Internat. J. Math. & Math. Sci.
Vol. 9 No. 4 (1986) 721-730
721
AN APPLICATION OF THE RUSCHEWEYH DERIVATIVES
SHIGEYOSHI OWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577, Japan
SEIICHI FUKUI
Department of Mathematics
Wakayama University
Wakayama 640, Japan
XOICHI SAKAGUCHI
Department of Mathematics
Nara University of Education
Nara 630, Japan
and
SHOTARO OGAWA
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577, Japan
(Received January 15, 1986)
ABSTRACT.
Let
Df(z)
be the Ruscheweyh derivative defined by using the Hadamard
I+.
are introduced
and
Certain new classes
z)
f(z) and z/(l
The object of the present paper is to
by virtue of the Ruscheweyh derivative.
product of
S
establish several interesting properties of
J (f)
integral operator
of
f(z)
Further, some results for
and
are shown.
Ruscheweyh derivative, Hadamard product, starlike function,
convex function, integral operator.
KEY WORDS AND PHRASES.
Primary 30C45.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
denote the class of functions of the form
Let
INTRODUCTION.
f(z)
Y-
n=O
an+
z
which are analytic in the unit disk
0
{z:
Izl
<
belonging to
(l.l)
I)
(a
I}.
Let
consisting of univalent functions in the unit disk
of
U
n+l
denote the subclass of
A function
f(z)
is said to be starlike with respect to the origin in the unit disk
if it satisfies
. .
Re
for all
z
{zf’(z}
>
f(z)
We denote by
the origin in the unit disk
(1.2)
0
the class of all starlike functions with respect to
A function
f(z)
belonging to
is said to be
722
S. OWA, S. FUKUI, K. SAKAGUCHI AND S. OGAWA
convex in the unit disk
if it satisfies
zf"(z)}
"(z) >
Re {I +
z U.
for all
we
K S’c- S.
1,2) in A
f.(z)
n=O
and that
n+l
(a
jz
I).
j
fl*f2(z)
(or convolution product)
product
Hadamard
Se
,
be given by
an+
l
3
Then the
zf’(z)
if and only if
f.(z) (j
Let
the class of all convex functions in the unit disk
denote by
f(z)
We note that
(1.3)
0
fl(z)
of
and
f2(z)
is defined by
n=OZ an+l,lan+l,2
fl *f2(z)
z
n+l
(1.5)
By the Hadamard product, we define
z
Daf(z)
z) 1+
(1
the
A
f(z)
for
The symbol
Ruscheweyh derivative of
Def(z)
>-
(a
*f(z)
(1.6)
l)
was introduced by Ruscheweyh
[I], and
is called
f(z).
To derive our results, we have to recall here the following lemmas.
LEMMA
#(0)
satisfy
and 6
(161
be analytic in the unit disk
Let #(z) and g(z)
([2]).
O. Suppose that for each
g(O)
O, #’(0) # O, g’(O)
I), we have
(z)*
0 (z
U),
C(z)
lw(z)l
z 0,
<
[z[ <
(.7)
,)
and satisfying
analytic in the unit disk
,(z) > o
"*g(z)"
(z
Re
U ).
{F(z)}
(t.8)
attains
Let
w(z) be regular in the unit disk
its
maximum value on the circle
U
Izl
with
r (0
<
w(0)
r < I)
O.
Then,
at a point
we can write
ZoW’ (z O)
where
(0
0
F.g(z).
LEMMA 2 ([3]).
if
g(z)
I)
we have
Re
where
+ 6z
-z
F(z)
Then for each function
>
and
(II
m
is real and
LEMMA 3 ([4]).
m
>
mW(Zo),
I.
For a real number
z(Daf(z))
(a +
a (a >-I),
we have
l)Da+If(z) -aDaf(z).
(1.9)
723
AN APPLICATION OF THE RUSCHEWEYH DERIVATIVES
Note that (1.9) holds true for
REMARK.
LEMMA 4 ([5])
(u,v)
Let
complex plane) and let
u
u
=-I.
:
be a complex function,
+ iu2, v
+
v!
iv
CC(C is
Suppose that
2.
the
satisfies the
following conditions
(i)
(u,v)
(ii)
(!,o)D
and
(iii)
Re{(iu 2,
vl)} <
.
p’())
.
D
fo= al
for
z
2.
PROPERTIES OF
THEOREM I.
D f(z)
Re{(l,O)} > 0;
0 (0
Df(z).
<
z
and such that v
the
in
Re{(p(z), zp’(z))}
If
>
*Df(z)
[!),
>
then Re p(z)
0
and satisfy the condition
Z
z(Df(z))
Def(z)
De(zf’(z))
Def(z)
z)
(I
is also in the class
z/(l
z)
l+e,
(2.1)
*f(z)
z)
f(z),
g(z)
*
(z))
I+ *(zf’
z
(
6 =-I, #(z)
2
u2)/2"
such that (p(z),
unit disk
0 (z
(! +
We note that
PROOF.
Setting
<
Applying Lemma I, we prove
f(z) be in the class
I) for a __>- I. Then
Let
<
regular
be
D
(iu2,v I)
0 for all
+ pl z + p2 z 2 +
p(z)
Let
D;
is continuous in
F(z)
and
zf’(z)/f(z) in Lemma
1, we have
z(Df(z))’}
Re
a
which implies
D
<
Izl
<
I)
3.
e
for
PROOF. Since
De(zf’(z))
f(z)
$
THE CLASSES
>__-I.
f(z)
Then
Def(z)
AND
(2.2)
and satisfy the condition
,
Theorem
In view of Theorems
(f(z)
{f(z)
E A:
e
D f(z).
e
=> -1}
i A:
f(z) (
K,:
Now, we derive:
THEOREM 3.
PROOF
For
For
f(z)
>_ O,
we have
e+l’
z(Def(z))’
Def(z)
derives
z(Def(z))
and 2, we can introduce the follow-
and
K
0
Def(z)
ing classes;
S"
De(zf’(z))
is also in the class
if and only if zf’(z)
Hence we have
U),
(z
0
Let f(z) be in the class
THEOREM 2.
(0
.
>
Def(z)
+I
we define the function
+ w(z)
w(z)
(w(z)
w(z) by
1).
S. OWA, S. FUKUI, K. SAKAGUCHI AND S. OGAWA
724
Then, with Lemma 3, we have
z(Def(z))’
Def(z)
f(z)
D
+
e
Def(z)
(I + e) + (I
(I + e)(l
+ e}
(3.2)
w(z))
Differentiating both sides of (3.2) logarithmically, it follows that
zD
D
e+l
e+1
Suppose that for
+ w(z)
w(z) + (I
f(z))’
f(z)
z
2zw’(z)
w(z)){(l + e) + (!
(3.3)
e)w(z)}"
0
(w(z
O)
+ 1).
(3.4)
Then it follows from Lemma 2 that
z0w’ (z 0) row(z0)
where
m
is real and
m
Re{
i60
>=
I.
z0(D
w(z 0)
Setting
e+l
we obtain
e
f(z 0))’
De+ f z0
+ w(z 0)
w(z 0)
2mw(z0)
,}+Re{
w(z0)){(1
(1
cos8 O)
m(1
<
+ ) + (i
a)W(Zo)))
O,
(3.5)
M
where
M
{e(l
cosS0) + (I
This
contradicts
that
I(,.)I
<,
f(z)
THEOREM 4.
For
Re
Def(z)
@
-)cosS0}2sin280
e+l"
{II" + w(’z
w(z. >
0,
S
>
e
O,
we have
{id},
e
where id is the identity function
Note that
+ {a + (I
f(z)
Dez
z
Therefore, w(z)
6as
to satisfy
uha
z(Def(z))’’}
S*
PROOF.
that
zU.
fora
Re
which implies
-e)sin280 }2
the hypothesis
f(z)
for all
z.
,
and that
(3.6)
AN APPLICATION OF THE RUSCHEWEYH DERIVATIVES
Re
for all
a.
{}
D z
>
(zf_U)
0
id
Consequently, we conclude that
.
a
for all
f(z)
For the converse, we assume that the function
S
class
Z
n=O
a
>
0.
f(z)
Is in the
S*.
+ I)
r(n +
n!r( + I)
an+IZ
n+1
"
S*
It is well known that
lan+ll
for
.
belonging to
Then we have
D f(z)
for all
725
<
n
(n> I)
+
This implies
=
r(n +
+ I)
n!r( + I)
an+l <
n +1
(n
>
1),
(3.7)
(n
>
1)
(3.8)
or
llan+ll
<
(n + l)!r(a + I)
r(n + a + I)
= > 0. Therefore, we have f(z)
By virtue of Theorem 3, we prove:
_> 0, we have
THEOREM 5. For
for all
z.
Ka+
PROOF.
K.
By Theorem 3, it follows that
f(z)
Da+If(z) K
z(Da+if(z))
Da+l(zf (z))
Ka+l
zf’ (z)
E
Sa+ I
====zf’ (z)
(5
=== ma(zf
S
(z))
.z(Dafz))
This asserts the result of the theorem.
THEOREM 6.
FOR
a
>
O,
S*
S*
we have
S*
S
S. OWA, S. FUKUI, K. SAKAGUCHI AND S. OGAWA
726
Ka
{id},
f(z)
where id is the identity function
z.
The proof of Theorem 6 is similar to that of Theorem 4.
Furthermore, an application of Lemma 4 to the classes
THEOREM 7.
f(z) be in the class
Let
{I Df(z))
z
Re
<
where
<
8
>
Daf(z)}
+ I/2(8-I).
A
*
f(z)
p(z)
u
#(u,v)
+
u
is
>
Re{#(l,0)}
!
(3.9)
p(z)
by
(3.1o)
Differentiating (3.10) logarithmically, we have
zp’ (z)
+ i.
p(z) + (A- I)
8-
Zg’(z)
iu
2
and zp’(z)
v
(3.11)
in
u
(u,v)
by
(3.13)
(C -{I-A})xC,
Moreover, for all
(3.12)
+ I.
+ (A- I)
D
U).
(z
0
and define the function
iv2,
V
8-
continuous
0.
+
v
>
+ I}
p(z) + (A- I)
8-
(u,v)
Then
(ze
28
it follows that
a’
Re
Let
Then
p(z) + (A- I)
z(Daf(z))
Daf(z)
Since
>-I.
gives:
3/2.
A
where
with
and
8-I
We define the function
PROOF.
*
a
together
and
(iu2, Vl) D
such
(I,0)
with
that
v
<
0,
<
-(I +
D
and
u22)/2,
we can show that
v
Re
{dp(iu2,vl)}
<
for
<
8
<
3/2.
Re p(z)
Re
AIDaf(z)l
z
2
+ (A- I)
0
,
.} +
2
2)
"2)
2{u22 + (A-
+
z
for
-(A- I)}
>
8
COROLLARY I.
3/2
Let
in Theorem
f(z)
(z
0
7, we have:
be in the class
*
It
that is,
This completes the assertion of Theorem 7.
Taking
(3.14)
satisfies the conditions in Lemma 4.
#(u,v)
>
u
(A- I)(1 + u
-I
8-
Hence the function
follows from this fact that
Re {.
8-
with e
>
-I.
Then
).
(3.15)
AN APPLICATION OF THE RUSCHEWEYH DERIVATIVES
I/2
af()
COROLLARY 2.
f(z)
Let
<
where
<
PROOF.
(3.16)
be in the class
"i[Daf[z))’# B-I
Re
727
>
>
a
with
-I.
Then
(z
2B
-E. U),
(3.17)
3/2.
Note that
Ks ====> Df(z)
f(z)
====
K
S*
(z)) E S*
z(Def( z))’
D(zf
<==== zf’ (z) E S*,
which implies
D(zf’(z))
(Df(z)),.
Therefore, we have the corollary with the aid of Theorem 7.
INTEGRAL OPERATOR J (f). We define the integral operator
c
4.
j
(f)
z.j
+
c
z
for
A
f(z)
[6].
Livingston [8].
Bernardi
.
THEOREM 8.
the class
The operator
f(z)
tc_if(t)dt
J (f)
the
by
(4.1)
(c >-1)
when
be in the class
Define the function
z(DJ (f))’
DJ(f)
N
c
operator
$,
PROOF.
c
0
In particular,
Let
J (f)
Jl
$
{i, 2, 3,
(f)
was
with
studied
=>
0.
was studied by
by Libera
Then
J(f)
[7]
and
is also in
w(z) by
+ w(z)
-w(z)
(w(z)
I).
(4.2)
Then, by taking the differentiation of both sides logarithmically, we have
2(DJ(f))" + z(DJ(f))
z(DJ(f))
DJ
(f)
z2(D=f(z))
+
z(DaJa(f) )’
2zw’ (z)
(I + w(z))(l
w(z))
(4.3)
Since
z(z(Def(z))’)
z(Def(z)) ’,
(4.4)
we can see that
z2(DC*f(z))
((, +
l)z(DC*+If(z))
(c, +
l)z(Daf(z))
(4.5)
728
S. OWA, S. FUKUI, K. SAKAGUCHI AND S. OGAWA
by Lemma 3.
J(f)
Furthermore, it follows from the definition of
a+l
Daf(z)
D
that
Ja( f)-
(4.6)
By using (4.5) and (4.6), we have
z
2
Daja (f))"
(a +
l)z(Sa+IJa(f) )’
(a +
l)z(Daf(z))
(a +
(a +
l)z(Daja(f))
1)z(DaJa(f)) ’.
(4.7)
With the aid of Lemma 3, we have
z(DaJe(f))
-aDaJe(f).
l)Daf(z)
(a +
(4.8)
Consequently, from (4.3), we obtain
<a +, al)z(.Daf(z))
z(D
l)Daf(z),
(a +
Ja(f))’
D
a
Ja
2zw’(z)
(1 + w(z))(1
(f)
w(z))’
(4.9)
or
(a
[z(Daf(z))
+_l)Daf(z)_
z(DaJa (f))’
DaJa
D af(z)
z(DaJa(f))
2zw’ (z)
(I + w(z))(l
(f)
(4 I0)
w(z))"
Since (4.8) implies
(a +
z
it follows from
l)Daf(z
DaJa
aD
+
(f) )’
z
Ja (f)
DaJ
( + a) + (
+ w(z)
(f) )’
(4.11)
(4.10) that
z(Daf(z))’
Daf(z)
+ w(z) +
(l
w(z)
2zw’(z)
w(z)){(l + a) + (I
a)w(z)}"
(4.12)
By assuming
Izl lzol
for
z0 U
I <z)l
and using the same technique as in the proof of Theorem 3, we can show that
Re {.
z(DaJa (f))’
Daja
Thus we conclude that
COROLLARY 3.
function.
Re
{I
+
w(z)}
w(z) >
f
Ja(f)
(z
0
U).
(4 13)
is in the class
Let f(z) be in the class
(Zp+iFp(a+l
where
I-<zo)l
a+l
,I
p+IFp(a ,..., ap+ 81 ,..., 8p;Z)
Sa
a+2
with
.....
a
>
O.
Then, for
pN
,
a+2; z))*f(z)
denotes
the
generalized
hypergeometric
729
AN APPLICATION OF THE RUSCHEWEYH DERIVATIVES
It is easy to see that
PROOF.
t-I
+
a
Ja (f)
In +++
n=0
an+l z
E
for
.
f(z)
an+ t n+l)
E
n=0
0
z
dt
n+l
(4.14)
(z2F l(a+l ,I ;a+2;z))*f(z)
Therefore, by Theorem 8, we have
(Z2Fl(+l l;+2;z))*f(z)
*
Repeating the same manner, we conclude that
f(z)
,
,
Sa
(z2Fl(a+l, l’a+2"z))*f(z)
Sa
(z3F2(a+l,a+l,1;a+2,a+2"z))*f(z) Sa
.(Zp+iFp(a+l,...,a+l,l’a+2,-.-,a+2;z))*f(z) a.
Finally, we prove
THEOREM 9.
Let
f(z)
be
in the class
a
Then
with
J=(f)
is also
in the class
PROOF.
In view of Theorem 5, we can see that
f(z)
z(Daf(z))’
Ka
D(zf’ (z))
<-_--
*
f’ (z)
Da(j(zf’))
<=F
z(DJa(f) )’
W=2DaJ
(f)
.
S
J.(zf’(z))
S*
S
.
K
which completes the proof of Theorem 9.
COROLLARY 4.
(Zp+iF
(+I
...,
Let
f(z)
+I I;+2
with
be in the class
a+2;z))*f(z)
K
a
p
denotes the generalized hypergeometrlc function.
where
>= O.
Then, for P
p+IFp(a I,
’=p+l ;B
N,
...,8 p ;z)
S. OWA, S. FUKUI, K. SAKAGUCHI AND S. OGAWA
730
ACKNOWLEDGEMENT:
Victoria while
The
the
present
first
investigation was
carried
author was on study leave
out
at
the University of
from Kinki University,
Osaka,
Japan.
REFERENCES
I.
RUSCHEWEYH, S. New Criteria for Univalent Functions, Proc. Amer. Math. Soc. 49
(1975), 109-115.
2.
RUSCHEWEYH, S. and SHEIL-SMALL, T.
3.
JACK, I. S. Functions Starlike and Convex of Order a, J. London Math. Soc. 2
(1971), 469-474.
4.
FUKUI, S. and SAKAGUCHI, K. An Extension of a Theorem
Bull. Fac. Edu. Wakayama Univ. Nat. Sci. 29(1980), I-3.
5.
MILLER, S. S.
Hadamard Products of Schllcht Functions and
the Polya -Schoenberg Conjecture, Comment. Math. Helv. 48 (1973), 119-135.
Math. Soc. 81
6.
7.
8.
of
S. Ruscheweyh,
Differential Inequalities and Caratheodory Functions, Bull. Amer.
(1975), 79-81.
BERNARDI, S. D. Convex and Starlike Univalent Functions, Trans. Amer. Math. Soc.
135 (1969), 429-446.
LIBERA, R. J. Some Classes of Regular Univalent Functions, Proc. Amer. Math. Soc.
16 (1965), 755-758.
LIVINGSTON, A. E. On the Radius of Univalence of Certain Analytic Functions,
Proc. Amer. Math. Soc. 17 (1966), 352-357