de Branges's theorem
/ duh BRAHNZH /
Some theorems are famous for what they say; this one is also famous for ending a 69-year wait. De Branges's theorem is the resolution, in 1985, of the Bieberbach conjecture: the long-suspected ceiling |a_n| <= n on the coefficients of normalized univalent functions is true after all. It closed one of the marquee open problems of complex analysis, and it did so by a route few had expected.
The statement is exactly the old conjecture: for every f in the class S, f(z) = z + a_2 z^2 + ... , one has |a_n| <= n for all n, with equality only for rotations of the Koebe function. Louis de Branges proved it not by attacking the coefficients head-on but by proving a logically stronger statement, the Milin conjecture, about the logarithmic coefficients of f, which by inequalities of Lebedev and Milin implies the Bieberbach bound. The engine was the Loewner differential equation: he embedded f in a Loewner chain, derived a system of functions evolving along it, and showed a cleverly chosen combination was monotone, the monotonicity reducing to the positivity of certain special-function sums (a positivity Askey and Gasper had earlier established for Jacobi polynomials).
It is a landmark in two senses: the conjecture itself was a touchstone the whole field had been measured against, and the proof fused geometric function theory with special-function inequalities in a way that surprised everyone. An honest footnote on history: de Branges's first manuscript was long and at first met with scepticism; the proof was checked, simplified, and confirmed by a seminar in Leningrad, and the streamlined version is now standard. The result is secure. It also illustrates a recurring lesson: proving a stronger, better-structured statement can be easier than the original.
Before 1985 one could only assert |a_n| <= n for small n (n <= 6) and weaker uniform bounds like |a_n| <= e n (Littlewood). De Branges's theorem upgraded this to the full sharp |a_n| <= n for ALL n at once, e.g. it guarantees |a_100| <= 100 for any normalized univalent f, with the Koebe function the unique extremal.
De Branges (1985) turned the Bieberbach conjecture into a theorem via the Milin conjecture and Loewner chains.
The proof goes through the logarithmic coefficients (Milin), not the a_n directly, and leans on a positivity result for Jacobi polynomials — a striking case of special-function analysis settling a geometry problem.