The Bieberbach Conjecture and de Branges

From |a_2| <= 2 to a conjecture for every coefficient

Recall where the last two guides left us. A function f in the class S is a univalent (one-to-one) holomorphic map of the unit disk, normalized so that f(0) = 0 and f'(0) = 1, hence with a power series f(z) = z + a_2 z^2 + a_3 z^3 + ... . The very first non-trivial fact about those coefficients came out of guide 2's area theorem: a clean, almost magical argument squeezes the second coefficient down to |a_2| <= 2, the Bieberbach bound. And the bound is sharp — the Koebe function k(z) = z/(1 - z)^2 = z + 2 z^2 + 3 z^3 + ... hits it dead on, with a_2 = 2 exactly.

Now look at that Koebe series again: its coefficients are 1, 2, 3, 4, ... — the n-th coefficient is exactly n. Bieberbach, in the same 1916 paper that proved |a_2| <= 2, noticed this and dared to guess that the Koebe function is the worst case for every coefficient at once. That is the Bieberbach conjecture: for every f in S and every n, |a_n| <= n. The n = 2 case he had in hand. Everything beyond it — n = 3, 4, 5, and the infinite tail — he left as a conjecture, and a famously stubborn one.

Why it was so hard — and the slow march up the coefficients

Here is the difficulty in one sentence: the area theorem trick that pins |a_2| <= 2 does not iterate. It exploits a special relationship between the disk's outside and inside that is essentially a one-coefficient miracle; by the time you reach a_3 you are already tangled in cross-terms the simple argument cannot reach. Each new coefficient seemed to need a brand-new idea, and the ideas got harder fast. So the field advanced one painstaking rung at a time.

The honest timeline is humbling. Loewner proved n = 3 in 1923 — and to do it he invented the Loewner differential equation, a tool that would much later turn out to be the master key. Garabedian and Schiffer got n = 4 in 1955; Pederson and (independently) Ozawa reached n = 6 in 1968; Pederson and Schiffer settled n = 5 in 1972. After more than half a century of heroic effort, the conjecture was proved only for n up to 6. Worse, the methods were so specific that pushing to n = 7 looked hopeless, never mind the whole infinite sequence.

Meanwhile a parallel front attacked all n at once but only crudely. Littlewood showed in 1925 that |a_n| <= e n for every n — the right shape, n, but off by a factor of e (about 2.718). Later work shaved that constant down, and a related bound by Hayman showed the limit of |a_n|/n exists for each f and is at most 1 with equality only for Koebe. So the asymptotic truth pointed straight at the conjecture; the gap was always the finite, sharp, every-n statement.

de Branges's idea: prove something stronger

In 1985 Louis de Branges did something that, in hindsight, is a recurring lesson of mathematics: he stopped attacking |a_n| <= n directly and instead proved a logically stronger statement that turned out to be better-structured and, paradoxically, easier to handle. The detour ran through the logarithmic coefficients of f. Since f(z)/z is holomorphic and non-vanishing near 0, you can take a branch of its logarithm and write log(f(z)/z) = sum 2 gamma_n z^n; the numbers gamma_n are the logarithmic coefficients, and they encode f more smoothly than the raw a_n do.

The chain of implications is the heart of the story. Milin had conjectured a precise inequality on these logarithmic coefficients (the Milin conjecture). Through a set of inequalities due to Lebedev and Milin — algebraic identities relating the coefficients of e^g to those of g — the Milin conjecture implies an intermediate statement of Robertson about odd univalent functions, and Robertson's statement in turn implies the original Bieberbach bound |a_n| <= n. So a tower had been built: prove Milin, and Bieberbach falls out the bottom for free, for all n simultaneously.

Milin conjecture  (on logarithmic coefficients gamma_n)
        |
        |  via Lebedev-Milin inequalities (e^g coefficients vs g coefficients)
        v
Robertson conjecture  (on odd univalent functions)
        |
        v
Bieberbach conjecture     |a_n| <= n   for all n at once

The Loewner chain and a positivity nobody expected

How do you prove the topmost inequality? Here Loewner's old tool returns as the engine. The idea of a Loewner chain is to deform the given f continuously, growing the image region from a tiny disk out to the full image as a time parameter t runs from 0 to infinity, with each stage f_t still univalent. Differentiating along this evolution turns the static inequality into a question about how a certain quantity changes in time: you build a cleverly weighted combination of the logarithmic coefficients of f_t and ask whether it is monotone — always moving the right way as t increases.

When de Branges did the differentiation, the monotonicity he needed boiled down — astonishingly — to the positivity of a particular sum of special functions. And that exact positivity had already been proved, years earlier and for entirely unrelated reasons, by Askey and Gasper in their work on Jacobi polynomials. The two worlds, geometric function theory and classical special-function inequalities, had no obvious business meeting; that they fit together precisely was the surprise that made the proof go. Once the Askey-Gasper inequality supplies the sign, the weighted combination is monotone, Milin's conjecture follows, and the whole tower discharges down to |a_n| <= n.

What it means, and where it points

With de Branges's theorem the conjecture became a theorem: every f in S obeys |a_n| <= n for all n, and equality at any single n already forces f to be a rotation of the Koebe function. Concretely it guarantees, for instance, that |a_100| <= 100 for every normalized univalent map of the disk — a statement nobody could prove for n that large before 1985. The Koebe function, which extremized the area theorem, the one-quarter theorem, and the growth and distortion bounds of guides 2 and 3, turns out to be the universal extremal here too: the single function against which the whole class S is measured.

1. Recognize the prize: the sharp, every-coefficient bound |a_n| <= n, with the Koebe function as the unique extremal up to rotation.
2. Trade the coefficients a_n for the smoother logarithmic coefficients gamma_n, where log(f(z)/z) = sum 2 gamma_n z^n.
3. State the stronger Milin conjecture on the gamma_n; via the Lebedev-Milin inequalities it implies Robertson, which implies Bieberbach.
4. Embed f in a Loewner chain, differentiate, and reduce the needed monotonicity to a special-function positivity — supplied ready-made by the Askey-Gasper inequality.

Where does this point next? Two ways. First, the same Loewner-evolution idea — growing a region one boundary point at a time and watching a quantity evolve — was reborn decades later as the Schramm-Loewner evolution that revolutionized probability and conformal field theory, so the engine here outlived its first job. Second, within this rung the natural follow-up is structural: which subclasses of S are tame enough to handle by hand? That is exactly the starlike and convex functions and the Schwarzian story of guide 5, where univalence is detected by a sign or a derivative rather than chased through 69 years of coefficients.