Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

TL;DR

This paper proves Krzyz's 1968 conjecture on the bounds of coefficients for nonvanishing holomorphic functions in the unit disk, using complex geometry and pluripotential theory, establishing the sharp bound of 2/e for all coefficients.

Contribution

The paper introduces a novel approach based on complex geometry and pluripotential theory to prove Krzyz's conjecture for all coefficients, surpassing previous partial results.

Findings

Proved the sharp bound |c_n| ≤ 2/e for all n ≥ 1.

Identified the extremal functions achieving equality.

Established the conjecture for the first time in full generality.

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.