De Branges's Theorem - Schlicht Functions

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

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

A family of schlicht functions are the rotated Koebe functions

with α a complex number of absolute value 1. If f is a schlicht function and |an| = n for some n ≥ 2, then f 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 |an|≤n for all n, but it is not injective since f(−1/2 + z) = f(−1/2 − z).

Read more about this topic:  De Branges's Theorem

Famous quotes containing the word functions:

    Adolescents, for all their self-involvement, are emerging from the self-centeredness of childhood. Their perception of other people has more depth. They are better equipped at appreciating others’ reasons for action, or the basis of others’ emotions. But this maturity functions in a piecemeal fashion. They show more understanding of their friends, but not of their teachers.
    Terri Apter (20th century)