Bieberbach conjecture

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.