Functions of a complex variable Geometric function theory Theorem

Bieberbach conjecture

Builds on Univalent function Koebe function Loewner differential equation

Statement

Let $f$ be a univalent function on the unit disc $\mathbb{D}$ with $f(0) = 0$ and $f'(0) = 1$ , and write $f(z) = z + \sum_{n=2}^{\infty} a_n z^n, \qquad \abs{z} < 1.$ Then $\abs{a_n} \le n$ for every $n \ge 2$ . If $\abs{a_n} = n$ for some $n \ge 2$ , then $f(z) = e^{-i\theta} k(e^{i\theta} z)$ for some $\theta \in \R$ , where $k(z) = z/(1-z)^2$ is the Koebe function.

Remarks

Bieberbach proved the case $n = 2$ and stated the general bound as a conjecture. Loewner proved the case $n = 3$ with his differential equation. Louis de Branges proved the full statement, which is therefore also called de Branges's theorem. The conjecture was known to follow from the Robertson conjecture on odd univalent functions, which in turn follows from the Milin conjecture on the logarithmic coefficients of $f$ . De Branges proved the Milin conjecture, using the Loewner equation and an inequality on sums of Jacobi polynomials that had been proved by Askey and Gasper. The bound is sharp for every $n$ at once, because the Koebe function has $a_n = n$ .

Going deeper

For a student

The case $n = 2$ , Bieberbach's inequality $\abs{a_2} \le 2$ , is a short consequence of the Grönwall area theorem and implies the Koebe quarter theorem and the distortion theorems. For larger $n$ the problem resisted for a long time, and partial results such as $\abs{a_n} < e\,n$ and the proofs for small $n$ produced many methods of univalent function theory. The final proof goes through the stronger Milin inequality and the Loewner equation.

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Tag	`ZMRFSYVWQT`
URI	`m2://ZMRFSYVWQT`
Kind	statement, theorem
Writing	W0 stub
MSC	30C50, 30C55
Status	This copy of the site has no build data. Run `m2 status ZMRFSYVWQT`.
About	Univalent function
Uses	Univalent function, Koebe function, Loewner differential equation
Same as	Wikidata Q857089 Wikipedia De Branges's theorem
Changed	2 November 2026

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