I, f)
A0(l,f)
where
Let the function
< i;
is
(3.2)
l-w (z)
an analytic function in the unit disk
U
< 1
We note that the linear transformation
p(z)
1 + Bw(z)
1 + Aw(z)
wl
maps the disk
z
&
and satisfies
w(O)
O
(3.3)
onto the disk.
(3.4)
It follows from (3.4) that
(3.5)
l-’zl
fz,
This completes the proof of Theorem 3.1.
COROLLARY 3.1.
S*(0,I) ( < i).
arg
where
AO(l,f)
f(z)
=<
f(z)
defined by (i.i) be in the class
’=< sin
(z
(3.6)
U)
zl/IDl+tf (Z)
CO.’.OLLARY 3.2
S*(e,l) (0
Let the function
Then
Let the function
c < i; I < i).
Then
f(z)
defined by
(i i)
be
in
the class
460
S. OWA AND C. Y. SHEN
Izl
l-(l-2e)
l+Izl
A(A,f)
where
PROOF.
IA(l,f)
l+(l-2e)Izl
llf(z)
(z
l-lz
U)
(3.7)
is given by (1.8).
The proof is clear from (3.5).
Moreover it is easy to show that
THEOREM 3.2.
A0(A
A0(A,f)
z
Let the function
A0(A
f(z)
Z
f)
(z
defined by
2!z
l+Izl 2
sin- 1
Let the function
< i; A < i).
l-(l-2e)
l+Iz
4.
Izl 2
Izl
u)
(3.8)
(i.i) be in the class
(z u)
(3.9)
zl+ADl+Af(z)
z
COROLLARY 3.4.
A(A,f)
where
2(i-)
+(-)
A0 (A0 (A,f))
A0(A,f
K(,A) (0 _-<
<- sin -I
f)
Then
arg
where
defined by (I.i) be in the class
I+ADI+A
f (z)
z
COROLLARY 3.3.
K(0,A) (A < i).
f(z)
Then
A0 (A0 (A, f)
arg
w,efe
Let the function
< i; A < i).
K(,A) (0
f(z)
defined by (i.I) be in the class
Then
Izl <--I A(A(I,f) ]& l+(l-2e)Izl
A(A-f)
l-lz
U)
(z
(3. i0)
is given by (1.8)
MODIFIED HADAMARD PRODUCT.
Let
f.(z)
1,2)
(j
be defined by
7. a. z
n=2 3’n
z
f.(z)
n
(a.
fl*f2 (z)
We define the modified Hadamard product
fl*f2 (z)
7. a
z
n=2
0)
(4.1)
3,n
z
I ,na2 ,n
of
fl(z)
and
f2 (z)
by
n
(4.2)
We recall here the following two lemmas due to Owa
[14]
before state and
prove our results of this section.
LEMMA 4.1.
the class
Let the function
<=
T*(,I) (0
A <
< I;
F (n+l) F (l-A)
for
0
<=
< i
and
LEMMA 4.2.
the class
C(,A)
(0 -<
E
n=2
for
0
< 1
A < 1
and
< I;
A
F(n+I) F(1-A)
F(n-l)
A < 1
n
The result
Let the function
f(z)
< I)
Then
f(z)
is in
if and only if
a
l"(n-A)
n=2
be defined by (1.9).
f(z)
i)
_<- 1
(4.3)
(4.3) is sharp.
be defined by (1.9).
Then
f(z)
is in
if and only if
F(1-A)
{F(n+I)
F(n-l)
}
an -<
The result (4.4) is sharp.
1
(4.4)
CLASSES OF STARLIKE AND CONVEX FUNCTIONS
the aid of Lemma 4. i, we can prove
With
THEOREM 4.1
Let the functions
T*(Q,A) (0 _<-
class
is
461
T*(Z(Q,A) ,l)
the class
in
A <
< i;
2
(1-)
1
defined by (4.1) be in the
fl*f2(z)
(l-l)
C2-+l)
(Q,A)
1,2)
(j
where
2(l-Q)
1
f. (z)
Then the modified Hadamard product
i).
2
2
(1-1)
(2-e+l)
(4.5)
2
2
The result is sharp.
PROOF.
We use a technique due to Schild and Silverman
[20].
It is
sufficient to prove that
F (n+l) P (l-A)
7
n=2
for
8
8} al,na2, n -<-
(n-A)
8
1
(4.6)
<- 8(Q,l).
By using the Cauchy-Schwarz inequality, we know from (4.3) that
(n+l) F (l-)
<_- I-Q
7.
Q
/a
(4.7)
a
F(n-l)
I ,n 2 ,n
n= 2
8
Therefore we need find the largest
7.
n=2
F(n+I) F(I-A)
F (n-l)
such that
8} al, na2,n /(1-8)
F (n+l) F (1-7,)
F (n-l)
7.
<=
Q}
al,na2,n / (l-Q)
(4.8)
n=2
or
F(n+l) F(1-1)
F(n-l)
F(I-I)
(l-Q) {F(n+I)
F (n-A)
}
(i-)
6al’na2’n
(n >- i)
(4.9)
}
In view of (4.7), we observe that it suffices to find the largest
F (n+l) F (1-1)
note that (4.10)
F (n-k)
F(n+l)F(1-1)
r Cn-l)
(l-Q)
r (n-l)
we
F(n+l)? (l-A)
(1-8)
I-Q
8
such that
Q}
(4.10)
8
gives
1
I-Q
H(n)(H(n)_ )2
i-
(n
i)
(4.11)
2
where
H (n)
Since, for fxed
a
F (n+l) F (i-I)
F (n-l)
(4.12)
462
S. OWA AND C. Y. SHEN
l-e
2
H (n)-
H (n)
1
(n)ia
H
is an increasing function of
(H--)_--)
I-C
H(2)-C
1
we have
n
i-
H(2)
1
}2
I-C
i-
2
2 (l-s)
i
)2
(i-I)
2
2 -o+al
2
2
(l-a) (i-I)
2
1
(4.13)
(2-0+(]1)
f. (z)
Finally, by taking the functions
(j
1,2)
defined by
(l-s) (1-1) 2
z
2-a+C
z
f.(z)
2
(4.14)
we can see that the result is sharp.
COROLLARY 4.1.
Let the functions f. (z)
(I < i).
the class
T*(0,1)
the class
T*(8(I) ,I)
(j
1,2)
defined by (4.1)
Then the modified Hadamard product
be In
fl*f2(z)
is in
where
2(1+I)
8(I)
3+21-I
(4.15)
2
The result is sharp.
By using the same technique and Lemma 4.2, we have
THEOREM4.2.
class
C(d .)
Let the functions
< i; I < I)
(0
is in the class
C(y(e,1) ,k)
f. (z)
Then the modified Hadamard product
(e,1)
i
(l-a)
2
COROLLARY 4.2.
the class
C(y(1) ,i)
Let the functions
y(1)
2
f.(z)
(l-e) (1-I)
2 (2-e+el)
z-
(I < i).
C(0,I)
(1-I)
2
2-e+ I
3
2
(l-a) (1-I)
2
2 (2-e+el)
The result is sharp for the functions
the class
fl *f2 (z)
where
1
f (z)
1,2) defined by (4.1) be in the
(j
2
z
(4.16)
1,2)
(j
defined by
2
f. (z)
(4 17)
(j
1,2)
defined by (4.1)
Then the modified Hadamard product
fl*f2(z)
be in
is in
where
2 (3+21-I
2
(4.18)
7+31-312+I 3
The result is sharp.
5.
THE CLASS
S* (e,l;a,b)
Let the function
f(z)
defined by (1.1)
P(f(z);e,l;a,b)
(A(l,f____)_
f(z)
be in the class A and let
a
s)
(A(A(I,f)
A(l,f)
b
(5.1)
463
CLASSES OF STARLIKE AND CONVEX FUNCTIONS
a
where
S*(,l;a,b)
if
Then we say that
are real numbers.
b
and
f(z)
Re{P(f(z) ;,l;a,b)
for
I
0 -< e < i
< i
is in the class
0 <-
Let
(5.2)
> 0
z 6 U
and
S*(,l)
S*(e,l;l,0)
We observe that
THEOREM 5.1.
f(z)
satisfies
(
S*(e,l;a,b)
and
I < I, and
< i,
K(e,l)
S*(,I;0,1)
0 _<- t <- 1
Then
S*(,I;I,0)
c S*(,l; (a-l)t+l,bt)
PROOF.
Let the function
S*(,l;a,b) D S*(e,l;l,0)
0
z E U
for
(a-l) t+l
whlch shows that
l-t
(V(z))
(V(z))
t
(5.6)
t
(a+b-l) t+l
> 0
(l-t)larg(U(z))
+
(5.7)
and
tlarg(V(z))
maps the unit disk
F (z)
U
-<
(5.8)
onto a domain whlch is contained in
Hence we complete the proof of Theorem 5.1.
the right half-plane.
COROLLARY 5.1.
l-t
bt
A (A (l,f))
by
(l-e)
&
It follows from (5.4) and (5.5) that
e)
(U(z))
F(z)
larg(F(z))
Further, setting
(5.5)
z E U
for
F(z)
F(0)
(5.4)
f(z)
(U(z))
Then we have
)
A(l,f)
(A(l,f)
f(z)
Define the function
be in the class
b
A(A(I,f)
A(,f)
)
f(z)
Re(V(z))
Re(U(z)) > 0
a
A (l,f)
u(z)
we have
defined by (1.1)
f(z)
and let
v(z)
Then we have
(5.3)
Let
< i,
0
I < 1
and
0 -< t -< i.
Then
S*(e,l;a,b) N S*(,l;0,1)
c S*(,I; at,
6.
J (f)
c
J (f).
GENERALIZED LIBERA OPERATOR
For a functlon
f(z)
(5.9)
(b-l)t+l).
c
belonging to the class
A
we define the operator
by
Jc (f)
c+ic
z
J (f) when c
c
In particular, the operator
This operator
I tc-lf(t)dt
z
(c
>=
0).
(6.1)
0
is a natural number was studied by Bernardi [21].
Jl(f)
and Mocanu, Reade and Ripeanu [24].
was studied by Libera [22], Lvingston [23],
It follows from (6.1) that
S. OWA AND C. Y. SHEN
464
=
fzJ
c+l
J (f)
c
7. a
0
n=l
(c---l)
7.
n+c
n=l
n
a z
n
tn+c_l
(a
dt
i)
I
n
(6.2)
In order to prove our theorem, we recall here the following theorem due to
Jack [25].
Then, if
lw(z)l
a point
z
w(z)
Let
LEMMA 6.1.
m
w(0)
with
Izl
0
r (0 _<- r < i)
at
we can write
0
z0w’(z 0)
where
U
be regular in the unit disk
attains its maximum value on the clrcle
mw(z
(6.3)
0)
m->_ 1
is real and
Furthermore, we need the following lamina by Pascu
[26].
J (f) ( S*.
c
and
6.2, we prove
With the aid of Laminas 6.1
LEMMA 6.2.
If
(0 -<
S*
S*, then
Let the function
THEOREM 6.1.
S*(,I)
f(z)
< I; i < i).
f(z)
be in the class
(i.i)
defined by
J (f)
c
Then the functional
is in the class
s* (,l).
PROOF.
Define the function
A(l,J (f))
w(z)
by
i+(i-2) w (z)
l-w(z)
c
J (f)
c
Then
w (z)
(w(z)
is a regular function in the unit disk
U
w(0)
with
(6.4)
i)
0.
Differentiating both sides of (6.4), we obtain
z(A(l,J (f)))
j
c
A(l,J (f))
z(J (f))
c
j (f)
c
c
(f)
c
j
c
(f)
2 (l-s) zw’ (z)
2
(l-w (z))
(6.5)
Note that
z(J (f))
c
(c+l)
z( 7
n+c
n=l
a z n)
n
7. n (c+l)
n+c
n=l
a z
n
7.
(c+l)
n=l
(a
z(A(l,J (f)))
c
c(c+l)
a z
n
n+c
n
(6.6)
z[F(l-l) zl+iD zl+IJ c (f) ]’
z[F(1-1) z
z( 7.
n=l
ItIDl+l
z
i)
n
(c+l) f(z) -c J (f
c
and
n
(c+l)
a z n)
n
7.
n+c
n=l
(c+l)F(n+l)F(l-l)_
(n+c) F (n-l)
a
n
zn)
]’
CLASSES OF STARLIKE AND CONVEX FUNCTIONS
F
(i-I)
7. n (c+l) (n+l) F
(n+c) F (n-l)
n=l
n=l
7.
(c+l)
n=l
n
anZ
F(n+I)F(I A)
a z
F (n-l)
n
c(c+l).}
n+c
7. {(c+l)
465
F (n+l) F (l-l)
F (n-A)
anZ
n
Z
c
n=l
(c+l) F (n+l) F
(n+c) F (n-l)
(c+l) A(A,f) -cA(A,J (f))
c
a z
n
n
(6.7)
Applying (6.6) and (6.7) to (6.5) we have
(c+l)
A(A, f)
J (f)
c
A(A,J (f))
,f(z)
(c+l)
2 (l-s) zw’ (z)
c
J (f)
c
J (f)
c
(l-w (z))
2
(6.8)
or
A(A J (f))
A(I f)
J (f)
c
(c+l) f(z)
2 (l-o) zw’ (z)
(l-w (z)) 2
c
J (f)
c
f(z)
(6.9)
Since
z(J (f))
c
J (f)
c
(c+l) f (z)
J (f)
c
+ c
(6 .i0)
equatlon (6.9) becomes
A(I, f)
i+ (i-20) w (z)
f (Z)
l-w (Z)
2 (l-S) zw’ (z)
2
l-w (z))
1
z(J (f))
c
J (f)
(6.11)
+ c
c
z
Assume that there exists a polnt
U
0
such that
(w(z
n%ax
0)
i)
(6.12)
Then, Lemma 6.1. implies that
ZoW’(Z O)
where
m
is real and
i@
w(z
O)
e
0
m w(z
m->
i
O)
Applying Lemma 6.1
we know that
i+ (i-2) w
Re(
w(z)
to
f(z
Re
l-w (z
O)
2 (l-S) mw
+Re{
(l-w (z
e
(Zo)
O)
(l-z) m
1-cos @
(Zo)
2
1
Zo (Jc (f))
Re
0
i
Zo (Jd (f))
J (f)
c
We note that Lemma 6.2 gives, for
+ c
J (f)
c
f(z)
S*
/
c
and putting
S. OWA AND C. Y. SHEN
466
1
Re
-}
{z 0 (J c (f))%
+ c
J (f)
c
z
Re(
0
(Jc (f))
+c
J (f)
c
z0(J c (f))’
>
2
z (J (f))’
0 c
)}
J (f)
2
+ c}
J (f)
c
{I
+
m
c
0
(6.14)
Consequently, we conclude that
A(I, f)
_-< C
(f-)
Re
(6.15)
which contradicts the condition
c
J (f)
c
COROLLARY 6.1.
PROOF.
Hence we have completed the proof of the theorem.
0
S*() c S*
for
defined by (1.1) be in the class
f (z)
J (f)
c
is also in the class
S*(e).
in Theorem 6.1, we have
S*(a) Q S*
f(z)
Noting
I
>
l-w (z)
Then the operator
Taking
This proves from (6.4) that
U
(l+(l-2)w(z)
Let the function
(0 _<- a < i).
S*(a)
Re
S*(a,l).
or, that J c (f)
z
for all
A(I,J (f))
Re
S*(,I)
f(z)
< 1
w(z)
Thus we have
J (f)
c
0 <-
< 1
S*(a)
we have the corollary.
Finally, we state and prove
THEOREM 6.2.
K(,I)
with
J (f)
operator
PROOF.
Let the function
(0 _-<
< I)
f (z)
defined by (i. i) be in the class
l(l < i)
such that
S*(,l) c S*
Then the
K(,l)
also in the class
Is
S*(,l) for
J (f)
c
From the definition of the class K(,I), we obtain that
In view of Theorem 8, we observe that
S*(a,l) c S*
f(z)
and
S*(a,l)
A(l,f)
K(a,l)
f(z)
J (ACl,f))
S*Ce,l)
A(I,J (f))
S*(e,l)
c
c
J (f)
K(e,l)
Thls completes the proof of the theorem.
Let the definition
COROLLARY 6.2.
K()
(0
<=
PROOF.
i.
< i).
f (z)
defined by (i.I) be in the class
is also in the class K(e).
J (f)
c
in Theorem 6.2, we can easily obtain the result.
Then the operator
Letting
i
0
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