Worked solution: De Branges's proof of the Bieberbach conjecture via Milin's conjecture (1984)

Step 1 of 8: State the conjecture and the extremal Koebe function

In plain words

A function $f$ is univalent (one-to-one) on the unit disk if it never sends two different points to the same place, like a rubber sheet stretched and bent but never folded onto itself. Bieberbach's 1916 conjecture said that for such functions, normalized to start as $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , the size of each coefficient $a_n$ can never exceed $n$ .

The bound is tight: the Koebe function $k(z) = \frac{z}{(1-z)^2}$ , which stretches the disk onto the whole plane minus a slit along the negative real axis, achieves $a_n = n$ exactly for every $n$ , so no bound better than $n$ could ever hold.

|a_n| \le n \ \text{for univalent} \ f(z) = z + \sum_{n=2}^{\infty} a_n z^n \ \text{on} \ |z|<1, \quad \text{equality only for rotations of} \ k(z) = \frac{z}{(1-z)^2}

Detailed analysis

In 1916, Ludwig Bieberbach conjectured that every univalent (injective) holomorphic function $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ on the open unit disk $\{z \in \mathbb{C} : |z| < 1\}$ satisfies $|a_n| \le n$ for every $n \ge 2$ , with equality only for rotations of the Koebe function $k(z) = \frac{z}{(1-z)^2}$ , which maps the disk onto the entire plane minus a slit along the negative real axis from $-\infty$ to $-\tfrac{1}{4}$ .

Direct attacks on the $n$ -th coefficient succeeded only case by case: Bieberbach himself for $n=2$ (1916), Karl Loewner for $n=3$ (1923, inventing his differential equation for the purpose), Garabedian and Schiffer for $n=4$ (1955), and by 1972 the cases up to $n=6$ ; no method in sight could handle every $n$ simultaneously, and the conjecture stood open for 68 years.

The eventual proof, found by Louis de Branges in 1984, does not attack $|a_n| \le n$ directly for each $n$ ; instead it proves a stronger, uniform conjecture of Isaak Milin about a different set of coefficients, which forces all the Bieberbach bounds at once — the subject of the next step.

Terms in this step

univalent function: A holomorphic function $f$ on a domain (here the unit disk) that is injective: $f(z_1) = f(z_2)$ only when $z_1 = z_2$ , so it never maps two different points to the same image.
Koebe function: The function $k(z) = \frac{z}{(1-z)^2} = \sum_{n=1}^{\infty} n z^n$ , the unique (up to rotation) extremal example achieving equality $|a_n| = n$ in the Bieberbach conjecture for every $n$ .

Knowledge used in this step

Functions of a complex variable