I ntnat. J. Math. & Math. Sci.
Vol. 5 No. 3 (1982) 537-543
537
ON THE ASYMPTOTIC BIEBERBACH CONJECTURE
MAURISO ALVES and ARMANDO J.P. CAVALCANTE
Department of Mathematics
Universidade Federal de Pernambuco
Recife, Pe, 50.000 BRASIL
(Received December ii, 1981)
ABSTRACT.
The set S consists of complex functions f, univalent in the open unit
f’ (0)
disk, with f(0)
1
We use the asymptotic behavior of the positive
0.
semidefinite FitzGerald matrix to show that there is an absolute constant N
that, for any f(z)
z
+
n=2
n>N
an
z
n
S with
la31
<- 2.58, we have
[an
such
o
< n for all
o
KEY WORDS AND PHRASES.
Univalent
functions.
1980 AMS SUBJECT CLASSIFICATION CODES.
30A32, 30A34.
IN TRODUCT ION.
1.
Let S denote the class of all normalized univalent functions f(z)
z
+
a z
k
k=2
in the open unit disc D.
functions in S, one has
anl
The Bieberbach conjecture states that, for
_< n for all n
N.
The best known estimate for all coefficients is
On the other hand,
Hayman’s
It is known to be true for
lan
-< (I.066)n (Horowitz [i]).
a
Regularity Theorem (Hayman [2]) states that lira
JNI
JanJ
(l-nz)
n.
This implies that
lanl
-< n for n ->
Hayman [3] also proved that A /n tends to a limit, where A
n
anl
to i.
An
for all f
S.
n
..lanl-
1
2’
no(f).
is the maximum of
It is still an open question as to whether this limit is equal
The asymptotic Bieberbach conjecture asserts that lim A /n
n
max
feS
_<
z
and that equality holds only for the Koebe function K(z)
i, for which
n
n
n-o
for each f e S
<- 6.
n
1, where
M. ALVES AND A.J.P. CAVALCANTE
538
Ehrig [4] has proved via the FitzGerald Inequality [5] that if f
laB1
< C <
2.43, then
lanl
No,
< n for all n _>
(as in Hayman’s Regularity Theorem) on f.
where
No
S and
depends only on C and not
This result is a proof of the Asymptotic
Bieberbach Conjecture for a subclass of S.
In this paper, we apply the Asymptotic FitzGerald Inequalities to get, by ele-
Ehrig’s result (Theorem i) and the result in [6],
mentary means, an improvement of
(see Remark 2).
2.
.
PRELIMINARY RESULTS.
THEOREM A.
(FitzGerald Inequality, [5]
f(z)
z
+
Let
ak(f)
k=2
z
be in S and define
qmn (f)
qnm (f)
j (m,n)
b
j=l
where
bj(f)
.laj(f) I; j(m,n) j(n,m),
j (m,n)
j
2
J
2
bm(f)
f
N, and for
lj-nl
m-lj-n
for
0
if otherwise.
2
b (f)
n
m < n:
< m
Then the FitzGerald matrix
(qmn (f))m,n
Q(f)
N
is positive semi-deflnite.
THEOREM B.
(Asymptotic FitzGerald Inequalities [7]).
Let {f
n
n e
N, be
a
sequence of functions in S, such that
converges locally uniformly to f
a)
f
b)
lim inf b
c)
d)
n
(fn)/n < 8
n-o n
lira b (f)/n
(f)
n-o n
d
lira
n-o
Then A
< lira sup b
n-o
S
n(fn )/n
a(fn ).
Q(jl,J2
’Jr-l’
(f)
,(j),8,d
d) (f), defined below, is a
positive semi-deflnite matrix.
Denote by E
the m x n matrix whose elements are all equal to one.
Moreover,
539
ASYMPTOTIC BIEBERBACH CONJECTURE
let H
(f) be the m
mn
hst(f
n matrix defined by its elements
j2t
b
2
Jt
(f).
We
use the notation
Q(JI’
’Jr-1
(f)
(qjsj t (f))l<s
t<p
,Mp(X)
(rest (x)) l<s
t<p
where
7x2/6- x 4
x2(l- x2)
rest(X)
and 6
lit sup 6
n
n-o
where 6
(f)HTr_ I q r(f),
2HTr l,p_q(f)
suPk bn(fk)/n.
2(f))El,
s
for
s
#
t
d2(l-2(f))Eq r,p-q
B4)EI,I 2(I B2)El,p_q
r i
(762/6
q r’
t
Then matrix A has the following form:
82(l-e2(f))Eq_
Mq r(e(f)),
B2(I
2HTr-l,l(f)’
d
B2 Ep_q, i, Mp_q (d)
2
d (i
d2(l e2(j)) Ep-q, q-r
T is
the transposed matrix of H.
where H
[6].
THEOREM C.
3.
n
for
Let f
S; if
laB1
lanl
-< 2.042, then
n for all n
>- 2.
MA IN RESULT S.
For the proof of the Theorem i, we need the following lemmas:
LEMMA i.
Suppose that n > i and that
n
(H)
where H is a compact subclass of S.
]anl n(H).
sup
a
H
f
Let f(z)
2
{f’(z) }2
(v)
a
n
z
+
n-I
(v)
a
n
Here,
I,
a2z
2
+
be in H with
Then f(z) satisfies the differential equation
n
z
n
v=2
n
f(z)
-v-i
n-i
va
+
a
v=l
are the coefficients of f(z)
v,
f(z)V
z
-n+v+ vav zn-v)
a
n
where
a
n=v
n
v
(V)
n
g
n
The proof of this lemma is completely similar to that of Theorem i in
Schaeffer and Spencer ([8], p. 612).
i._._.
M. ALVES AND A.J.P. CAVALCANTE
540
As an application of the Lemma i, we have the following:
LEMMA 2.
If f(z)
+
z
a
n=2
such that a > 0, then 2a
3
3
z
n
a
n
e H is the extremal function maximizing
+ 2.
The proof of this lemma is completely similar to that in Garabedian and
Schiffer
([9], p. 118).
Hayman [2] showed that for each f e S, the limits
lira (l-r)
e(f)
a
2
M(r,f)
r- 1
exist, where
Moo(r,f
If(z)
is the maximum of
lira
n
(f)
n-o
on
Izl
The number (0 -<
r.
-< I)
is called the Hayman index of f.
LEMMA 3.
If f(z)
[i0].
z
+
a
n=2
a (f)
n
n
--(f)
lira
n-o
where b
2
LEMMA 4.
la21 )1/2
(2
If f(z)
a31
+
z
lira
a
fan (f)
z
n
n
2-
PROOF.
(la3,-
4C
_<
n
n-oo
la?l
and
exp(2-4b
is given
then
-i
a21
2
satisfies the conditions of the Lemma 2
-2
exp(2- 4C
-I)
i)I/2 I/2.
By Lemma 3, we have
_<
(f)
2
consider e
-iOf(ei0z)
b
-2
exp(2
4b
i)
e
H,
Since we may assume a real positive (otherwise, we
3
arg a 3
< 2), we obtain that
where 0 < 0
2
la21) I/2
2- (2-
c(f)
Let {f
2-
/(la3l-
2-
LEMMA 5. [7].
45
a21) I/2.
(2
where b
Hence,
-2
S
> i, then
(f)
where C
< 4b
n
and this inequality is sharp for 0
n=2
with
z
n
n
},
n
_<
4C
-2
N, be
i)
(la31- i
i121112
exp (2
i12]i12
c.
4C-i).
a sequence of univalent functions in
S,
541
ASYMPTOTIC BIEBERBACH CONJECTURE
that converges locally uniformly to a function f in S and suppose that (f) > O.
Then
762 2(f) >_ 64
and 6 are chosen as in theorem B
where
Consider the (q
PROOF.
Q((f)
+ l)
r
(q
r
+ i) principal
minor
Mq_r ( (f)
B2 (i_2 (f))Eq_r,
2 (i-2 (f))El,q-r
(762/6-4) El
(f) ,B)
of the matrix A in theorem B.
i
A well-known result about positive semidefinite
quadratic form is that all principal minor determinants of the matrix of the coef-
ficients of the quadratic form are non-negative.
(f) and n
Let
If we
q-r.
use induction, we obtain:
Det Q(
(2/6)n (i_2)
,)
hence, for 0 -<
<
Let f(z)
is an absolute constant N
64)
6(I-2)B 4
6n (i- 2) +i
+
resultlows.
Since n is arbitrary, the
PROOF.
(72_6B4/2)
O;
I
(762 2
THEOREM I.
(762_64/2 + 2n 6-(n+l)
z
+
a
The case
z
k
> 0o
k
be in S.
i is immediate.
(independent of f), such that
o
Suppose the contrary and take a sequence
la31
If i <
lan
{gk },
<
2.58, then there
< n for all n > N
o
k e N, of univalent
functions in S such that
i)
ii)
iii)
iv)
{gk },
k e N, converges locally uniformly to a function
bnk(gk)
REMARK.
-> n
k
k
Lemma 2 to the subclass
We pick for each n
Hk,
k
S; i -<
bnk
b3(g)
< 2.58 and
bnk(g)
> n
k}
la3(gk)
of S.
in
Applying
we obtain condition (iii).
one of the functions of
{gj },
which maximizes
going to infinity.
k
are the extremal functions maximizing
{g
the compact subclass H
2,
for sequence n
The functions
S,
<- 2.58,
1 <
b3(gk
a3(gk
2
2a 3(gk
+
a 2(gk
go
and denote it by
j
fnk;
0,I
precisely, let
{fnk },
k
N, be a sequence
M. ALVES AND A.J.P. CAVALCANTE
542
of the functions in
{gj},
0,i
j
such that
sup b
j
bnk (fnk).
nk (gj
}, k N, converges locally uniformly to a function f S.
k
< 2.58 and
Otherwise, we pick a subsequence of {f }, k e N. Evidently, i -<
n
k
2
+ 2. For this sequence {f }, k N, we have
n
k
We may assume that {f
n
b3(f)
a2(f)
2a3(f
sp bnk fnj )/nk
nk
bnk fnk) /n
k
Thus
lim sup b
n
k-oo
k
6 in Theorem B.
We take
This excludes (f)
Let f(z)
Q(Jl
-> i.
0 because
664
This implies, by Lemma 4, that
0
b3(f)
z
+
d)(f) is
By Lemma 5, we have
2(f)
or
>
662/7
6/7.
>
> 2.58 which contradicts the assumptions.
=a
k
laB1
If
be in S.
k=2
an absolute constant N (independent of f), such that
o
n
z
la
PROOF.
In fact, the determinant
Jr-l’ (f) ,(f),8,d
64(f)) (762 64)/36_ 64(1_ 2(f))2 >_ 0.
72(f)62
COROLLARY.
i.
First we show that (f) > 0.
of the 22 submatrix Q((f),6) of A
(72(f)
(fnk)/nk >
<
2.58, then there is
< n for all n > N
o
The proof of corollary follows immediately from Theorem i and Theorem C.
ACKNOWLEDGEMENT.
This research was supported in part by FINEP and CNPQ.
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ASYMPTOTIC BIEBERBACH CONJECTURE
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