Let f be an injective holomorphic function in the unit disc, with f(0) = 0 and f' (0) = 1. If we write f(z) = z + a2z2 + a3z3 ··· ,than shows that |a2| ≤ 2. Bieberbach conjectured that in fact |an| ≤ n for all n ≥ 2; this was proved by deBranges. This problem outlines an argument to prove the conjecture under the additional assumption that the coefficients an are real.
(a) Let z = reiθ with 0 iθ), then
an rn = 2/Π 0 ∫0θ v(r,θ ) sin nθ dθ.
(b) Show that for 0 ≤ θ ≤ π and n = 1, 2,... we have | sin nθ| ≤ n sin θ.
(c) Use the fact that an ∈ R to show that f(D) is symmetric with respect to the real axis, and use this fact to show that f maps the upper half-disc into either the upper or lower part of f(D).
(d) Show that for r small,
v(r,θ) = r sin θ [1+O(r)],
and use the previous part to conclude that v(r, θ) sin θ ≥ 0 for all 0
(e) Prove that |anrn| ≤ nr, and let r → 1 to conclude that |an| ≤ n.
(f) Check that the function f(z) = z/(1 - z) 2 satisfies all the hypotheses and that |an| = n for all n.