In
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a
necessary condition
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
on a
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
in order for it to map the
open unit disk
In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1:
:D_1(P) = \.\,
The closed unit disk around ''P'' is the set of points whose di ...
of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
ly to the complex plane. It was posed by and finally proven by .
The statement concerns the
Taylor coefficient
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
s
of a
univalent function
In mathematics, in the branch of complex analysis, a holomorphic function on an open subset of the complex plane is called univalent if it is injective.
Examples
The function f \colon z \mapsto 2z + z^2 is univalent in the open unit disc, ...
, i.e. a one-to-one holomorphic function that maps the unit disk into the complex plane, normalized as is always possible so that
and
. That is, we consider a function defined on the open unit disk which is
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivativ ...
and injective (''
univalent'') with Taylor series of the form
:
Such functions are called ''schlicht''. The theorem then states that
:
The
Koebe function In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:
Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk
\mathbf onto a subset of the complex plane con ...
(see below) is a function in which
for all
, and it is schlicht, so we cannot find a stricter limit on the absolute value of the
th coefficient.
Schlicht functions
The normalizations
:
mean that
:
This can always be obtained by an
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
: starting with an arbitrary injective holomorphic function
defined on the open unit disk and setting
:
Such functions
are of interest because they appear in the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected space, simply connected open set, open subset of the complex plane, complex number plane C which is not all of C, then there exists a biholomorphy ...
.
A schlicht function is defined as an analytic function
that is one-to-one and satisfies
and
. A family of schlicht functions are the
rotated Koebe function In complex analysis, a branch of mathematics, the Koebe 1/4 theorem states the following:
Koebe Quarter Theorem. The image of an injective analytic function f:\mathbf\to\mathbb from the unit disk
\mathbf onto a subset of the complex plane con ...
s
:
with
a complex number of
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
. If
is a schlicht function and
for some
, then
is a rotated Koebe function.
The condition of de Branges' theorem is not sufficient to show the function is schlicht, as the function
:
shows: it is holomorphic on the unit disc and satisfies
for all
, but it is not injective since
.
History
A survey of the history is given b
Koepf (2007)
proved
, and stated the conjecture that
. and independently proved the conjecture for
starlike functions.
Then
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sig ...
() proved
, using the
Löwner equation. His work was used by most later attempts, and is also applied in the theory of
Schramm–Loewner evolution
In probability theory, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE''κ''), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensiona ...
.
proved that
for all
, showing that the Bieberbach conjecture is true up to a factor of
Several authors later reduced the constant in the inequality below
.
If
is a schlicht function then
is an odd schlicht function.
showed that its Taylor coefficients satisfy
for all
. They conjectured that
can be replaced by
as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by , who showed there is an odd schlicht function with
, and that this is the maximum possible value of
.
Isaak Milin Isaak Moiseevich Milin, (Исаак Моисеевич Милин); * February 16, 1919, Oster, Ukrainian Soviet Socialist Republic – † November 17, 1992 Saint-Petersburg (former Leningrad), Russian Federation) was a prominent Soviet/Russian mat ...
later showed that
can be replaced by
, and Hayman showed that the numbers
have a limit less than
if
is not a Koebe function (for which the
are all
). So the limit is always less than or equal to
, meaning that Littlewood and Paley's conjecture is true for all but a finite number of coefficients. A weaker form of Littlewood and Paley's conjecture was found by .
The Robertson conjecture states that if
:
is an odd schlicht function in the unit disk with
then for all positive integers
,
:
Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for
. This conjecture introduced the key idea of bounding various quadratic functions of the coefficients rather than the coefficients themselves, which is equivalent to bounding norms of elements in certain Hilbert spaces of schlicht functions.
There were several proofs of the Bieberbach conjecture for certain higher values of
, in particular proved
, and proved
, and proved
.
proved that the limit of
exists, and has absolute value less than
unless
is a Koebe function. In particular this showed that for any
there can be at most a finite number of exceptions to the Bieberbach conjecture.
The Milin conjecture states that for each schlicht function on the unit disk, and for all positive integers
,
:
where the logarithmic coefficients
of
are given by
:
showed using the
Lebedev–Milin inequality that the Milin conjecture (later proved by de Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture.
Finally proved
for all
.
De Branges's proof
The proof uses a type of
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s of
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s. The study of these spaces grew into a sub-field of complex analysis and the spaces have come to be called
de Branges space In mathematics, a de Branges space (sometimes written De Branges space) is a concept in functional analysis and is constructed from a de Branges function.
The concept is named after Louis de Branges who proved numerous results regarding these spac ...
s. De Branges proved the stronger Milin conjecture on logarithmic coefficients. This was already known to imply the Robertson conjecture about odd univalent functions, which in turn was known to imply the Bieberbach conjecture about schlicht functions . His proof uses the
Loewner equation In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (c ...
, the
Askey–Gasper inequality In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by and used in the proof of the Bieberbach conjecture.
Statement
It states that if \beta\geq 0, \alpha+\beta\geq -2, and -1\leq x\leq 1 then
:\sum_^n \fr ...
about
Jacobi polynomial
In mathematics, Jacobi polynomials (occasionally called hypergeometric polynomials) P_n^(x)
are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight
(1-x)^\alpha(1+x)^\beta on the interval 1,1/math>. The ...
s, and the
Lebedev–Milin inequality on exponentiated power series.
De Branges reduced the conjecture to some inequalities for Jacobi polynomials, and verified the first few by hand.
Walter Gautschi
Walter Gautschi (born December 11, 1927) is a Swiss- American mathematician, known for his contributions to numerical analysis. He has authored over 200 papers in his
area and published four books.
Born in Basel, he has a Ph.D. in mathematics fr ...
verified more of these inequalities by computer for de Branges (proving the Bieberbach conjecture for the first 30 or so coefficients) and then asked
Richard Askey
Richard Allen Askey (4 June 1933 – 9 October 2019) was an American mathematician, known for his expertise in the area of special functions. The Askey–Wilson polynomials (introduced by him in 1984 together with James A. Wilson) are on the t ...
whether he knew of any similar inequalities. Askey pointed out that had proved the necessary inequalities eight years before, which allowed de Branges to complete his proof. The first version was very long and had some minor mistakes which caused some skepticism about it, but these were corrected with the help of members of the Leningrad seminar on Geometric Function Theory (
Leningrad Department of Steklov Mathematical Institute) when de Branges visited in 1984.
De Branges proved the following result, which for
implies the Milin conjecture (and therefore the Bieberbach conjecture).
Suppose that
and
are real numbers for positive integers
with limit
and such that
:
is non-negative, non-increasing, and has limit
. Then for all Riemann mapping functions
univalent in the unit disk with
:
the maximinum value of
:
is achieved by the Koebe function
.
A simplified version of the proof was published in 1985 by
Carl FitzGerald
Carl Fitzgerald is a professional Canadian Football fullback who is currently a free agent. He was drafted by the Winnipeg Blue Bombersin the third round of the 2013 CFL Draft.
At the conclusion of the 2015 CFL season, Fitzgerald was re-signed ...
and
Christian Pommerenke
Christian Pommerenke (born 17 December 1933 in Copenhagen) is a mathematician known for his work in complex analysis.
He studied at the University of Göttingen (1954–58), achieving diploma in mathematics (1957), Ph.D. (1959) on the dissertati ...
(), and an even shorter description by
Jacob Korevaar
Jacob "Jaap" Korevaar (born 25 January 1923) is a Dutch mathematician. He was part of the faculty of the University of California San Diego and University of Wisconsin–Madison, as well as the University of Amsterdam (Korteweg-de Vries Institute ...
().
See also
*
Grunsky matrix
In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pai ...
*
Fekete–Szegő inequality In mathematics, the Fekete–Szegő inequality is an inequality for the coefficients of univalent analytic functions found by , related to the Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theo ...
*
Schwarz lemma
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...
References
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* Koepf, Wolfram (2007),
Bieberbach’s Conjecture, the de Branges and Weinstein Functions and the Askey-Gasper Inequality'
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* (Translation of the 1971 Russian edition)
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Further reading
*{{cite journal , doi=10.1007/s11425-015-5016-2, title=A proof of a weak version of the Bieberbach conjecture in several complex variables , year=2015 , last1=Liu , first1=Xiaosong , last2=Liu , first2=Taishun , last3=Xu , first3=Qinghua , journal=Science China Mathematics , volume=58 , issue=12 , pages=2531–2540 , s2cid=122080390
Theorems in complex analysis
Conjectures
Conjectures that have been proved