173
Internat. J. Math. & Math. Scl.
(1985) 173-177
Vol. 8 No.
ON THE HADAMARD PRODUCTS OF SCHLICHT
FUNCTIONS AND APPLICATIONS
H.S. AL-AMIRI
Department of Mathematics
Bowling Creen State University
Bowling Green, Ohio 43403
(Received April. 25, 1984)
We show that each of the schlicht classes of starlike, convex, close-to-
ABSTRACT.
convex and strongly starlike with respect to symmetric points is invariant under the
Hadamard product with the class of convex functions.
The influence of certain operators
over these classes is also investigated.
KEY WORDS AND PIIRASES. Hadamard products, schlicht functions, and starlike, cone.n,
close-to convex and stron@ly starlike with respect to symmetric points.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE.
i.
INTRODUCTION.
Let
By
Primary 30C45
S
A
we denote those functions in
f’(O)
Let
O.
1
C, S*
and
A
S
I}.
z
which are univalent and normalized by
be the usual subclasses of
K
{z:
U
be the class of analytic functions on the unit disk
f(O)
which are consisting
of the convex, starlike and close-to-convex functions, respectively.
zn
Z a
The Hadamard product (convolution) of two power series f(z)
0 n
n
b z n. Polya and
Z b z
Z a nn
zs defined as the power series (f’g) (z)
0
0 n
Schoenberg [9] have conjectured that f * g is in C whenever both f and g are in
C. This conjecture was finally shown to be true by Ruscheweyh and Shell-Small [12].
g(z)
THEOREM A (Ruscheweyh and Shell-Small):
h * f
then
C, S*
is in
or
Let
be in
h
C.
If
f
is in
C,S*
K
or
K respectively.
The proof of Theorem A requires a key lemma that turned out to be beautiful in its
simplicity and powerful in its applicability to various problems.
We state this key
lemma of Ruscheweyh and Shell-Small in the form stated and proved by Barnard and
Kellogg in [I].
LEflA A.
Let
and
’(0)g’(0) # 0.
g
Izl
be analytic in
Suppose that for each
a(la
i
with
i) and
0 and
g(O)
have
I) we
4(0)
o(Io
+ aoz) g](z) 4 0
[4*( i’--az
1
on
0
zl
r
"’.
i.
Then for each
F
in
A
the image of
(I.I)
zl
r
under
H. S. AL-AMIRI
174
( * Fg)/( * g)
REMARK i.
C
is in
A function
U,
it was shown that condition [i] is satisfied for all
[12],
In
whenever
denoted
F(U).
is a subset of the convex hull of
and
A
f
z
U
in
S*.
is in
is said to be starlike with respect to symmetric points in
S *,
f
g
if and only if
s
zf’ (z)
f(z)
f(-z)
Re
for all
0,
z
(1.2)
U.
in
This class was introduced by Sakaguchi [13].
C-S *cK.
REMARK 2. Sakaguchi has shown that
(f(z)
f(-z))/2
We say that
U, denoted
points in
*
s
then
%, with respect
is strongly starlike of order
f
S
f
Further, if
s
S*.
is in
S
f
*(),
s
-
to symmetric
if and only if
[arg{f(z)z f’(z)f(-z) }[
0
(1.3)
U
z
i
This class was introduced by Padmanabhan and Thangamant [8].
Das and Singh [4]
respect to symmetric points in
U, denoted
C
f
and those which are close-to-
s
convex with respect to symmetric points, denoted
which are convex with
f
introduced the classes of function
K
f
as follows:
s
C
f
s
if
and only if
z(zf’(z))’
zf’(z) + zf’(-z)
Re
and
K
f
[4],
In
furthermore,
it was shown that both
C
f
S
zf’ (z)
K(z)
K(-z)
Re
REMARK 3.
K
if and only if there exists
s
O,
zf’
if and only if
s
*
0,
C
such that
(1.5)
U.
z
and
s
(1.4)
U,
z
K
s
s
The purpose of this note is to investigate these subclasses of
U
symmetric with respect to points in
S
are subsets of
s
and
S *.
S
which are
In section 2, we
via the Hadamard products.
state our main theorem which shows that these classes are invariant under the
Hadamard product with convex functions.
In section 3, we use the results of section
2 and an observation made in [i] to obtain results concerning the radii of these
classes under the influence of certain well known operators.
2. MAIN RESULTS
The elegant technique and results of Ruscheweyh and Shell-Small allow us to
produce a theorem similar to Theorem A for the classes
THEOREM i.
is in
S *
S
S
PROOF.
S
Let
*
S
C.
h
K
C
or
S
If
f
S *(%)
S
is in
S *, C
S
S
K
S
S
or
S
K
s
*(),
S
S
and
function
<_
0
S
*(X),
g
Let
S
s*
and
h
then
h
* f
C.
Then
g(z)
f(z)
f(-z)
2
In order to
and the starlike
for the different cases.
f
I,
respectively.
The proof of the four cases requires an application of Lemma A.
apply this lemma, we need to identify the convex function
Case i.
C
I.
0
,
S
by Remark 2.
Thus
HADAMARD PRODUCTS OF SCHLICHT FUNCTIONS
175
condition(l.l)is satisfied according to Remark i, i.e.,
for all
z
U.
in
+
(i
[h *
Applying Lemma A with
*
*
(h
g)
0
Now let
z f’ (z)
v(z)
h
oz) g](z) @
F
as given in(2.1)we obtain
(f(z)
h(z) *
2
2-f(-z))
z(h * f)’(z)
(h * f)(-z)
(h * f)(z)
h(z) * z f’(z)
(h * f)(-z)
(h * f)(z)
F
which is contained in the convex hull of
in U.
Z(] * f)’(Z)
Re (h
(h * f)(-z)
* f)(z)
or
h * f
CASE 2.
S
Let
s
* according
f
C
f
K
s
h
and
h
This implies
O,
U,
Z
(1.2).
to
and
f(-z))
(f(z)
z f’(z)
h(z) *
f(-z))
(f(z)
(z)
(1.6)
f(-z)
f(z)
This case follows immediately from Remark 3 and
C.
Case 1 above.
CASE 3.
Let
s
h
satisfied for
and
g
K(z) be the associated function appearing in
K(z)
K(-z)
S*. Thus condition (|.I) is
Let
C.
E
2-
g(z)
(1.5). Then according to Remark 2
Applying Lemma A with
of this case.
z f’ (z)
F(z)
K(z)
K(-z)
we easily get after some simplification as in Case 1 that
z(h * f)’(z)
(h * K)(z)
(h * K)(-z)
is contained in the convex hull of
F.
That is
z(h * f)’(z)
Re (h
(h * K)(-z)
* K)(z)
Thus
h
* f
K
s
Let
f
S *()
S *
s
for all 0
s
_
h * K
provided
CASE 4.
S *(i), 0
s
S *.
s
I <- i, and
i.
Let
g
0,
U.
z
However, the latter is true as seen by Case I.
h
C. It is clear from (1.3) and (1.2) that
and
F
as defined in Case i,
then, as in the
proof of Case i,
z(h * f)’(z)
(h * f)(-z)
(h * f)(z)
F.
lies in the convex hull of
[arg
Thus
3.
h
* f
S *(I)
s
(h *
This means
z(h * f)’(z)
(h * f)(-z)
f)3
X/2,
0
X
by (1.3). This completes the proof of Theorem i.
APPLICATIONS
For
X
A, let
r (X)
c
denote the largest positive number so that every
X
H. S. AL-AMIRI
176
is convex in
for
h
zl
Consequently, the conclusion of Theorem I remains valid
r (X).
c
in the disk
zl
* f
r (X)
to unify and
X
h
l _+n z
y + i
n= I
H(z)
X
n=l
P(z)
X
Y
y
(I
Rey
Ixl
0,
I,
# I
x
I
+ i
+ y
z
n
-i +
+ i
+n
x
n=l
z
n
z
1
n
i
z
i
+
afztY-if(t)dt
Y
B3f(z)
jz
f(xt)
f(t)
t
0
xt
For example, the operator
Libera [5] when
y
(H
0
z
I
B
K
s
s
this section.
or
B
while
B.,
S *() up to r (h), r (H)
c y
c y
s
However,
i, Rey -> 0
i
I
l+y
1,2,3,
r (P)
or
# I
y
i, B 2
z
[Ii], and
r (h).
c y
Thus
z h "(z)
i+-
2 +z
z
i
’
h ’(z)
Y
i
i
+
i
Ic]r Izl
cz
This implies
z h "(z)
Re(l + _____Z
h ’(z)
This shows that
r (h)
c y
2
r
I + r
i
is the smallest positive number
Iclr 2
or
r (h)
c
y
2(1 +
Icl)r +
i
0
[2],
preserve the classes
l+cz
(i-z)
z)
(i
is due to
see also
as mentioned at the start of
We need only to calculate
i
+ ly
[I0],
c
was shown by Ruscheweyh
rc(Hy)
hy ’(z)
c
x
i
was first used by Pommerenke in
r (P)
i was shown by Pommerenke [i0].
c
definition of h
we get
Y
where
Rey > 0
is due to Livingston [7] when
I
1,2,3,
O,
Ix[
(P * f)(z)
dt
B.,
The operators
Let
* f)(z)
T
3
73], [6], [ii]. Consequently, the operators
C
U.
in
y
B2f(z..
z) 2
I
(h * f)(z), Rey
l+y
z
(I
i -xz
z
log
x
and for all
f(z) + z f’(z)
y
Blf,z,,
*
s
Let
and
n
x)n
y
defined below have been studied by many authors.
S
in the given four
f
and every
reduce the amount of work usually associated with such problems.
h (z)
y
where
for each
c
This observation was first used effectively by Barnard and Kellogg in [i]
classes.
r
r
satisfying
From the
HADAMARD PRODUCTS OF SCHLICHT FUNCTIONS
177
Finally, we have the following theorem.
THEOREM 2
If
f
is in
S
S
i)
Bof
is in
S
*
C
ii)
Blf
is in
S *
S
C
Izl
s
*
s
s
i/(1 +
r (h)
c y
C
K
K
s
s
K
S
or
S *()
s
S
*(%),
,’
% _<_
I,
then
+
2,3, and
respectively for
S
+
0
*(>,), respectively for each
or
S
Icl
S
or
S
Ic[ +
Icl), =
l+y’
P,y -> O.
l’heorem 2 part (i) was shown in [8] for the class
REMARK 4.
with
and
y
x
-i
was shown in [4] for the classes
S *(I) only.
Part (i)
S
S *, C
S
S
and
K
S
RE FERENCES
I.
BARNARD, R. W., KELLOGG, C., Applications
2.
5.
BARNARDI, S. D., Convex and Starlike univalent functions, Trans. Amer. Math. Soc.
135 (1969), 429-446.
CAUSEY, W. M., The close to convexity and univalence of an integral, Math. Z. 99
(1967), 207-212.
DAS, R. N. and SINCH, P., On subclasses of schlicht mappings, Indian J. Pure AppI.
Math. 8 (1977), 864-872.
LIBERA, R. J., Some classes of regular univalent functions, Proc. Amer. Math. Soc.
6.
(1965), 775-758.
LIBERA, R. J. and LIVINGSTON, A. E., On
3.
4o
of convolution operators to problems in
univalent function theory, Michigan Math. J., 27 (1980), 81-94.
16
the univalence of some classes of regular
functions, Proc. Amer- Math. Soc. 30 (1971), 327-336.
7.
8.
LIVINGSTON, A. E., On the radius of univalence of certain analytic functions, Proc.
Amer. Math. Soc. 17 (1966), 352-357.
PADSANABHAN, K. S. and THANCAMANI, J., The effect of certain integral operators on
some classes of starlike functions with respect to symmetric points, Bull. Math.
de la Soc. Sci. Math. de la R.So de Rawmanie 2__5 (1982), 355-360.
9.
Remarks on de la Valle’e Poussin means and convex
conformal maps of the circle, Pacific J. Math. 8 (1958), 295-334.
POLYA, G. and SCHOENBERG, I. J,
i0.
POMMERENKE, C., On close-to-convex analytic functions, Trans. Amer. Math. Soc. 114
(1965), 176-186
II.
RUSCHEWEYH, S., New
criteria for univalent functions, Proc.
Amer. Math. Soc. 49 (1975),
109-115.
12.
RUSCHEYH,
13.
SAKAGUCHI, K., On
S and SHEIL-SMALL, T., Hadamard products of schlicht functions and the
Polya-Schoenberg conjecture, Comment. Math. IIelv. 48 (1973), 119-135.
a certain univalent mapping, J. Math. Soc.
Japan
ii
(1959),72-80.