the Bieberbach conjecture
/ BEE-ber-bahkh /
One of the most famous problems in twentieth-century complex analysis grew from a single observation. The hero example, the Koebe function, has the suspiciously clean coefficients a_n = n. Bieberbach proved in 1916 that a_2 can never beat 2, and he wondered out loud whether the same ceiling held for every coefficient: could it be that no normalized univalent function ever has a coefficient larger than its index? That guess — simple to state, fiendishly hard to prove — became the Bieberbach conjecture.
Precisely, the conjecture asserts that for every f in the class S, with f(z) = z + a_2 z^2 + a_3 z^3 + ... , the coefficients obey |a_n| <= n for all n >= 2, with equality (for any n) only for rotations of the Koebe function. It is a statement about every coefficient at once, and that is what made it so resistant. Mathematicians chipped away for decades: |a_3| <= 3 (Loewner, 1923, using his new differential equation), |a_4| <= 4 (Garabedian and Schiffer, 1955), then |a_5| and |a_6|, plus broad asymptotic and average results (Littlewood, the Robertson and Milin conjectures) that hemmed it in without finishing it.
The conjecture stood open for 69 years and was finally proved in 1985 by Louis de Branges, so it is now de Branges's theorem; the proof routed through Milin's stronger conjecture and the Loewner differential equation. It is a model story of how a clean question drives the invention of deep machinery. The crucial honesty: the bound is for the class S — normalized, univalent, on the disk. Drop univalence and it fails instantly; for merely bounded holomorphic functions the coefficients can be far larger. The miracle is that one-to-one-ness alone forces the Koebe ceiling.
The conjecture is tight at every n: the Koebe function k(z) = z + 2 z^2 + 3 z^3 + 4 z^4 + ... has a_n = n for all n, so no class-S function can exceed it. The first hard case beyond Bieberbach's own |a_2| <= 2 was |a_3| <= 3, which Loewner proved in 1923 with the Loewner chain method.
Conjectured 1916, fully proved 1985 (de Branges); Koebe is the equality case for every coefficient.
The bound depends essentially on univalence — it is false for general bounded holomorphic functions. And it is no longer a conjecture: since 1985 it is a theorem (de Branges), so 'Bieberbach conjecture' now names a settled result.