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 |a_n| = 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 |a_n|≤n for all n, but it is not injective since f(−1/2 + z) = f(−1/2 − z).