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

Step 2 of 8: Reducing the coefficient bound to Milin's inequality on logarithmic coefficients

In plain words

Instead of controlling infinitely many coefficients $a_n$ one at a time, Isaak Milin proposed controlling a different, related sequence of numbers — the logarithmic coefficients $\gamma_k$ , obtained by taking a logarithm of $f(z)/z$ first — and showed that a single weighted inequality on all the $\gamma_k$ at once forces every Bieberbach bound simultaneously.

Taking a logarithm turns a delicate, individual coefficient problem into a more forgiving 'averaged' statement, the same way it can be easier to control the total brightness of a light source than the exact shape of every individual ray.

|a_n| \le n \quad\text{follows from}\quad \sum_{k=1}^{n} (n+1-k)\,k\left(|\gamma_k|^2 - \frac{1}{k^2}\right) \le 0

Detailed analysis

Isaak Milin conjectured in 1971 that the logarithmic coefficients $\gamma_k$ defined by $\log(f(z)/z) = 2\sum_{k=1}^\infty \gamma_k z^k$ for a univalent function $f(z) = z + \sum_{n=2}^\infty a_n z^n$ satisfy $\sum_{k=1}^{n} (n+1-k)\,k\left(|\gamma_k|^2 - \frac{1}{k^2}\right) \le 0$ . By the Lebedev–Milin exponentiation inequality, this bound implies M. S. Robertson's 1936 conjecture for odd univalent functions, which in turn implies the Bieberbach bound $|a_n| \le n$ for every univalent $f$ .

Direct approaches to the $n$ -th coefficient bound had succeeded only for small $n$ ; Milin's reformulation turns the problem into a single monotonicity-type statement that can be attacked uniformly for all $n$ at once, provided one can control the logarithmic coefficients $\gamma_k$ through some dynamic process rather than a static algebraic identity.

That dynamic process is supplied by Karl Loewner's 1923 theory of slit mappings, the subject of the next step.

Terms in this step

logarithmic coefficients $\gamma_k$: The coefficients $\gamma_k$ in the power series $\log(f(z)/z) = 2\sum_{k=1}^\infty \gamma_k z^k$ of a univalent function $f$ ; a different bookkeeping of the same function that turns out to be more tractable than the coefficients $a_n$ directly.

Knowledge used in this step

Functions of a complex variable