I nternat. J. ath. & ath. Sci.
427
Vol. 2 #3 (1979) 427-439
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
H. SILVERMAN*
Department of Mathematics
College of Charleston
Charleston, South Carolina 29401
E. M. $1LVIA
Department of Mathematics
University of California, Davis
Davis, California 95616
(Received December 4, 1978)
ABSTRACT.
The extreme points for prestarlike functions having negative coef-
ficients are determined.
Coefficient, distortion and radii of univalence, star-
likeness, and convexity theorems are also obtained.
KEY WORDS AND PHRASES.
,
stake
Prtake of order
univalence, radius of convexly, ereme points.
AMS (MOS) SUBJECT CLASSIFICATION (1970) CODES.
Secondar
, radius o f
of order
Pimary 30A32, Primary 30A34,
30A40.
I. INTRODUCTION.
Let S denote the class of functions normalized by f(0)
are analytic and univalent in the unit disk U.
f
Given
,
f’ (0)
0 _<
S is said to be in the class of functions starlike of orde_r
_<
I,
,
1
0 that
a function
denoted by S*(a),
428
H. SILVERMAN and E. M. SILVlA
if
Re{zf’(z)/f(z)} >
U),
(z
a, denoted by K(a), if
and is said to be in the class of functions conve_x o_.f
Re{l
+ zf"(z)/f’(z)}
(z
>
U).
Further, let T and T*[a] denote the subclasses of S and S*(a), respectively, whose
elements can be expressed in the form f(z)
I
z
n=2
..lanlZn
The convolution or Hadamard product of two power series f(z)
In= 0 bnzn is
g(z)
defined as the power series (f’g)(z)
f’ (0)
function normalized by f(0)
prestarlike of order
z/(-z) 2(-)
S*(a).
,
En= 0 anZ
In= 0 anbn zn"
n
and
An analytic
0 is said to be in the class of functions
1
0 < a < i, denoted by R
if f*s
S*() where s (z)
is the well-known extremal function for the class
The function s
In the sequel, we let
n
(k-2a)
(n-l)
k=2
C(,n)
so that s
(n=2, 3,
),
(I. I)
can be written in the form
sa(z)
z
+
Zn= 2
n
C(a,n)z
Note that C(a,n) is a decreasing function of a with
-
llm C(a,n)
n
The class R
,
a <
112
I,
a
1/2
0,
>
(1.2)
1/2
was introduced by Ruscheweyh [3], who showed that a necessary and
sufficient condition for f to be in R
is that the functional
429
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
s (z)
f(z),
G(=,z)
l’Z
fCz) *
S
OCz)
satisfy
Re G(e,z) > 1/2
Sl(Z)
Since
(z
(1.3)
U).
z, we say that f is prestarlike of order 1 if and only if
Re{f(z)/z} > 1/2
U).
(z
K(0) and
Note that R0
RI/2
S*(I/2).
In [3] it was
shown that
Rs
c
R
< 8 <
for 0 <
8
I,
which generalizes the well-known result that K(0) c
S*(I/2).
In Section 2, we obtain a sufficient condition in terms of the modulus of the
coefficients for a function to be in R
and show that this condition is also nec-
essary for the subclass
R[]
n T.
R
(1.4)
In Section 3, we obtain the extreme points for the closed convex hull of R[] and
use them to prove distortion and covering theorems.
In Section 4, we determine the
radii of univalence, starlikeness, and convexity for R[e].
8(e) for which T*[]
smallest 8
2.
c
Finally, we find the
R[8], 0 < e < I.
COEFFICIENT INEQUALITIES FOR THE CLASS RIll.
We first obtain a relationship between the order of prestarlikeness of a
function and the modulus of its coefficients.
THEOREM i.
Zn=2
then f
R.
+
Let f(z)
z
(n-e)C(en)
lanl
i
-’ e
Zn= 2 anZ
< I
n
If
0 <e < I,
H. SILVERMAN and E. M. SILVIA
430
REMARK.
expression
(n-)C(,n)/(l-)
For s (z)
z
1
is taken to be 2, its limit as
+
+
En= 2
Ek= 2
C(,n)z
C(e,k).
n
(z)/(l-z)
we have s
/
i
I/G(,z)- I
An equivalent formulation of (1.3) is
PROOF.
o(,n)
1 in the theorem and all subsequent results, the
For the case
z
+
Y.n= 2
< i, 0
!
<_
i.
n
o(,n) z where
Thus
anzn
En= 2 o(,n-l)
z +
En= 2 o(e,n)an zn
En= 2
<
o(,n-l)lanl
Y.=n=2
i
( n)
lanl
which is bounded by 1 whenever (2.1) is satisfied.
The converse of Theorem 1 is also true for the class RIll, defined by (1.4).
Since a necessary and sufficient condition [4] for
f(z)
z-
En= 2 Ibn Iz
n
to be in T*[] is that
n=2(n-
Ibnl
A function f(z)
z
,
<_1
(2.2)
we obtain
THEOREM 2.
Zn=2= (n-)C,(l_ ==’n)
COROLLARY.
lan
z
If f(z)
<_ (l-)/(n-)C(,n),
z
En= 2 lan Iz
lan
r.n=21anlzn
<
n
is in R[a] if and only if
,
R[],
0 <o<I.
0
<_
<_ I,
then
with equality only for functions of the form
(l-a)zn/(n-)C(a,n).
We now determine the extreme points of this class.
431
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
3.
EXTREME POINTS OF R[].
,
.
For any compact family of analytic functions
it is well known that the real
is maximized (minimized) at one of
part of any continuous linear functional over
the extreme points of the closed convex hull of
The solutions to several extremal
problems in R[] follow easily from the extreme points for this class.
Theorem 2, we see that R[] is a closed convex family.
In view of
Thus, the extreme points are
obtained in
THEOREM 3.
RIll, 0
Then f
f(z)
Y.
n I
PROOF.
Set
fl(z)
_<
_<
z and
(l-)zn/(n-)C(,n),
z
n=2, 3,
i, if and only if it can be expressed in the form
where % > 0 and Y.
%
n
n I n
%nfn(Z)
In=I %nfn (z)
Suppose f(z)
i.
Then
n
(n-) C (,n)
n=2
fn(Z)
In=2 %n
(n-) C (,n)
i
i_
%1
< i"
Therefore, f e R[] by Theorem 2.
Conversely
lan
<
%1
1
f(z)
suppose f(z)
(l-)/(n-)C(n)
In==2 %n"
n--2
z
y.n= 2 fan
n
z
Set %
3
R[], 0 <
I.
Then
(n-)C(,n)lanl/(l-)
n
From Theorem 2, it follows that
Y.n= 1%nfn(Z),
<
%1
> 0.
and
Since
the proof is complete.
As an immediate consequence of Theorem 3, we obtain distortion theorems for the
class RIll.
THEOREM 4.
r-
If f(z)
1
2"2--------r
with equality only for
2
n
z
Y.n= 2 lanlZ
<
If(z) <_ r +
f2(z)
2
I
4-2
z -z
R[], 0
1
2(2-a)
r
<_
2
z=+r.
<_
i, then
(Izl
r),
432
H. SILVERMAN and E. M. SILVIA
PROOF.
From The@rem 3, we have
r
max
.__
I
=nlnC(’’,
It suffices to show that A(,n)
r
n
<_ If(z)
< r
+
(l-)/(n-)C(,n)
max
(n-) C (,n) r
is a decreasing function of n.
From (I.i), it follows that
n+l-2
C(,n+l)
n
C(,n).
(3.1)
Therefore, we have A(,n+l) < A(,n), n=2, 3,
whenever (n+l-)(n+l-2) > n(n-e)
This is equivalent to (l)[l+2(n-)] > 0, which proves the result.
COROLLARY.
lwl
<
zl
The disk
(3-2e)/(4-2e) for any f
< 1 is mapped onto a domain that contains the disk
E
R[], 0 <
< i.
The result is sharp, with extremal
function
f2(z)
PROOF.
Let r
/
1
z
2(2-e)z
R[].
in Theorem 4.
As a second application of Theorem 3, we have
THEOREM 5.
(Izl
If f
E
R[], 0 < e < I, then 1
M(e,r)
_<
If’(z) ! I +
r), where
M(s,r)
max
n
COROLLARY.
If f
1
PROOF.
I
-2_--r
:,n.
(n-)C(on)
R[e] with either r
<
r
2/3 or
I
If’(z)l <_ I + 2_-L-r
It suffices to show that
n-
=
<lzl
<
I/2, then
=>.
M(s,r)
433
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
(1-a)nrn-1
g(a r,n)
is a decreasing function of n.
g(a,r,n+l) < g(a,r,n)
h(,,r,n)
(n-a) C’(a,n)
In view of (3. I), the inequality
is equivalent to
2
(1-r)n + [2-3c-(1-s)r]n + (1-)(1-2) + ar > O.
Since, for n fixed, h is a decreasing function of r, we have
(l-2a)n + (l-a)(l-2a) + a 0
h(,r,n) > h(a,l,n)
for s < 1/2.
Since h is also a decreasing function of s, it follows for r < 2/3
that
h(,r,n) > h(l,r,n)
I.
REMARKS.
a
I/2.
2
n
Since h(l,r,2)
g(l,r,2) < g(l,r,3).
2.
(l-r)n
2
+
r
h(l,2/3,n)
3r < 0 for r
h(1,2/3,2)
2/3,
we have
Thus the corollary will not be true for all
when r > 2/3.
We next show that the corollary will not be true for all r when
For each a, 1/2 < a < I, we must find an r
M(u,r) > (I/ (2-u))r.
(l-u)n/(n-u)C(u,n)
It suffices to show for n
I/(2-u), which
r(a) such that
n(u) sufficiently large that
is equivalent to
C(,n) < ,(I-,) (2-)n
1/2 and the right hand side of (3.2)
Since C(s,n)
/
(l-s)(2-)
0, the result follows.
4.
0.
0 for
(3.2)
is bounded below by
RADII OF UNIVALENCE, STARLIKENESS, AND CONVEXITY.
The functions in R[] for 0 <
bound is given in
=
< 1/2 are starlike of a positive order.
The
H. SILVERMAN and E. M. SILVIA
434
-< 1/2, then f
If f e R[], 0 -<
THEOREM 6.
T*[2(I-)/(3-2)].
The result
is sharp, with extremal function
f2(z)
PROOF.
i
z
2(2_-----z
2
From (2.2), it suffices to show that
En=2
(n-) C(n)
i-
anl
implies
2(i-)
n
En= 2
3-2
I- 2(I-)
3-2
an
This will follow if
(3-2)n
2(i-)
I
<
-
I
< C(,n)
or, equivalently, when
[(3-2)n
2(l-=)](1-a)
(n-=) C (,n)
g(,n)
Since
g(,2)
I,
it suffices to show that
g(,n)
is a decreasing function of n.
In view of (3.1), the inequality g(,n+l) < g(,n), n=2, 3,
2+n
(3-2) n
(n+l-a) (n+l-2a)
<
is equivalent to
(3-2) n-2
n-
This holds if and only if, for each fixed a, we have
h(n)
Note that h(1)
h(n+l)
(4a2-8a+3)n 2
+
(62-4a 3 -l)n
+ 4a 3
10a
2
+ 8a
2 > 0.
0 and
h(n)
2(4a2-8a+3)n + 2(l-4a+52-2a 3)
A()n + B(a)
435
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
Since
A() > 0 and B() > 0 for 0 <
REMARK.
K[0]
R[0]
When
n
c
T*[2/3].
>
1/2, R[]
2C(
z
1/2, the result follows.
0, Theorem 6 reduces to the known result [4] that
For
n)zn/n.
gn(Z)
We will show that
S.
If n e 4, then
n() sufficiently large.
gn*S
<
gn
R[I].
Taking (1.2) into account
2C( n)/n < (l-)/(n-) for n sufficiently large
gn
Therefore,
2zn/n
z
R[]
If 1/2 <
<
S for
I, then
we have
so that
gn*S
T*[].
e R[].
We next determine the largest disk in which R[] is univalent.
The radius of univalence and starlikeness for R[], i/2 <
THEOREM 7.
<
I,
is
r()
PROOF.
valid for
Iz
For f(z)
<_
min
n
(n-a)n(lC(a,n)
-a)
Y.n= 21anlz
z
r whenever Y.
n= 2
n
nlanlr
}
I/(n-l)
in R[], the inequality
n-I
<_
Izf’/f
In view of Theorem 2, this is true
i.
if
r <
Hence, f is starlike for
f’ (z)
n
i
I/(n-1)
r(n_)C(,n)
]
n(l-)
L
Izl
]
<_ r().
On the other hand, for some n we have
n-I
n(l-)
z
(n-) C (,n)
Thus f is not univalent (or starlike) for
COROLLARY.
i/
m
0
<_ r,
when z
r().
r > r().
The radius of univalence and starlikeness for R[I] is
794.
PROOF.
Izl
We must show that
i is
H. SILVERMAN and E. M. SILVIA
436
1/(n-l)
r(1)
1
mlnn
(2/x)I/(x-l)
g(x)
This can be found by differentiating
decreasing for 2 _< x _< x and increasing for x >
0
x0,
and observing that g is
where x
0
satisfies 4 < x
0
< 5.
Since g(4) < g(5), the result follows.
,
Thus, for all
REMARK.
for R
I
is
f in R[] is univalent and starlike when
Izl
<
i’/.
MacGregor showed [2] that the radius of univalence and starllkeness
I/v
.707.
We will now obtain the radius of convexity for R[e].
THEOREM 8.
R[e], 0
If f
zl
<
r(e)
inf
n
< i, then f is convex in the disk
<
I(n_)C(e,n)I
11 (n-l)
n2 (1-)
The result is sharp, with the extremal function of the form
f (z)
n
z-
1-x
(n-=) C(,n) z
for some n.
PROOF
For f(z)
z
In=2[anl zn
Izf"(z)If’(z)l -< 1 for Izl -< r(e).
we
RIll,
E
have
n-I
-r.n__2 n(n-1)lan Iz
I
hold whenever
Y’n=2
In=2 n lanl zn-I
which is bounded by I whenever
En= 2 n
2
it suffices to show that
1-
a
n
z
n(n-l)lan
Izl n-1
-1
En__2 =1,n Iz[
n-I <
I.
From Theorem 2, this will
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
n
2
[z[ n-1
(n-a) C (.a,n)
<’
n=2
l-a
437
3
or
(4.1)
The result follows upon setting
Izl
r(a) in (4.1).
Using arguments similiar to those in the corollary to Theorem 7, we have the
following
COROLLARY.
The radius of convexity for R[I] is
The radius of convexity r.c. for R
REMARK.
.395 < r.c. < .40.
1
/3
is known to satisfy
See [I].
We conclude with the determination of the-smallest
T*[m]
c
B
B() for which
R[].
THEOREM 9.
f
.471.
T*[a] then f
If f
R[I/23 for 1/2 < a < I.
z
R[(2-3a)/2(l-a)3 for 0 < a
1/2, and
The result is sharp, with extremal function
l-a 2
for 0 < a < 1/2
-2_--z
and of the form
for 1/2 < a < I
z----- z
PROOF.
From Theorem 2 it suffices to show, for 0 _< a s 1/2, that
implies
8
8(a)
(2-3a)/(2-2a).
g(a,8,23
I,
<
I, where
This will follow if we can show that
g(a,B,n)
Since
Zn__2(n-B)C(B,n)lanl/(1-B)
I- n-S
n-a
l-B
C(B,n) < 1
it is sufficient to show that
g(a,B,n)
is a decreasing function
H. SILVERMAN and E. M. SILVlA
438
In view of (3.1), g (,8,n+l) < g(e, 8,n) whenever
of n.
(n+l-8) (n+i-28)
n(n+l-s)
<
i-2
2
n-8
n-=
which is equivalent to
p(,n)
2(I-)
(2+2s_2) n 2)
(2(l-e) n +
2
> 0
However, since
IU2’
p(e,n)
p(e,n+l)
2(-)
=
is nonnegative for 0 <
>
1/2.
I]
c
T*[2]
for e
I
2)
is proved.
> e
2
shows that T*[]
c
R[I/2]
But for any e < 1 and y < 1/2, we will show that
f(z)
for n
(4(l-s)n+e
1/2, the first inclusion
<
The inclusion relation T*[e
for
2
z----- z
[e]
n(e,y) sufficiently large.
f,s
z
y
in-
R[y]
R[y], then
If f
C(y,n) z
n
T*[7].
This is true if and only if
I-___ c(,n)
<
n-
I----!
n-
or, equivalently, if
C(y,n) <
Since
C(7,n)
/
.n-o) (11_)
[n--
for y < i/2 and the right hand side of (4.2) is bounded, the in-
equality is not true for n sufficiently large.
REMARK.
(4.2)
T*[I]
{z}
c
R[0].
PRESTARLIKE FUNCTIONS WITH NEGATIVE COEFFICIENTS
439
REFERENCES
I.
Campbell, D. M., A Survey of Properties of the Convex Combination of Univalent
Functions, Rocky Mt. J. 5, no. 4__ (1975) 475-492.
2.
MacGregor, T. H., The Radius of Convexity for Starlike Functions of Order 1/2,
Proc. Amer. Math. Soc. 14 (1963) 71-76.
3.
St. Ruscheweyh, Linear Operators Between Classes of Prestarllke Functions, Comm.
Math. Helv. 52 (1977) 497-509.
4.
Silverman, H., Univalent Functions with Negative Coefficients, Proc. Amer. Math.
Soc. 51 (1975) 109-116.
research of the first author was supported in part by grants from the
American Philosophical Society and Sigma XI, and was completed while he was
visiting at the University of California, Davis.
* The