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.
References
For n>6 the Zalcman conjecture remains an open problem.
In 1960, Zalcman proposed the conjecture $\vert a_n2 - a_{2n-1} \vert \leq (n-1)2 $ for $f \in \mathcal{S}$. This conjecture has its own interest, but the main point is that it provides the Bieberbach conjecture $ |a_n|\leq n$. Both conjectures attracted considerable attention and were extensively studied by many authors. The Bieberbach conjecture has now been proved for all $n$, whereas Zalcman's conjecture is proved only for $n\leq 6$ and for certain subclasses of univalent functions.
In 1999, Ma proposed a generalized Zalcman conjecture for $f\in \mathcal{S}$ that $$ \vert a_n a_m - a_{n+m-1} \vert \leq (n-1)(m-1) \quad \forall\; m,n\geq 2,$$ which is still an open problem, however he proved it for the classes $\mathcal{S}*$ and $\mathcal{S}\mathbb{R}$, where $\mathcal{S}\mathbb{R}$ contains univalent functions with real coefficients.