Zalcman conjecture for indices n > 6

Prove the Zalcman conjecture for indices n > 6: for each normalized univalent function f(z) = z + ∑_{k=2}^∞ a_k z^k in the class S on the unit disk, show that |a_n^2 − a_{2n−1}| ≤ (n − 1)^2, with equality for the Koebe function K(z) = z/(1 − z)^2 or its rotations.

Background

The Zalcman conjecture asserts that for f in S and n ≥ 2, |a_n^2 − a_{2n−1}| ≤ (n − 1)^2 with equality for the Koebe function or rotations. This conjecture implies the Bieberbach conjecture and is a central open problem in the theory of univalent functions.

The paper notes that the conjecture is settled for n = 2 (via the Bieberbach theorem), for n = 3 (Krushkal, 1995), and for n = 4, 5, 6 (Krushkal, 2010). Beyond these cases, the general statement remains unresolved.