De Branges' theorem

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In complex analysis, a branch of mathematics, de Branges' theorem, named after Louis de Branges, formerly 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.\,

The normalizations a0=0 and a1=1 mean that f(0) = 0 and f '(0)=1; this can always be assured by 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.

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.

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. The theorem was proved in generality in 1984 by de Branges, with a proof that was subsequently much shortened by others.

De Branges's 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.
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