De Branges's theorem
In
The statement concerns the Taylor coefficients of a univalent function, 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 and injective (univalent) with Taylor series of the form
Such functions are called schlicht. The theorem then states that
The
Schlicht functions
The normalizations
mean that
This can always be obtained by an affine transformation: 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.
A schlicht function is defined as an analytic function that is one-to-one and satisfies and . A family of schlicht functions are the
with a complex number of absolute value . 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 by Koepf (2007).
Bieberbach (1916) proved , and stated the conjecture that . Löwner (1917) and Nevanlinna (1921) independently proved the conjecture for starlike functions. Then Charles Loewner (Löwner (1923)) proved , using the
Littlewood (1925, theorem 20) 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. Paley and Littlewood (1932) 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 Fekete & Szegő (1933), who showed there is an odd schlicht function with , and that this is the maximum possible value of .
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 Garabedian & Schiffer (1955) proved , Ozawa (1969) and Pederson (1968) proved , and Pederson & Schiffer (1972) proved .
Hayman (1955) 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
Milin (1977) 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 de Branges (1987) proved for all .
De Branges's proof
The proof uses a type of
De Branges reduced the conjecture to some inequalities for Jacobi polynomials, and verified the first few by hand.
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 maximum value of
is achieved by the Koebe function .
A simplified version of the proof was published in 1985 by Carl FitzGerald and Christian Pommerenke (FitzGerald & Pommerenke (1985)), and an even shorter description by Jacob Korevaar (Korevaar (1986)).
See also
References
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- FitzGerald, Carl; Pommerenke, Christian (1985), "The de Branges theorem on univalent functions", Trans. Amer. Math. Soc., 290 (2): 683, JSTOR 2000306
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Further reading
- Liu, Xiaosong; Liu, Taishun; Xu, Qinghua (2015). "A proof of a weak version of the Bieberbach conjecture in several complex variables". Science China Mathematics. 58 (12): 2531–2540. S2CID 122080390.