De Branges's Theorem - History

History

Bieberbach (1916) proved |a₂| ≤ 2, and stated the conjecture that |a_n| ≤ n. Loewner (1917) and Nevanlinna (1921) independently proved the conjecture for starlike functions. Then Charles Loewner (Löwner (1923)) proved |a₃| ≤ 3, using the Löwner equation. His work was used by most later attempts, and is also applied in the theory of Schramm–Loewner evolution.

Littlewood (1925, theorem 20) proved that |a_n| ≤ en for all n, showing that the Bieberbach conjecture is true up to a factor of e = 2.718... Several authors later reduced the constant in the inequality below e.

If f(z) = z + ... is a schlicht function then φ(z) = f(z2)1/2 is an odd schlicht function. Paley and Littlewood (1932) showed that b_k ≤ 14 for all k. They conjectured that 14 can be replaced by 1 as a natural generalization of the Bieberbach conjecture. The Littlewood–Paley conjecture easily implies the Bieberbach conjecture using the Cauchy inequality, but it was soon disproved by Fekete & Szegö (1933), who showed there is an odd schlicht function with b₅ = 1/2 + exp(−2/3) = 1.013..., and that this is the maximum possible value of b₅. (Milin later showed that 14 can be replaced by 1.14., and Hayman showed that the numbers b_k have a limit less than 1 if φ is not a Koebe function, so Littlewood and Paley's conjecture is true for all but a finite number of coefficients of any function.) A weaker form of Littlewood and Paley's conjecture was found by Robertson (1936).

The Robertson conjecture states that if

is an odd schlicht function in the unit disk with b₁=1 then for all positive integers n,

Robertson observed that his conjecture is still strong enough to imply the Bieberbach conjecture, and proved it for n = 3. This conjecture introduced the key idea of bounding various quadratic functions of the coefficients rather than the coefficients themselves, which is equivalent to bounding norms of elements in certain Hilbert spaces of schlicht functions.

There were several proofs of the Bieberbach conjecture for certain higher values of n, in particular Garabedian & Schiffer (1955) proved |a₄| ≤ 4, Ozawa (1969) and Pederson (1968) proved |a₆| ≤ 6, and Pederson & Schiffer (1972) proved |a₅| ≤ 5.

Hayman (1955) proved that the limit of a_n/n exists, and has absolute value less than 1 unless f is a Koebe function. In particular this showed that for any f there can be at most a finite number of exceptions to the Bieberbach conjecture.

The Milin conjecture states that for each simple function on the unit disk, and for all positive integers n,

where the logarithmic coefficients γ_n of f are given by

Milin (1977) showed using the Lebedev–Milin inequality that the Milin conjecture (later proved by de Branges) implies the Robertson conjecture and therefore the Bieberbach conjecture.

Finally De Branges (1985) proved |a_n| ≤ n for all n.