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 |a_n| ≤ 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 a_n are real.

(a) Let z = re^iθ with 0 iθ), then

a_n rⁿ = 2/Π 0 ∫₀^θ 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 |a_nrⁿ| ≤ nr, and let r → 1 to conclude that |a_n| ≤ n.

(f) Check that the function f(z) = z/(1 - z) ² satisfies all the hypotheses and that |a_n| = n for all n.