n - a

THE BIEBERBACH CONJECTURE AND ITS IMPACT ON
THE DEVELOPMENTS IN GEOMETRIC FUNCTION THEORY
O.P. Ahuja
(received 25 March 1986)
1.
Introduction
The study of Geometric Function Theory is one of the most fascinating
aspects of the theory of analytic functions of a complex variable.
In this
field, we are mainly concerned with the power series of the form
f(z)
=
in the complex variable
b Q + b^z + b2z2 + ... + b^zn + ...
z
that are convergent in a domain
power series may be interpreted as amappingof the donain D
some range set
(i)
E
in the
given the sequence
u-plane.
{fc0,
o
{fco>^i>...} ?
E ,
coefficients, what can be said
E ;
and
given some geometric
A nice geometric property from the point of view of conformal
We recall that a function
to be univalent in
f(zx) t f{zz)
(ii)
what can be said about the sequence
mapping possessed by an analytic function
D .
Such a
There are two natural questions to ask:
f
about the geometry of the range set
property of the range set
D .
in the z-plane onto
if
D ,
/(z)
/O)
is that of univalence in
that is analytic in
D
is said
if it never takes the same value twice, that is,
z l t z2 ,
whenever
z l , z2 e D .
Univalent functions are the simplest analytic functions from the
geometric point of view.
The theory of univalent functions is so vast and
complicated that certain simplifying assumptions are necessary.
take the unit disk
A = {z : \z\ < 1}
First is to
in place of arbitrary domain
Second is to take normalization conditions:
D .
/(0) = 0 , /'(O) = 1 .
Invited address presented at the 50th Annual Conference of the
Indian Mathematical Society at Sardar Pael University (India), February 6,
1985 .
Math. Chronicle 15(1986), 1-28 .
1
With these assumptions, we can rewrite
/(z)
in the form
00
v
n
) a nz ,
L
»
n =2
z +
/(*)
z e A .
(1)
It may be noted that the normalization does not disturb the univalence of
the function because if
/(z)
is analytic and univalent in
A ,
so also is
the function
We may also add that
/'(O) t 0 .
could not be univalent in
A .
U
=
{f
: f
5
=
{/
: / e U
If
/'(O)
vanishes in
A ,
then
/(z)
Further, let
is analytic and normalized in
A)
and
Example 1.
/
is univalent in
It is easy to see that the function
univalent in
Example 2.
and
A ,
and
^(A)
is the half-plane
A} .
g(z) = (1+z)/ (1-3)
is
Re g(z) > 0 .
One can prove that the function
( 2)
is univalent in
A .
This is called the Koebe function and it maps
A
onto
the entire complex plane except the slit along the negative real axis from
-«
to
-1/4 .
In an intuitive sense, the Koebe function is the largest
function in the family
S
because we cannot adjoin to
without destroying univalence.
property.
fc(A)
any open set
(This is not the only domain with this
However, the maximal nature of the domain
and the size of the coefficients of fe(z) .)
2
fc(A)
The functions
is its symmetry
2
also belong to
S
z +
i(n-1)0 n
\ ne!
Z
n=2
(3)
and are referred to as the rotations of the Koebe function.
The serious study of univalent functions began in 1907, when Koebe
published a paper [58] in which he proved the existance of a positive
constant
a
such that
In 1916, Bieberbach [13] proved that
a = 1/4 .
result that the open disk
is always covered by the map of
any function
f e S .
|zj| < 1/4
Thus we have an interesting
A
of
Furthermore, Bieberbach observed that the Koebe
function or any of its rotations are the only ones whose image domain
contains a boundary point on the circle
|u| = 1/4 .
In the same paper, he
proved that the absolute value of the second coefficient of any function
in
S
is never more than
function, we have
2 .
|a^| = n 2 2 .
showed that the class
S
By proving a distortion theorem, Koebe
is compact.
Thus for every positive integer
there is a uniform bound on the magnitude of the coefficient of
f
in
S .
f
He further observed that for the Koebe
n ,
z 1 for all
These observations encouraged Bieberbach to make the famous
conjecture [13] in 1916 .
The Bieberbach Conjecture.
Among all the functions
z +
I a z1
(zeA)
71=2
in
S ,
the Koebe function has the largest coefficients, that is
S
for all
n ,
n > 2
f e S .
Ludwig Bieberbach, a German mathematician, is well known in the
mathematics community.
His conjecture has challenged the best investigators
3
of the world for about 70 years.
This conjecture has been considered so
difficult to prove or disprove that some eminent mathematicians believed it
to be false.
A great number of researchers have devoted their careers in
trying to resolve it.
The Bieberbach conjecture has inspired several
developments in geometric function theory by imparting many powerful new
methods and generating a large number of related problems, some of which are
open while others have been solved completely.
In all likelihood, it has
already contributed more to mathematics as a challenging problem than it will
ever contribute as a theorem.
In the mid-summer of 1984, Louis de Branges from the University of
Purdue greatly excited and surprised the mathematics world [59] by making a
claim that he had resolved the Bieberbach conjecture.
Thus he ended the
efforts of many mathematicians of almost seventy years.
In this expository survey, we mention only those results of the theory
of univalent functions which bear directly on the Bieberbach conjecture or
on the problems resulting from an effort to resolve the conjecture.
also emphasize some recent results and related open problems.
is an immense literature
surveys
[24,40,57]
[12,20,39] ,
books
We shall
Since there
[41,54,74,85,97]
and several
on theory of univalent functions, we shall make a
selection of the results relevant to our precise objective.
2.
The Bieberbach conjecture for individual coefficients
Let
1
be the class of functions
q(z)
J
=
g e \ .
0
|2 | > 1
analytic and univalent in
residue 1 .
2 + i> + b z
1
-1
+ bz
-2
2
+ ...
except for a simple pole at
“
with
Grownwall [43] in 1914 proved that
\ n\b |2 £ 1 , whenever
n= l
This elementary result is known as the area theorem. Applying
this result to the function
=
(/(I/s2))"'5
=
z - a2/2z + ... ,
one can easily prove the Bieberbach conjecture for
4
n = 2 .
There are many
simple proofs available to prove the conjecture for
n = 2
In 1923, Lowner [68] proved the conjecture for
n = 3
type of functions which map the disk
Jordan arc terminating at
“ .
A
by using the
onto the full plane slit along some
These functions are dense in
topology of uniform convergence on compact subsets of
A .
S
in the
Lowner showed
that they can be generated (along with certain other univalent functions)
by a differential equation of prescribed form.
respresentation of the coefficient
k{t) , |k(t)| = 1
This leads to a parametric
in terms of integrals of a function
given by
a3
-2j e~2t[k(t)]Zdt - 4^|
=
It can be shown from
function.
a3
(4)
that
|a3| £ 3
.
(4)
with equality only for the Koebe
By using variational methods, Schaeffer and Spencer [96] in 1943
gave another proof of
Schiffer
|a3| £ 3 .
[98,99,100]
developed a Calculus of variations for the class
S
and had used it to prove that each
A
onto the plane slit along a system of analytic arcs satisfying a certain
differential equation.
f z S
which maximizes
|a^|
must map
In 1955, Garabedian and Schiffer [33] used the
variational method developed by Schiffer to establish the Bieberbach
conjecture for
n = 4 .
Their proof was extremely complicated.
However,
in 1960, Charzynski and Schiffer [21] provided an elementary and simple proof
of
lajJ £
4 .
Their proof was based on the Grunsky inequalities [44]
which can be established by a slight generalization of the method used to
prove the area theorem.
The Bieberbach conjecture for
[81] and Ozawa
[77,78] .
[44] to prove that
n = 6
was proved in 1968 by Pederson
These researchers have used Grunsky inequalities
|a& | £ 6 .
Their proof is difficult and too long.
Applying a variational method, Garabedian and Schiffer 1967 [34]
generalized the Grunsky inequalities to yield certain relations known as the
Garabedian - Schiffer inequalities.
The inequalities involve auxiliary
parameters which can be choosen to optimize the estimates.
Using the
Garabedian - Schiffer inequalities, Pederson and Schiffer [83] in 1972
5
settled the conjecture for the fifth coefficient.
No other case of the Bieberbach conjecture has been verified.
Ozawa and Kubota 1972 [79] have shown that
Re a2 > 0 .
provided that
By using a computer, Horowitz 1977 [50] has shown that if
r
k
p(z) = 2 + \a.vz
is a polynomial in
1
then |a^| S k , 2 < k < 27 .
3.
Re aQ < 8
However,
S
of degree not greater than
27 ,
Some estimates for all the coefficients
In the meantime, mathematicians got closer and closer to the Bieberbach
estimate for all the coefficients.
The first good estimate for all the
coefficients was given by Littlewood [65] in 1925 .
estimation of the Cauchy integral formula for
\an \
-
r
(r >f) ,
By using the crude
:
0 < r < 1 ,
where
1
'a
|d0 ,
1o
he proved that
logarithms.
\an \ < en
f°r
n »
where
e
is the base of the natural
In 1951, Bazilevic [ll] improved Littlewood's estimate for
to prove that
1974 [9] proved that
|a^| < en/2 + 1.51 , n = 2 , 3 , ... .
Baemstein
\an \ < en/2 , n = 2 , 3 , ... .
Milin [72] developed a new method in 1965 to obtain information from the
Grunsky inequality and thus proved that
|a^| < 1.243 n , n > 2 .
result was improved by FitzGerald [30] in 1972 .
His
Using a method involving,
roughly speaking, exponentiation of the Grunsky inequalities [44] ,
he
obtained the inequalities
n
,
2n - 1.
l k \ a k \Z *
I {2n-k)\ak \Z
k= 1
k=n+1
6
>
|a | \
n = 2 , 3 ......
(5)
FitzGerald [30] used the inequalities
(5)
to prove
|a | < (7/6)'s n < 1.081rc ,
n > 2 .
Horowitz 1978 [49] further improved the estimate of FitzGerald by using a
stronger form of
(5) ,
again due to FitzGerald [30] .
|a | 5 (1,659,164,137/681,080,400) 1/1I+n
S
His estimate was
1.0657n.
The Bieberbach conjecture is true for functions with sufficiently small
second coefficient.
We list the estimates on the second coefficient as
follows:
For each function
f(z) = z +
Jv 1
) a z
L n
n= 2
in
S , a
< n
’ 1 n1
for all
n > 2
if
\a2 \ < 1.05 ,
Aharonov 1970 [l] and Illina 1973 [52]
\a \ < 1.15 ,
Ehrig 1974 [28]
|a2 | < 1.55 ,
Bishouty 1976 [15]
|a2 | < 1.61 ,
Bishouty 1977 (unpublished).
Bieberbach Conjecture for subclasses of
4.
S
Failure to settle the Bieberbach conjecture for all the coefficients
has led many researchers to introduce and investigate properties of several
subclasses of
S .
We give below some of the most important subclasses of
S .
Convex univalent functions.
These are the functions which map the unit disk
A
A domain
onto the convex domains.
segment joining any two points of
E
A
studied hy E. Study [1063 in 1913 .
z e A .
f t S
is convex in
is said to be convex if the line
lies wholly in
of all convex univalent functions in
that function
E
by
K .
K .
In fact, we have
Denote the class
Roberston [87] , in 1936, observed
A
if and only if
Robertson [87] further proved that
in the class
E .
These functions were first
\a \ £ 1
7
Re(l+z/"(s)//' (z)) > 0,
|a^| 2 n , n 2 2
for all
f z K
for functions
and
z/{l-z) = 2 +
of
la I
1 n1
\ z 1 is an extremal function.
n= 2
for K can be lowered.
Starlike univalent functions.
Thus the Bieberbach estimate
Another important subclass of
S
consists of
the starlike functions first treated by Alexander [7] in 1916 .
functions map
A domain
E
A
These
onto a domain starshaped with respect to the origin.
containing the origin
w = 0
is said to be starshaped (or
starlike) with respect to the origin if the line segment joining
any other point of
E
lies completely in
starlike univalent functions in
is starlike because it maps
-00 < y < _i/4 .
A
S*
by
S* .
w - 0
2 /(1-2)2
The Koebe function
f z S*
The inequalities
Spiral-like univalent functions.
if and only if
\a^\ i n ,
and are sharp for the Koebe function
n > 2
hold for
[87] .
A class wider than the class
S*
is the
class of spiral-like functions introduced by Spacek [105] in 1932 .
analytic normalized function
if
all
f
for
some rj such that
A .
Sj5acek [ 105] proved that
and that the Bieberbach conjecture holds for
Univalent functions with real coefficients.
S
|r]| = 1 .
SB
H^
of
c S
H^ .
Let
SR
for which all the coefficients
The Bieberbach conjecture holds for
H
A
Taking
Libera [64] in 1967 introduced the class
\-spiral-like functions in
functions in
An
is said to be spiral-like function in
Re{r)2 / '(2 )//(2 ) } > 0 , 2 e A
r) = exp(£V) , |\| £ it/2 ,
to
Denote the class of all
onto the entire complex plane minus the slit
Robertson [87] showed that
Re{zf'[z)/f(z)} > 0 , 2 e A .
the class
A
E .
denote the subset of
of
f
are real.
and it was proved by Dieudonne [22]
in 1931 .
Close-to-convex functions.
In 1952, Kaplan [56] introduced the class
all close-to-convex functions in
A .
is said to be close-to-convex in
A
g z S*
A function
f ,
analytic in
C
of
A ,
if there exists a starlike function
such that
Re
zf'(z) \
.g
W
/
2 z A
(6 )
Geometrically, the condition
\z\ = r < 1
(6)
implies that the image of each circle
is a curve with the property that as
0
the tangent vector does not decrease by more than
[©1,02] .
increases, the angle of
-u
in any interval
Thus the curve cannot make a "hairpin bend" backward to intersect
itself.
Kaplan [56] proved that
C c S .
Reade [86] in 1955, proved that the
Bieberbach conjecture holds for the family
C .
Typically real functions.
f(z) = z +
in
A
If the function
and if for every non-real
z
in
/
that the upper half of
interval
>
0 ,
is said to be a typically real function in
the lower half of
(-1,1)
A
A
A .
This merely means
must go into the upper half-plane under
must go into the lower half-plane under
goes into the real axis.
typically real functions in
Rogosinski [93] in 1932 .
is analytic
A
(Im z)(Imf(z))
then
\ a zn
n= 2
A
by
TR .
f ;
/ ,
and
the
Denote the class of all the
The class
TR
was introduced by
He also proved the Bieberbach conjecture for
TR .
From the definitions of the above subclasses of univalent functions, it
follows that
K c S* c C c S , K c S* c HX c S
and
SR c TR c S .
We remark that various subclasses of the above classes have been introduced
and studied by many researchers including the author.
[2,3,4,5,6,41,53,64,70,80,86,87] .
See for instance
Usually, one studies distortion
theorems , coefficient estimates, bounds, radius of univalence, convexity,
starlikeness,
spiral-likeness, transformations
and extremal results for such subclasses of
9
S .
which are class preserving
5.
Local form of Bieberbach Conjecture
The local Bieberbach conjecture states that the Koebe
the maximum of
Re{a^}
at least for those functions
f
function yields
in the family
which are close enough to it in some appropriate topology.
S
There are
several interesting results on the local form of the Bieberbach conjecture.
Using the Grunsky inequality, Garabedian, Ross and Schiffer 1965 [35] proved
the local Bieberbach conjecture for the case of even index n = 2m .
1967, Garabedian and Schiffer [34]
In
derived a generalization of the Grunsky
inequality through a differential equation of the form
[f’(z)f E
where
E(t)
is almost quadratic
=
R (2 )
and
R{z) ,
is a rational function; and
they used generalized Grunsky inequality to prove
Theorem.
[34]
If a function
function, for example, if
e > 0 ,
m
a 2m+\
f
in
S
\a2-2\ <
is close enough to the Koebe
for an appropriate small number
then the Bieberbach conjecture is true for the odd coefficients
>
wi-th equality holding for the Koebe function.
Using the Lowner method together with the variational method developed by
Duren and Schiffer 1962 [27] , Bombieri 1967 [16] proved the local
Bieberbach conjecture in the following interesting form
(
where
and
6^
n - a (2-Recz )
n
2
if
n
even,
12 - a I < 6
1
2'
n
n - a^(3-Rea3)
if
n
odd,
|3 - ^ 3 | < 6^ ,
are some positive constants.
In 1969, Pederson [82]
established the equivalence of various topologies near the Koebe function.
6.
Successive coefficients of univalent functions
Let d = lla
I - la II , n = 2 , 3 , ... denote the difference of the
n
rc+i1
1 n" ’
’
moduli of successive coefficients of functions in S .
The problem of
estimating
d
has attracted the attention of many researchers.
following results have been obtained:
10
The
dn = 0 (n1^
log n ) ,
Goluzin 1946 [37]
d-n = 0( (log n)3//2) ,
dn < K
Biemacki 1956 [14]
for some absolute constant, Hayman 1963 [47]
d < 9 ,
n
’
Milin 1968 [73]
d < 4.17 ,
n
I1ina 1968 [51] .
Milin [74] was able to obtain bounds for the moduli
d
for coefficients of
odd univalent functions more precisely.
Leung 1978 [61] proved a conjecture made by Pommerenke [84 , Problem 3.5];
namely,
He further observed that equality occurs for fixed
2/(1-y3)(1-^2)
for some
y
K
with
n
only for the functions
|yj = |^| = 1 .
Robertson [90]
showed
M a J
for the class
C
with
Leung 1979 [62] that
n
and
m .
m - n
(7)
-
-
(7)
|m2-nZ I
being an even integer.
holds for the class
C
It has been shown by
for all positive integers
However, Jenkins 1968 [55] showed that the inequality
cannot hold for the whole class
S .
For the class
S ,
(7)
Duren 1979 [25]
has shown that
with Hayman index
7.
2 e6a_3i <
n+l
n
a > 0 ,
where
1.37a”5* ,
5 < 0.312
n > 2
is the Milin's constant.
Various conjectures related to Bieberbach conjecture
We now investigate several conjectures given so far to resolve the
Bieberbach conjecture.
the class
S
Our inability to prove the Bieberbach conjecture for
suggested that perhaps we should attack somewhat weaker conjectures.
11
Littlewood conjecture (1925).
number
w ,
If
f z S
and
/(z) ? w
for some complex
then
\an \ £ 4|u|n ,
n > 2 .
(8)
It is well known that for such an omitted complex number
|u| 2 1/4
and that if
|u| = 1/4 ,
its rotation, and in this case
hand, if
|u| > 1/4 ,
then
(8)
f
is true for all
then the right side of
w ,
we have
must be the Koebe function or
(8)
n .
exceeds
On the other
n
and hence
the Littlewood conjecture is weaker than the Bieberbach conjecture.
The
above conjecture was proposed by Littlewood [65] in 1925 .
Littlewood - Paley conjecture (1932).
g(z)
in the class
S ,
=
|c | £ 1
For each odd function
[ / O 2)]'5 =
for
...
2 + c 32 3 +
n = 3 , 5 , ... .
This conjecture was proposed by Littlewood and Paley [66] in 1932 .
In the same paper, they showed that
|c | £ A
absolute constant (their method gives
conjecture to the equation
Bieberbach conjecture.
S* [66] .
for all
A < 14) .
n ,
where
A
is an
If we apply their
/(z2) = [g(z)]2 , f z S ,
it easily implies
the
The Littlewood-Paley conjecture is true for the class
However, the conjecture is false in general because Fekete and
Szego 1933 [29] obtained the sharp inequality
|c | £ 1/2 + e"2^3 = 1.013 .
Using a variational method, Schaeffer and Spencer 1943 [96] proved that the
equality
|e | = 1.013
is attained for odd functions with real coefficients.
Moreover, for odd functions in
has shown that
S
\c7 \ £ 1090/1083 ,
Robertson conjecture (1936).
is any odd function in
S ,
If
then
with real coefficients, Leeman 1976 [60]
and this bound is sharp.
g(z) = z + b 3z
+ ... + b2k_lZ
+ ...
If we apply the Robertson conjecture to the odd function
[<7 (2 ) ] 2 = f(z2) >
the Bieberbach conjecture immediately follows.
conjecture was proposed by Robertson [88] in 1936 .
conjecture is equivalent to
\a^\ < 2 .
For
The above
n = 2 ,
the
Using the Lowner1s variational method,
Robertson [88] proved his conjecture for
n = 3 .
After a period of thirty-
four years, Friedland 1970 [32] proved the conjecture for
n = 4 .
He used
the Grunsky inequalities.
Rogosinski conjecture (1943).
analytic in
A ,
If a function
g(z) = b xz + b 2z2 + ... ,
is subordinate to a function
f
belonging to
S ,
then
\b | £ n , n > 1 .
1 n1
This conjecture was advanced by Rogosinski [95] in 1943 .
Since any
function is subordinate to itself, it follows that the Rogosinski conjecture
implies the Bieberbach conjecture.
For
is contained in the Schwarz lemma.
Earlier, Littlewood [65] proved it for
n = 2 .
and
n = 1,
Rogosinski [95] proved it for all
SR .
n
the Rogosinski conjecture
for the classes
Robertson 1965 [89] established this conjecture for
S* , K ,
C .
In
1970, Robertson [92] observed that the Rogosinski conjecture is implied by
the Robertson conjecture.
Mandelbrojt-Schiffer conjecture.
function
J ,
If
f
and
g
are in
S ,
then the
defined by
J{Z)
=
/ (2)
«
5 (2 )
=
3
+
I
Zn ,
n=2
is in
S .
If the above conjecture was true, we could immediately prove the
Bieberbach conjecture.
J(z) = z + (an/n2)zn .
In fact, we merely set
g(z) = z + z 1/n
and then
For many years, this conjecture was an open problem
but in 1959 it was disproved by Lowner and Netanyahu [69] .
conjecture was proved to be true for various subclasses of
Asymptotic Bieberbach conjecture (1958).
If
However, the
S[4l] .
This weaker conjecture than the Bieberbach conjecture was proposed by Hayman
[46] in 1958 .
In the same paper, he proved that
A^/n
tends to a limit.
Nehari [76] proved that this conjecture implies the Littlewood conjecture.
00
Robertson second conjecture (1970).
to
n
f{z) = z + \ a z
n- l
where
f
If
is in
g(z) =
\ b zn
is quasi-subordinate
n= 1 ”
then |b | £ n , n > 1 .
n
S ,
The concept of quasi-subordination was introduced by Robertson
[91,92]
CO
in 1970 .
We say that a function
g(z) = \ b z
1 n
if both functions are analytic in
ftz) = \ a z
1 ”
$( 2 ) , analytic and bounded in
is subordinate to
f(z)
in
A
with
The above conjecture is true for
all
n
if
is in
5*
and there is a function
|$(3) | £ 1 ,
such that
n = 1 , 2 , 3 , 4
n > «o (/)[9l] .
are all real
/
A
g{z)/${z)
A .
sufficiently large values of
coefficients
is quasi-subordinate to
{b
or
and for all
It is true for all
may be complex) [9l] .
or in
TR [9l] .
n
if the
It is true for
Further, it is observed
by Robertson [92] in 1970 that the Robertson second conjecture (1970) ,
if
true, implies the truth of the Bieberbach conjecture and Rogosinski
conjecture, and is true if the Robertson first conjecture (1936) is true.
Thus all these conjectures are ture if the Robertson first conjecture is true.
Mil in conjecture (1971).
For every
I k{n+l-k)\ak \2 <
k«i
where
c^
I
-L'- - ,
k= 1
K
=
2
g(z) = I &,z2<~1
2 £ c, 2 k t
k= l K
n 2 1
k
are the logarithmic coefficients of
log
If
f z S ,
f e S
given by the equation
z e A .
is an odd power series such that
then
J
y
k ~'
-
Sl e x p f j a / )
14
.
\,g{z)~\Z = f(.z2) ,
In 1967, Lebedev and Milin [75] obtained the inequality
n+i
n
I
n + 1 k=Il
exp
n + 1 k= l
(9)
n+l-k
Equality holds if and only if a complex number w of absolute value one
v
exists such that c^ = w /k for k = 1 , 2, ..., n .
Because of the
Lebedev-Milin inequality
(9) ,
the Milin conjecture implies the Robertson
conjecture and, so implies the Bieberbach conjecture [74] .
[42] has verified the Milin conjecture up to
Sheil-Small conjecture (1973).
of degree
For each
Grinspan 1972
n = 3 .
f e S
and for each polynomial
P
n ,
IIP * / I I . 2 llPll.
where
||*||
denotes the maximum modulus in
and
P * f
stands for the
convolution (or Hadamard product) of
P
This conjecture of Sheil-Small [104]
lies between the Robertson and
Rogosinski conjectures.
Taking
and
A ,
P(s) = zn ,
f .
it can be easily seen that the
Bieberbach conjecture is implied by the Sheil-Small conjecture.
Relationships between various conjectures
Interestingly, seven of the above mentioned conjectures are related as
follows:
Milin conjecture (1971)
=»
Robertson conjecture (1936)
Robertson conjecture (1936)
=«=» Sheil-Small conjecture (1973)
Sheil-Small conjecture (1973)
=
Rogosinski conjecture (1943)
==>
Bieberbach Conjecture (1916)
Rogosinski conjecture (1943)
Bieberbach Conjecture (1916)
=»
Asymptotic Bieberbach conjecture (1958)
Asymptotic Bieberbach conjecture (1958)
=»
Littlewood conjecture (1925) .
All these seven conjectures have remained open till as late as mid-summer
1984 .
In fact, the Bieberbach conjecture was considered so difficult to
prove that some eminent mathematicians believed it to be false.
15
8.
Discovery of the proof of Bieberbach conjecture
At last, the Bieberbach conjecture that has stumped several hundred
mathematicians for about 70 years has now been solved by Louis de Branges.
De Branges, 52 , comes from Purdue University.
He, in fact, has proved a
stronger conjecture that was proposed by Milin in 1971 .
settled all the seven conjectures as mentioned above.
that he worked on the conjecture for
He finally succeeded in May 1984 .
was not willing to read his
385
7
Thus he has
De Branges has claimed
years before he had any success.
But the American mathematical community
pages typed manuscript [59] .
in the Soviet Union that he finally got a hearing.
It was only
Milin and his colleagues
in Leningrad performed an extraordinary service to de Branges and mathematics,
by being a patient audience.
The proof has now been accepted by U.S.
mathematicians too, and the results have appeared in the Acta Mathematica
[18] .
In October 1984, FitzGerald and Pommerenke [31] circulated an
informal communication giving a shorter version of the de Branges theorem.
De Branges has proved the following result (Milin conjecture) which
implies the Bieberbach conjecture via the Robertson conjecture.
De Branges Theorem [18] .
Then, for
Suppose that
f e S
and write
n = 1 , 2 ,
n
I kin+l-k)
fc-l
Further, if
f e S
n + 1 - k
k
and if
f{z)
i
— £—
(l-ars)
then the strict inequality holds in
,
(10) .
16
CIO)
The proof of de Branges is very ingenous in many respects.
It makes use
of an important result of Askey and Gasper [8] on Jacobi polynomials
n
(namely,
£ P^,
(x ) 5 0) that was published in 1976 .
The other result
k- o
which was used is the Lebedev-Milin inequality (9) .
The proof of the Milin conjecture given by de Branges depends on a
continuous application of the Riemann mapping theorem which is due to
Lowner [68] .
Lowner used the method to prove the Bieberbach conjecture
for the third coefficient.
In this approach the problem is to propagate
information by means of a differential equation.
For this purpose,
information has to be coded in a convenient form and then carried from one
end of the interval to the other.
However, the classical theories do not
help because there is no fixed energy quadratic form which is preserved by
the propagation.
Thus de Branges had to develop some new techniques.
An
expository account of the new methods used by de Branges in proving this
theorem are to appear in [19] .
FitzGerald and Pommerenke [3l] have shown that the de Branges method
cannot work directly for the proof of the Bieberbach conjecture, and one has
to use the ingenious Lebedev-Milin inequality
(9)
which enables one to
prove first the Milin conjecture, leading to the solution of the Bieberbach
conjecture via Robertson's proceedure.
FitzGerald and Pommerenke [3l] have given a shorter version of the
de Branges proof.
The difference between these two proofs is purely technical.
De Branges deduces his theorem by first proving a more general result on
bounded univalent functions.
He uses the ordinary Lowner differential
equation which describes a contracting flow in the unit disk.
On the other
hand, FitzGerald and Pommerenke use the linear partial differential equation
of Lbwner which describes an expanding flow in the plane.
We close this section with a remark that a great deal of support work is
generally necessary for any scientific breakthrough.
More precisely,
mathematics progresses by an accumulation of insights - including, of course,
the major insights of Louis de Branges.
Above, we have witnessed that the
Milin conjecture was necessary for the proof of the Bieberbach conjecture.
But the Milin conjecture would not have been made without a few fundamental
17
results, including the Robertson conjecture, the Grunsky inequalities, and
Milin's earlier work showing
|a | 5 1.243n .
Finally, de Branges has
reduced his approach to the Bieberbach conjecture to an explicit question
concerning special functions, specifically the non-negativity of certain sums
related to Jacobi polynomials.
That there would be any connection between
univalent functions and these sums even now astonishes the mathematical
community.
These observations, indeed, speak of the interdisciplinary aspect
of the de Branges proof.
9.
What next after the Bieberbach conjecture?
The importance of the conjecture is mainly that it has proved so
difficult and so much useful mathematics was developed as researchers tried
to resolve it.
However, it is too early to predict whether de Branges'
method or his theorem will have any great significance for mathematics in
general.
We now discuss some of the related open problems as raised by earlier
researchers.
In view of the discovery of the proof of the Bieberbach
conjecture and de Branges1 method, one may now hope to solve some of these
challenging problems.
Goodman conjecture for multi valent functions
A function
f
is said to be a multivalent function of order
p-valent function) in
We let
^(p)
A
if it assumes no value more than
p
p
(or
times in
A .
denote the class of all functions of the form
/(z) = b^z + b2z2 + ... ,
that are analytic and
p-valent in
A .
Goodman
[38] in 1948 has made a conjecture analogous to the Bieberbach conjecture as
follows:
If
/(z) = biz + ... +
\b |
for every
For
I
<
n
+ •••
belongs to
F(p) ,
------------------- 2 k ( n + p ) l ------------------- | ^
then
(n )
k=l (p+fe)! (p-fc)! (n-p-1)! (n2-fc2)
n > p + 1 .
p = 1 = 2?1 ,
the Goodman conjecture yields the Bieberbach conjecture
which now holds for the class
few special subclasses of
S .
V(p) .
The conjecture is known to be true for a
Recently, Livingston [67] has shown that
18
(11)
is true for all
z e A ,
and
f(z)
find in
[39,40]
n
is
if
f{z)
has the form
f{z) =
p-valently close-to-convex in
^
A .
1 +b ^
+ ... ,
The reader will
some progress on the Goodman conjecture up to the year 1979 .
Coefficient conjecture for class E
Closely related to the class
g(z) = z + b Q + b^/z + ...
a simple pole at
class
E
z - 2z
was
S
is the class
°° with residue 1 .
|i>^| £ 2/[n+l) ,
/(rz+1) + ... .
some special cases.
E
of functions
analytic and univalent in
\z\ > 1
except for
The suggested conjecture for the
with equality for the function
This conjecture has been proved to be true [24] for
However, the general conjecture is false even for the
third coefficient.
Bazilevic [10] , in 1937, had shown that among all odd
functions
the sharp upper bound for
not
1/2
g e E ,
as asserted by the conjecture.
prove a coefficient conjecture for the class
the coefficient problem for the class
E
|Z?3 |
is
1/2 + e 6 ,
and
Thus the problem "Establish and
E"
is still open.
In fact,
is considerably more difficult
than the Bieberbach conjecture.
Extreme points of class
A point
r)
S
in a convex set
A
is called an extreme point of
is not an interior point of any line segment contained in
convex hull of a set
Let
E[S)
A
It is well known that
S
is a compact subset of a locally convex
U
theorem [23] ,
is contained in the closed convex hull of
Open problem.
E(S~)
of all analytic functions in
A .
By the Krein-Milman
S .
should provide tremendous information about
Determine all the extreme points of
We observe that the Bieberbach conjecture for
the conjecture is now known to be true for
S .
and its rotations are certainly extreme points but
points as well.
A .
S .
topological space
determination of
if it
is the smallest closed convex set that contains
denote the set of all extreme points of
S
A
A . The closed
Thus the
S .
S .
E(S)
is true because
Further, the Koebe function
S
has other extreme
In fact, it is known [17] that closed convex hull of the
Koebe function and its rotations contain
S*
but not all of
S .
Further,
the extreme point theory has been applied to a number of problems involving
19
special subclasses of
S
mappings to investigate
known to the author.
[7l] .
Hamilton [45] has now used quasiconformal
E(S) .
However, the results of Hamilton are not
But the author feels that quasiconformal mappings can
possibly be used to get
E(S) .
General coefficient problem
In most general form, the coefficient problem is to determine the region
Vn
of
°n~1 occupied by the points
The special problem of estimating
resolved by de Branges.
(a2,a3,...,an_1)
\a^\
But the
for all
V
is now being
general coefficient problem is still open
and is really a challenging problem for analysts.
about
for all . f z S .
f e S
A wealth of information
can be found in the book of Schaeffer and Spencer [97] .
Coefficient problem for class
Let
S^
analytic in
S^
be the class of homeomorphisms of the Riemann sphere which are
A
with normalization
A = {1 < |2 1 5 °°} .
^-quasiconformal in
k-quasiconformal
/(0) = 0 = /'(O) - 1
in
A
A function
f
and which are
is said to be
if relative to each standard rectangle in
A ,
/
is absolutely continuous on almost every horizontal and vertical line and
satisfies the dilatation condition
1
Since
£
141 + l%l
- 2------ ^
'
7-
1-quasiconformality is conformality,
2 / ( 1 -cz) , |c| £ 1 .
mappings
- -
-r
consists of the Mobius
On the other extreme, as
k ■* 00 the class
S, becomes dense in the familiar class S .
In 1976, Schiffer and Schober
k
[101] raised the following coeffient problem for S ^ :
Each
Find
f e
,
has the expansvon
r?
72
f(z) = z + 1 a^z ,
2 e A .
max Ia I .
f**k n
Since
f t
to find
max
fzsk
if and only if
e~taf{etaz) t
Re a
n
20
, a
real, it is equivalent
The problem of
[lOl] .
more
10.
max
Re a0
is completely solved by Schiffer and Schober
However the general problem, I believe, is still open.
recent open problems, a reader may refer to
For some
[107,108] .
Conclusion
In this brief survey, I have made an attempt to document the importanc
of various methods generated or furthered by the Bieberbach conjecture.
I
have been compelled to omit a number of related interesting problems - open
or solved.
However, I have tried to convey that the Bieberbach conjecture
has been an effective inspiration and testing ground for the developments i
geometric function theory in the last seventy years.
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University of Papua New Ginuea,
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