Hyperbolic geodesics, Krzyz's conjecture and beyond

TL;DR

This paper proves Krzyz's conjecture on bounds for coefficients of non-vanishing holomorphic functions in the unit disk, using two distinct methods, and extends the results to broader contexts.

Contribution

It offers two novel proofs of Krzyz's conjecture and generalizes the bounds to wider classes of functions, advancing understanding in complex analysis.

Findings

Proved Krzyz's conjecture for all n

Established sharp coefficient bounds for non-vanishing holomorphic functions

Extended bounds to generalized function classes

Abstract

In 1968, Krzyz conjectured that for non-vanishing holomorphic functions $f(z) = c_0 + c_1 z + \dots$ in the unit disk with $|f(z)| \leq 1$, we have the sharp bound $|c_n| \leq 2/e$ for all $n \geq 1$, with equality only for the function $f(z) = exp [(z^n - 1)/(z^n + 1)]$ and its rotations. This conjecture was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \leq 5$. We provide here two different proofs of this conjecture and its generalizations based on completely different ideas.