Bieberbach conjecture

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In complex analysis, de Branges' theorem, named after Louis de Branges, usually called the Bieberbach conjecture, after Ludwig Bieberbach, states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane.

The statement concerns the Taylor coefficients an of such a function, normalized as is always possible so that a0 = 0 and a1 = 1. That is, we consider a holomorphic function of the form

f(z)=z+\sum_{n\geq 2} a_n z^n

which is defined and injective on the open unit disk (such functions are also called schlicht functions). The theorem then states that

\left| a_n \right| \leq n \quad \mbox{for all }n\geq 2.\,

Contents

[edit] Schlicht functions

The normalizations

a0 = 0 and a1 = 1

mean that

f(0) = 0 and f '(0) = 1;

this can always be assured by a linear fractional transformation: starting with an arbitrary injective holomorphic function g defined on the open unit disk and setting

f(z)=\frac{g(z)-g(0)}{g'(0)}.\,

Such functions g are of considerable interest because they appear in the fundamental Riemann mapping theorem.

A particularly important family of schlicht functions are the rotated Köbe functions

f_\alpha(z)=\frac{z}{(1-\alpha z)^2}=\sum_{n=1}^\infty n\alpha^{n-1} z^n

with α being a complex number of absolute value 1. Indeed, one can show that if f is a schlicht function and |an| = n for some n ≥ 2, then f is a rotated Köbe function.

[edit] Condition not sufficient

The condition of de Branges' theorem is not sufficient, as the function

f(z)=z+z^2\;

shows: it is holomorphic on the unit disc and satisfies |an|≤n for all n, but it is not injective since f '(−1/2) = 0.

[edit] Special cases

The case n = 1 of De Branges' theorem is essentially the Schwarz lemma, which was known in the nineteenth century and is a consequence of the maximum modulus principle, applied to f(z)/z.

The conjecture was stated in 1916 by Bieberbach, after having proved the case n = 2. Subsequently the cases n = 3, 4, 5, 6 were settled, each requiring a different approach.

[edit] Proof

The theorem was proved in generality in 1984 by de Branges, with a proof that was subsequently much shortened by others. De Branges himself is of the opinion that the simplification came at the cost of substance[1].

His proof use a type of Hilbert spaces of entire functions. The study of these spaces grew into a sub-field of complex analysis and the spaces come to be called de Branges spaces and the functions de Branges functions.

[edit] Further reading

  • J. Korevaar (1986). Ludwig Bieberbach's Conjecture and Its Proof by Louis de Branges. American Mathematical Monthly, Vol. 93, No. 7 (Aug. - Sep., 1986), pp. 505-514
  • John B. Conway (1995). Functions of One Complex Variable II. Springer-Verlag, 1995.