Internat. J. Math. & Math. Sci.
VOL. 12 NO. 4 (1989) 641-644
641
SUFFICIENT CONDITIONS FOR SPIRAL-LIKENESS
H. SILVERMAN
Department of Mathematics
College of Charleston
29424
Charleston, SC
(Received December 13, 1988)
ABSTRACT.
Coefflcient
conditions
for
sufficient
are
splral-likenss
found
by
The order of starlikeness for such functions is also determined.
convolution methods.
KEY WORDS AND PHRASES.
Splral-llke, starlike, convolution.
1980 AMS SUBJECT CLASSIFICATION CODES. 30C45, 30C55.
INTRODUCTION.
A function f(z)
z +
analytic in the unit disk A
J2 anZ
<
z
I} is said to
starlike
of
order a, 0 < a < I, if Re {zf’/f} > a, z
A, and is said to
We
these
denote
.spiral-like, -/2 <
> O, z A.
< /2, if
classes, respectively, by S*(a)and Sp(). Note that Sp(0)
S*(0), the family of
starlike functions.
Functions in Sp(%)were shown by Spacek [4] to be univalent
in A and were later studied extensively by Llbera [I].
be
Re{eizf’/f}
be
A function f is in S*(a) if its coefficients are sufficiently small.
THEOREM A [2]. If the coefficients of f(z)
thenl(zf’/f)
11 <
l-a, z
z
+n,Z2 anZ
satisfy, the inequality
A, and hence f
It is our purpose here to find coefficient conditions guaranteeing that f is in
Sp(l). Our methods will involve convolution properties and will also furnish us with
an alternate proof that (I) is a sufficient condition for f to be in S*(a).
The convolution or Hadamard product of two power series f(z)
and g(z)
=n=ZO bnZ
shown that f
n
is defined as the power series
S*(a) if and only if
; (re
z
/
and f e Sp(X) if and only if
Now (z +
cz2) /(l-z) 2
-
(_,>2
i El,
Z
(l-z)
z
-
(jCx+2-l)l (2:2a)z
+2 (n +
z
(n-1)c)z
2
n
(f*g)(z)
,o
) ,o
(z
(
=n=ZO anZ
=n=EO anbnZ
n
In [3] it is
, , Ix)- >
, I1"
.
so e may restate these results as
H. SILVERMAN
642
The function f(z) -z
THEOREM B
+nr.2.
(n +
x
and
A
0 for all z
(n-l)c)an
n
a z E S* () (Sp(%)) if and only if
+nE2
A straightforward computation shows that
for the appropriate choice of c.
and we can conclude from Theorem B that condition (I) is sufficient for f to be in
The corresponding result for Sp(A) is computatlonally more involved.
S*(s).
2.
THE MAIN CLASS.
THEOREM I.
The result is
e
n
z +
fn(z)
sharp wlth__
anz
in Sp()
if and only. if
(x-e
c
+ ic 2,
n+- (n-1)((n-l)(.
c
Thu s
(n_l) (c
2
which is maximized when
8"
%,
+
2
c2)
g(8)
80
tan
+
2
that
real, we hav
+
--
sin 2
n
Setting
(I + cos 2X)_
cos(2t + 6) + cos
2(I + cos 2)
cos(2l + 8) + n cos 8 is maximized.
(-
I/Bn().
(I + n cos 2)
cos(2 + 8) + n cos
2(1 + cos 2)
2ncl
-I
lanl
c22) + 2ncl]
maximum when (n-l)(c21 + c2)2 + 2ncl does.
cos(2t + B)
and
c
+ cos 2
2
2
Cl + c2
this choice of
for
-2i)/(1+e-211)
and c
a computation shows that
its maximum when
an
2
n
Writing c
will attain its
+ (n-l)c
Bn()
co s A
From Theorem B, it suffices to show for c
In + (n-l)c Bn(X).
(2)
In + (n-1)cl
x
mlf n_Z2
2cos
max
Hence
Sp(A)
+nE2ffi anZ
(n-l.), + (n ,!)2+ 4n
B (A)
n
PROOF.
f(z) -z
The function
+
Co
2-),
with
g()
But g(8) attains
(n-l) 2 + 6n
co
s2 t
we have
(n-l)(c21 + c2)2 + 2ncl
-17
2
+ 4n
cos2 +
2 cos 2
(n-l)
2n cos
t
n
().
For
SUFFICIENT CONDITIONS FOR SPIRAL-LIKENESS
643
It now follows from (2) that
In +
max
(n2
(n-1)c
2
(n- I)_ + 2n co s
2
+
))L/2,
2
)_ (nY)_) + 4n cos2
+ (n-l)t
On-
/2
2
2 co s
which may be expressed as B ().
n
that according to Theorem B f (z)
fn(Z)
if
lanl
+
Sp() if
I/Bn( )"
z + a
[anl > I/Bn().
+ in
Sp() if z
In particular,
n-I
+ (n-l)c[
fn(Z)
n
n
B (), we see that
n
z +
an gn
Sp(%) if and only
n
This is a consequence of B
PROOF.
zn
n
n
Choosing c so that
has a solution for z e A.
To show sharpness, note
[(n + (n-l)c)a n]
This completes the proof.
n
(1) being an increasing function of
-.
will
for
In fact, any function that satisfies the conditions of Theorem
a sharp contrast to the inclusion properties for the
also be in Sp(A)for
general class Sp(A). The function
IAI IA01,
fl(z)
z(l-z)
other hand,
-2e
cos A is in Sp(A) but it is not in Sp(t) for any y
0
On the
Sp(A)
the upper bound on the modulus of the coefficients for f
decreasing
is
a
n
f e Sp(A) to be
n-I
lanl
H
Though Theorem
/(k-l) 2 + 4k
co
s2 (n-l)!,
with
f(z)
being extremal.
gives a sharp result, it is not aesthetically pleasing because
of the complicated nature of B (). A consequence of the inequality
n
+ (n-1)sec
B () is more palatable sufficient condition.
n
COROLLARY 2.
If
n_E2
(I + (n-l)sec
X)tta
n
I, then f(z)
z +
n.Z2
a z
n
Sp(l).
Using a different method, Corollary 2 will also be shown to follow from Theorem 3.
3.
ORDER OF STARLIKENESS.
Since B (A)
n, we see from Theorem A that a function satisfying the conditions
n
must be starlike. We can actually do better.
of Theorem
THEOREM 2.
a
0
+ 8
(3
f2(z)
z
PROOF.
every n.
+
If f satisfies the conditions of Theorem
cos2A)/2(l
S*(a0) for
The result is sharp, with extremal function
z2/B2(A).
In view of Theorem A
Since
B2(1)
x-I +
Setting
/(Xi 1-2 + 4x cos2v I
x-a
0
(n-a0)/(l-so) for
(l-aD)Bn(l)/(n-O) is
we need only show that B ()
n
it suffices to prove that
(2-a0)/(I-0),
an increasing function of n.
G(x)
+ cosA).
then f
will show that O’(x)
0 for x
2.
H. SILVERMAN
644
A d[fferentlatlon of G leads to
(l-a0-2cos
(X-ao)2G,(x) l-ao +
2
)x-
(l-a0+
+ 4x co s
(x-l)
2a0cos2)
2
H(x), say.
2
4sl n Xco s 2
Since H’(x)
((x-l)
2
X(x-a0)
)
cos2)3/2
4
cos 1
+ 4x
H(x) > H(2)
0, it follows for x
cos x-!
)
2 that
> 0.
41 + 8cos 2
Therefore G(x), and consequently
(1-o)Bn()/(n-a0),
is an increasing function.
This
completes the proof.
_
_
We have actually shown move according to Theorem A.
If f satisfies the conditions of Theorem I, then
< l-a0, z A.
COROLLARY.
lf’/f)
11
,
Functions in S (a) need not be in Sp() for 0.
The function
2(l-a)
z/(1-z)
the
half
e S (a) since zf’/f maps & onto
plane Re w > a.
Sp(),XO.
We next look at a subclass of S (a) whose functions are
However f
f (z)
splral-llke.
OR.
then f
f(z)
Sp() for
cos
(l-a).
We may write (zf’/f)-I
Re{eizf’/f}
cos
%
l(--f’/O-
tth
I < -f
z
,
The result is sharp, with extremal function
ze(l-a)z.
PROOF.
Thus
f( -+...
-cos
+ (l-a)
(l-a)) 0 for
COROLLARY.
PROOF.
Set a
cos
I(z’/
(l-a)m(z), where
Re{eim(z)}
cos
lo(z) <
A.
for z
-(1-a)leim(z)l
>
(l-a), and the proof is complete.
< =
’r
f
S(.
-cos ), in Theorem 3.
Finally an application of Theorem A, with a
cos A, to Theorem 3 provides us
with an alternate proof to Corollary 2 of Theorem I.
REFERENCES
I.
2.
3.
LIBERA, R.J., Univalent a-spiral functions, Canad. J. Math. 19 (1967), 449-456.
SILVERMAN, H., Univalent functions with negative coefficients, Proc. Amer. Math.
Soc. 51 (1975), 109-116.
SILVERMAN, H., SILVIA, E.M. and TELAGE, D.N., Convolution conditions for
convexity, starllkeness, and splral-llkeness, Math Zeltschrlft 162 (1978),
125-130.
4.
5.
’
SPACEK, L., P ispevek k teorll funckl pros tyk Csopls Pst Mat. a Fys.62
(1932), 12-19.
ZAMORSKI, J., About the extremal spiral schlicht functions, Ann. Polon, Math. 9
(196 2) 265-273.