Hyperbolic geodesics, Krzyz's conjecture and beyond

Abstract

In 1968, Krzyz conjectured that for non-vanishing holomorphic functions $f(z)=c_0+c_{1}z+\dots$ in the unit disk with $|f(z)|\le 1$, 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. This conjecture was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n\le 5$. We provide here two different proofs of this conjecture and its generalizations based on completely different ideas.

This is a corrected and expanded version of arXiv:0908.2587