A computer-assisted proof of Krzyż's conjecture for the sixth coefficient

Abstract

We prove Krzyż's conjecture for the sixth coefficient. Every holomorphic function $f(z)=\sum a_kz^k$ from the unit disk to the punctured disk satisfies $|a_6| \le 2/e$. Equality holds precisely for rotations of $\exp(-(1-z^6)/(1+z^6))$. Every extremal has a Herglotz representation with at most six atoms. By the reduction of Agler and McCarthy and the entropy theorem of Lei and Zhang, it suffices to exclude extremals with fewer than six atoms. We do this by a rigorous computer-assisted branch-and-bound, leaving the equality case entirely to the analytic argument. The main ingredients are explicit first-order conditions for extremals with fewer than the maximal number of atoms, which reduce to Chebyshev relations when one atom is missing, and a bound for the total mass obtained by convexity of the Poisson integral. The same method independently establishes the full-degree property for the known cases $n=3,4,5$.