Bieberbach conjecture

Solved, 1984Analysis

Statement

Let $f : \mathbb{D} \to \mathbb{C}$ be a holomorphic and injective (univalent / schlicht) function on the open unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$ , normalized so that $f(0) = 0$ and $f'(0) = 1$ , with Taylor expansion $f(z) = z + \sum_{n=2}^\infty a_n z^n$ . Then the coefficients satisfy $|a_n| \le n$ for every integer $n \ge 2$ , with equality if and only if $f$ is a rotation of the Koebe function $K_\theta(z) = \frac{z}{(1 - e^{i\theta} z)^2} = \sum_{n=1}^\infty n e^{i(n-1)\theta} z^n$ .

Solved by Louis de Branges in 1984 (published in Acta Mathematica in 1985). De Branges proved Isaak Milin's 1971 conjecture on the logarithmic coefficients $\gamma_k$ of $\log(f(z)/z) = 2\sum_{k=1}^\infty \gamma_k z^k$ —namely $\sum_{k=1}^n (n+1-k) k (|\gamma_k|^2 - 1/k^2) \le 0$ —which by Lebedev–Milin exponentiation implies M. S. Robertson's 1936 conjecture on odd univalent functions and hence the Bieberbach conjecture. Using Karl Loewner's 1923 differential equation for slit mappings, de Branges reduced Milin's inequality to the monotonicity of a system of weight functions $\sigma_k(t)$ whose derivatives turn out to be nonpositive by a 1976 positivity inequality for generalized hypergeometric functions ${}_3F_2$ (sums of Jacobi polynomials) proved by Richard Askey and George Gasper.

The theory developed around the Bieberbach conjecture catalyzed several pillars of 20th-century analysis: Loewner's 1923 differential equation was randomized by Oded Schramm in 2000 to create Schramm–Loewner evolution ( $\mathrm{SLE}_\kappa$ ), a central tool in two-dimensional statistical physics and conformal field theory. De Branges's theory of Hilbert spaces of entire functions also connects univalent function theory to spectral theory of canonical systems and the Riemann zeta function.

References

Ludwig Bieberbach (1916). Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln
Louis de Branges (1985). A proof of the Bieberbach conjecture · DOI:10.1007/BF02392821
Carl H. FitzGerald, Christian Pommerenke (1985). The de Branges theorem on univalent functions · DOI:10.1090/S0002-9947-1985-0800258-7

History of the problem

1916Ludwig Bieberbach proves $|a_2| \le 2$ using the area theorem (which yields the Koebe $1/4$ -theorem) and conjectures that $|a_n| \le n$ for all $n \ge 2$ .
1923Karl Loewner (Charles Löwner) introduces the Loewner differential equation for families of slit mappings and uses it to prove $|a_3| \le 3$ .
1955Paul Garabedian and Menahem Schiffer prove $|a_4| \le 4$ via variational methods, while Walter Hayman proves his asymptotic regularity theorem: $\lim_{n\to\infty} |a_n|/n$ exists and is at most $1$ for each univalent function.
1971Isaak Milin formulates the Milin conjecture on logarithmic coefficients of univalent functions, which implies Robertson's 1936 conjecture and the Bieberbach conjecture.
1984Louis de Branges proves the Milin, Robertson, and Bieberbach conjectures for all $n \ge 2$ , with verification by the Leningrad geometric function theory seminar and publication in Acta Mathematica in 1985.