Ienat. J. ath. & 4ath. S.
Vot. 6 No.
(1983) 59-68
59
CONVOLUTIONS OF PRESTARLIKE FUNCTIONS
O.P. AHUJA
Department of Mathematics
University of Khartoum
Khartoum, Sudan
Department of Mathematics
College of Charleston
Charleston, South Carolina 29424
(Received on October 2, 1982)
ABSTRACT.
The convolution of two functions
X
f(Z)
n=0
7.
g(z)
n= 0
b z
n
n
is defined as (f,g) (z)
n= 0
2(I-)
g(z)
z/(l-z)
order
y, we investigate functions
the unit disk.
8.
n.
and
For f(z)=z-
r
a z
n
n= 2
n
and
the extremal function for the class of functions starlike of
(zh’/h)-II/l (zh’/h)
inequality
a b z
n n
a z
n
Such functions
(f,g)(z), which satisfy the
h, where h(z)
+ (i-2)
< 8, 0
<_
< i, 0 < 8
<_
i, for all
are said to be 7-prestarlike of order
u
z
in
and type
We characterize this family in terms of its coefficients, and then determine ex-
treme points, distortion theorems, and radii of univalence, starlikeness, and convexity.
All results are sharp.
KEY WOS AN PHRASES:
Convolution, Starlike Functions, and Univalent Functions.
1980 AMS SUBJECT CLASSIFICATION CODES:
1.
0C45
INTRODUCTION.
Let
S
denote the class of functions of the form f(z)
z +
7
n=2
analytic and univalent in the unit disk E
said to be starlike of order
and type
{z
Izl
< i}
8 if the inequality
l(zf’/f)-ll/l (zf’/f)
+ (i-2*)
< 8
a z
n
A function
n
that are
f e S
is
O. P. AHUJA AND H. SILVERMAN
60
,8(0 < e < i, 0 < 8 < I) and for all
holds for some
such functions shall be denoted by
S*(e,8)
zf’/f
S*(), the class
is a subclass of starlike
(i-2e)82)/(i-82)
(i +
lie in a disk centered at
The class of all
S*(e,l)
f e S*(e,8), 0 <
For
functions studied by Padmanabhan [i].
(0,@)
E.
in
Note that
e, and that S*
of functions starlike of order
z
8 < I, the values of
whose radius is
28 (l-e) / (1-82 ).
Z, a z
n
n=0
The convolution or Hadamard product of ,o power series f(z)
b z
7.
g(z)
n
is defined as the power series
n
n=0
7.
(f,g) (z)
n=0
f, analytic in
E
and normalized by
f’ (0)-I
f(0)
s (z)
z/(l-z)
S* (y).
We say that a normalized analytic function
8
and type
with
respectively, by
8
7. a z
n
n=2
S*[e], S*[e,8], and
studied in [3] while the class
and
z
f
R (e,8), if
n
e S*(y), where
f,s
y-prestarlike of order
e S*(e,8)
S*(e), S*(,8),
in
S*[]
We begin with a characterization of the class
R (,8)
Y
R[e] was
R [e,l)
The class
was investigated in [4].
i, the class reduces to the family
or
We denote these classes,
a >0
n--
R [e,8].
S*[,8]
is
f,s
f
Our main interest will be with functions
f(z)
A function
is the well-known extremal function for the class
0<_y<l
(0<e<l, 0<8<1), denoted
that may be expressed as
n
and
is said to be in the
0,
class of prestarlike functions introduced by Ruscheweyh [2] if
2 (l-y)
a b z
n n
n
For
y
1/2
studied in [5].
R [e,8], from which we determine
Y
the extreme points, distortion properties, and radii of univalence, starlikeness, and
convexity.
2.
COEFFICIENT INEQUALITIES.
In the sequel, we set
n
C(y,n)
H
(k-2y)/(n-l)!
(n
2,3
(2.1)
),
k=2
so that
Note that
s
may be written in the form
C(y,n)
s (z)
Y
is a decreasing function of
z/(l-z)
2 (l-y)
y, 0<_y<l, with
z +
7. C(y,n) z
n=2
CONVOLUTIONS OF PRESTARLIKE FUNCTIONS
61
y<i/2
lim C(y,n)
n+
i, y=i/2
0, y>I/2
THEOREM i.
A function
f(z)
z
7.
a z
n=2
and only
n
n
a >0, is in the class R [e,8]
n-y
if
i_f
[(n-l)+ 8(n+l-2e)] C(y,n)a
< 1
n=2
PROOF.
g(z)
f e R [,8], then
If
(2.2)
28 (l-s)
Y
(f,s) (z)
y
7.
z
C(y,n)a
n= 2
n
e S*[,],
so that
E (n-l)C(y,n)a z
n
n=2
l(z,’/,)
z e E.
for all
of
z
+
(--o)I
2(l-e)
7.
(n+l-2e)C(y,n)a z
n
n= 2
< 8
n-I
(2.3)
Since the denominator in (2.3) is positive for small positive values
and, consequently, for all
(n-l)C(y,n)a
n= 2
n
z + i" to obtain
z, 0 < z < i, we let
< B[2(l-e)
7. (n+l-2e)C(y,n)a
n
n= 2
which is equivalent to (2.2).
Conversely, if (2.2) holds, we wish to show that
Iz
f,s
g
Y
is in S*[,8].
For
r < I, we have
7.
(zg’/g)_-.l,
1’1=2
(n-l)C(y,n)a z
n
(zg’/g) + (l-2e)
2(i-)
7.
n-1
(n+l-2)C( n)a z
n=2
n
7. (n-i) C (Y,n) a
n
n=2
(n+l --2) C (y,n) a
2 (l-e)- 7.
n
n= 2
The function
to (2.2)
g
Hence,
is in S* [,]
f e R [,8]
Y
if the last expression is
and the theorem is proved.
< 8
which is equivalent
62
O. P. AHUJA AND H. SILVERMAN
COROLLARY.
I__f
f(z)
z-
7.
n=2
+ 8(n+l)-2)]C(,n)
a z
n
e R [e,8], then
y
n > 2, with
fo__r
z-28(l-)zn/[(n-1)
f (z)
n
a
< 28(l-e)/[(n-l)
n
+
functions of the form
+ n+l-2)]C(y,n)
Tt follows from Theorem 1 that
R [:8] is a closed convex family. We shall now
y
show that the extreme points of the closed convex hull are those that maximize the
coe f ficients.
THEOREM 2.
Set
fl(z)
n
2,3
f e R [e,8], 0 <
Then
expressed as
7.
f(z)
I f (z)
Z
If
2B
n
> 0
C (y n)
i, if and only if it can be
7. 1
n
n=l
and
(2.4)
i
then
1n(28) (l-a)
(i-)
r..
(n-l) +8 (n+l-2) ]C (y,n)
)t
n=2
1-X
n
1
< 1
f e R [,]
Conversely, if
n
f (z)
n n
_<
y < i, 0 <
[(n-l)+8(n+l-2) ]C(y,n)
n=2
and
7.
n=l
f(z)
,
where
n n
n=l
PROOF
z-28(l-)zn/’ n-l) +8 (n+l-2)
z and f (z)
n
f(z)
7. a z
n
n=2
z-
n
e R [e,8],
y
[(n-l) + 8(n+l-2e)]C(y,n)a /28(i-e)
n
We see from Theorem i that"
> 0
i--
then set
2,3
n
f(z)
Since
11
and set
l
n=l
f (z)
n n
7.
1
n=2
n
the proof is corn-
plete.
3.
DISTORTION THEOREMS.
We may now find bounds on the modulus of
THEOREM 3.
If
f
f’
and
for
f e R [,8]
Y
f e R [e,8], 0 < e < i, 0 < 8 < i, and either
0 < y < (2+38-eS)/(2+48-2eS)
o__r
r
<_
(i+28-e8)/(i+38-28),
then,
fo__r
max{0,r-8(l-e)r2/[(l+8(3-2e)] (l-y) }<If(z)l<_r+8(l-)r2/[l+8(3-2e)] (l-y)
are sharp, with extremal function
f2(z)
2
z-B(l-e)z /[i+8(3-2e)](l-y)
Izl
<_
r
Th___e bg..unds
CONVOLUTIONS OF PRESTARLIKE FUNCTIONS
28(l-e)r
max{0,r-max
n
n
[(n_l)+8(n+l_2)]C(,n)
n
Under the constraints for
y
n.
(3.1)
From (2.1) we see that
n>_2.
so that (a,8,y,r,n) > (e,8,7,r,n+l) if and only if
(n+l-2y) [n+8(n+2-2e)]-rn[n-l+8(n+l-2)
(3.2)
> 0
and
is decreasing in 7
h
fixed, the function
Hence,
28 (l-e) r
(n-l) +8 (n+l-2e) ]C(y,n)
r, it suffices to show that
for
n
[(n+l-2y)/n]C(y,n)
h(e,8,y,r,n)
in
n
and
is a decreasing function of
For e and 8
r+max
28(l-e)rn/[(n-l)+8(n+l-2e)]C(y,n)
(e,8,y,r,n)
C(y,n+l)
If(z)I<
} <
h(,8,y,r,n) > h(,8,(2+38-eS)/(2+48-28), 1,2)
0<_7<_(2+38-8)/(2+4-28),
_>
r < I, and n
2.
0
for
Similarly,
h(e.8,y,r,n) > h(e,8,1, (I+28-e8)/(i+38-28), 2)
0<_y<l, r<(l+28-eS)/(l+38-2eS), and n>_2.
and increasing
r
0
for
Thus max (e,,y,r,n)
is attained at
n>2
n=2, and the proof is complete.
As a special case of Theorem 3, we get the result in [3] as a
f e R [e,l], 0 <
If
COROLLARY.
2
r-r /2(2-()
When
PROOF.
<_ If(z)
i, we have
8
< i, then
<
2
(Izl=r)
r+r /2(2-e)
so that the first con-
e < (5-)/(6-2e)
y
dition in Theorem 3 is satisfied.
REMARK.
z
The function
[i+8(3-2e)
If(z)
> r
f2(z)
0
Letting
(i-7)/8(i-)
8(l-e)r2/[l+8(3-2e)] (l-y)
in Theorem 3 when
z
1
for all
we thus have
z
in
E
if and only if
0<_y<[l+8 (2-e)]/[1+6 (3-2e) ].
Theorem 3 leaves open the question of an upper bound for
>(2+38-8)/(2+48-2e8)
THEOREM 4.
If
Set
r>(l+28-eS)/(l+38-2eS).
and
fl
We resolve this with
rn0(e,8,Y)=(n0+l-2y) [n0+8(n0+2-2)]/n0[n0-1+8(n0+I-2)]
f e R [e,8], 0<e<l, 0<6<1
(i+8)
Y0
when
n0+8 (l-s)
n0+8(n0+2-2e)
n0+8 (2-e)
I+ (i+8) n0+8 (3-2e)
i+ (i+8)
< Y <
Y1
(n =2,3
0
O. P. AHUJA AND H. SlLVERMAN
64
and
(,B,Y)
r
n
< r < i, then
0
If(z) <_
r + 28(I-)r
with equality for
imized for
n
O
+i
n
n
n0#l
+ 1 > 2.
n
< 1
n
o
attains its maximum value at
+i, which occurs for
occurs at
defined in (3.1), is max-
defined in (3.2), is negative for
h
if and only if
max (e,8,y,r,n)
n
(e,8,y,r,n)
The function
if the function
positive for
rn0(,8, Y)
given in (2.4).
f
O
(Izl=r)
/[n0+8(n0+2-2) ]C(y,n0+l)
It suffices to determine when
PROOF.
n
no+l
n0+l
n
n
and
Y>--Y0
(,8,Y) < r <
r
r
n
O
n
> i
+l(e,8,y)
+i
for
n
and
O
{,8,Y)
y<y
however,
There fore,
1
yn<<y<_y 1
(e,8,Y) < r < i and
r
for
rn0
O
n
0
and the
proof is complete.
We use similar methods to determine a distortion theorem for
THEOREM 5.
r
<_
If
f e R [e,8], 0 < e < i, 0 < 8 < i, and either
(2+48-2eS)/(3+98-6eS)
1-28(l-e)r/[l+8(3-2e) (l-y)
with
equality when
PROOF.
For
f’.
f2(z)
r
0 <’y < 1/2
or
then
0
If’ (z) <_ l+28(l-e)#u/[l+8(3-2e)] (l-y)
z-28(l-)z2/[l+8(3-2e)] (l-y).
<_
A(e,8,y, r,n)
28(l-)nr
n- 1
Izl
for
/[(n-l)+8(n+l-2e)]C(y,n)
r
we have,
according to Theorem 2,
1- max A(e,8,y,r,n) <
n>2
ing function of
hl(e,8,y,r,n)
Since
h
i
+ max A(,8,y,r,n)
But
A
is a decreas-
if and only if
(n+l-2y)[n+8(n+2-2e)]
I(e,8,Y,r,n)
0 < y < 1/2
< 1
n>2
is decreasing in
h
for
n
If’(z)
r
_>
h
and
y
for
(n+l)r[(n-l)+8(n+l-2e)] > 0.
<_ 1/2
y
I(,8,1/2,1,2)
and increasing in
i-8(I-2)
_>
n, we have
0
and
hl(e,8,y,r,n) _> hl(e,8,l,r0,2)
0
for
r
_<
r
0
This completes the proof.
REMARK.
The theorem is the best possible in that
hl(,8,1/2,r,2)
< 0
for
CONVOLUTIONS OF PRESTARLIKE FUNCTIONS
r > r
0
A(e,8,y,l,n) > A(e,8,y,l,2)
and
65
y > 1/2
for each fixed
and
n
n(y)
sufficiently large.
4.
RADII OF UNIVALENCE, STARLIKENESS, AND CONVEXITY.
As we have seen in Theorem 3, it is possible to have
for
f
in
R [e,8], which means that
0)
need not be univalent.
f
Y
f(z
0, 0 <
Iz01
< 1
We now determine
when the family contains only univalent functions.
THEOREM 6.
PROOF.
R [,8]
y
_-
z +
Since
S
7.
n=2
for
< 1/2
anZ
n
which is equivalent to
for
for
C(y,n)
/
if suffices to show
(4.1)
so we need only prove (4.1) for
1/2,
This last inequality is true
n > 2.
for y > 1/2
0
1
n=2,3
n[i+8-28(i-)] > i-8(I-2e).
n=2, and consequently for all
Conversely, since
_<
n
that
y < 1/2
for
1
n a
n=2
according to Theorem 1
C(y,n) > C(i/2,n)
7
if
S
[(n-l)+8 (n+l-2e) ]C(,n)/28(i-) > n
But
y < 1/2.
if and only if
f (z) defined by (2.4),
n
we take
and note that
n-i
f’(,z)
n
28 (l-e) nz
(n-l) +8 (n+l-2) ]C(y,n)
1
0
for
z
n-I
[(n-l)+B(n+l-2) ]C(y,n)/28(l-e)n
which is less than 1 for n sufficiently larg6.
y > 1/2 and n
Thus,
f (z)
n
is not univalent for
n(y) sufficiently large.
Since functions of the form
z
l a z
n
n=2
n
a > 0, are starlike if and only if
n--
they are univalent [5], we have shown that functions in
R [e,8], 0 < y < 1/2, are
all starlike.
We now determine the largest disk in which such functions are star-
like of order
6, 0 < 6 < i.
THEOREM 7.
0 <
< i, 0 <
If
f(z)
7. a z
n
n=2
z
< 1/2, then
f
n
e R [,8]
5"
0 <
i__s starlike of order 6,
< 1
0
_<
< i, in the disk
O. P. AHUJA AND H. SILVERMAN
66
Iz
< r
0
where
i/(n-l)
r
inf
n
0
[(1-6) [(n-l)+8(n+l-2)]C(y,n)]
28 (l-e) (n-d)
with equality for a function of the form (2.4).
(zf’/f)
It suffices to show that
PROOF.
I
ii
(zf,/f)
n=2
<
< 1-6
1
Izl
for
< r
But
0
n-i
nlzl n-I
n=2 nlzl
(n-i)a
< i- d
(Izl
r)
a
1
if and only if
n-d
Z
n=2
I--- a rn
n-i
_<
In view of Theorem i, we need only find values of
n-d
(i-) rn-i
<
[(n-l)+8(n+l-2e) ]C(y,n)
28 (l-a)
r < r
which will be true when
COROLLARY i.
r
PROOF.
Since
d < 1 in the disk
inf
n
I
z +
%
n=2
z +
%
n-2
na z
n
replaced by
n
a z
n
(l-d)
for which
(n=2,3
< i, 0 < 8 < i, 0 < y < 1/2, then
zl
< r_
(n-l)+8 (n+l-2e)
28(l-e)n(n-d)
f
is
where
I
is convex of order
is starlike of order
na
r
and the theorem is proved.
f e R [e,8], 0 <
If
6, 0
convex of order
0
(4.2)
i
]C(y,n) (n-l)
d
if and only if
d, the proof follows that of Theorem 7, with
n
0 in Theorem 7, we may determine the radius of univalence (and
By taking d
starlikeness) of R [e,8] when
> 1/2
a
n
67
CONVOLUTIONS OF PRESTARLIKE FICTIONS
univalent
an__d
f e R [e,8], 0 <
If
COROLLARY 2.
r
5.
zl
starlike for
inf
n
2
< r
8 < I, 1/2 < y < i, then
< I, 0 <
f
is
where
2
((n-l) +8 (n+l-2e)) C (y,n)
28n(1-a)
i/(n-l)
ORDER OF STARLIKENESS
R [e,8], 0 <
Since functions in
determine the order of starlikeness.
THEOREM 8.
< 1/2, are starlike, it is of interest to
Y
We do this in
< i, 0 <
f e R [,8], 0 <
If
8 < i, 0 < 7 < 1/2, then
f
is star-
like of order
[i+8(3-2)] (l-y)-28(l-e)
[i+8 3-2e) (l-y)-8 (l-e)
I
From Theorem 1 and [5], it suffices to show, for
PROOF.
f(z)
2
z-8(l-e)z /[i+8(3-2)](i-y)
f(z)
with equality for
E
z-
a z
n
n=2
n
y
n
and
8
fixed,
n
This will be true if
(n-l)+8 (n+l-2e) ]C(y,n) (1-I)
28(i-e) (n-l)
g(e,8,y,n)
For
E [(n-l)+8(n+l-2e)]C(y,n)a /28(i-e) < 1 implies
n=2
< i.
[(n-l)/(1-1) ]a
n= 2
e R [,8], that
g(e,8,y,n) > g(,8, 1/2,2)
8
1
and
(n=2,3
can be shown to be an increasing function of
g
0 < y < 1/2, and an increasing function of
Choosing
> 1
1 for
y
e
n, n > 2, so that
0 < y < 1/2
and n > 2.
This completes the proof.
in Theorem 8, we get the following result proved in
[3] as a
COROLLARY.
If
f e R [e,l], 0 <
<
1/2, then
f
is starlike of order
(2-2) / (3-2).
University of
This work was completed while the first author had a grant from the
of CharlesCollege
the
from
leave
sabbatical
on
was
author
second
the
Khartoum, while
Michigan.
of
University
the
at
Scholars
Visiting
ton, and while both authors were
68
O. P. AHUJA AND H. SILVERMAN
RE FE RENCES
i.
PADMANABHAN, K.S. On Certain Classes of Starlike Functions In the [nit Disk,
J. Indian Math. Soc. 32 (1968), 89-103.
2.
RUSCHEWEYH, ST. Linear Operators Between Classes of Prestarlike Functions,
Comm. Math. Helv. 52 (1977), 497-509.
3.
SILVERMAN, H. and SILVIA, E.M. Prestarlike Functions with Negative Coefficients,
Inter. J. Math. & Math. Sci. 2 (1979), 427-439.
4.
GUPTA, V.P. and JAIN, P.K. Certain Classes of Univalent Functions with Negative
Coefficients, Bull. Aust. Math. Soc. 14 (1976), 409-416.
5.
SILVERMAN, H. Univalent Functions With Negative Coefficients, Proc. Amer. Math.
Soc. 51 (1975), 109-116.