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

Step 3 of 8: Embed the problem in Loewner's theory of slit mappings

In plain words

In 1923, Karl Loewner realized that a univalent map onto the plane minus a growing slit can be built up continuously in time, starting from the identity map and evolving according to a simple differential equation driven by a single point $\kappa(t)$ moving around the unit circle.

This turns a static question about one function's coefficients into a dynamic question about how a whole family of functions evolves over time, opening the door to calculus-based tools that a purely algebraic approach could never use.

\frac{\partial f}{\partial t}(z,t) = z f'(z,t)\,\frac{1-\kappa(t)z}{1+\kappa(t)z}, \qquad |\kappa(t)| = 1

Detailed analysis

De Branges embedded the extremal problem in Karl Loewner's 1923 theory of slit mappings: every candidate extremal function arises as the limit of a Loewner chain $f(z,t)$ satisfying $\frac{\partial f}{\partial t}(z,t) = z f'(z,t)\,\frac{1-\kappa(t)z}{1+\kappa(t)z}$ with $|\kappa(t)| = 1$ , starting from $f(z,0) = f(z)$ and converging to the identity-like map $e^t z$ as $t \to \infty$ .

Milin's inequality is equivalent to a monotonicity statement for an auxiliary system of weight functions $\sigma_k(t)$ attached to this flow: if $\sigma_k(t)$ can be shown to be non-increasing in $t$ for every $k$ , then evaluating at the endpoints $t=0$ and $t \to \infty$ recovers exactly Milin's inequality.

Defining these weight functions precisely, and showing what their monotonicity reduces to algebraically, is the task of the next step.

Terms in this step

Loewner chain: A family of univalent functions $f(z,t)$ evolving continuously in a time parameter $t \ge 0$ according to Loewner's differential equation, starting at the function of interest and ending at a simple map, used to turn a static extremal problem into one about a differential equation.

Knowledge used in this step

Functions of a complex variable