Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

Abstract

The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z) = c_0 + c_1 z + ...$ in the unit disk with $|f(z)| \le 1$, we have the sharp bound $|c_n| \le 2/e$ for all $n \ge 1$, with equality only for the function $f(z) = \exp [(z^n - 1)/(z^n + 1)]$ and its rotations. The problem was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \le 5$. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains in complex Banach spaces.