Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture

The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z)=c_0+c_{1}z+\mathrm{.}\mathrm{.}\mathrm{.}$ in the unit disk with $\mathrm{\mid }f(z)\mathrm{\mid }\le 1$, we have the sharp bound $\mathrm{\mid }{c}_n\mathrm{\mid }\le 2\mathrm{/}e$ for all $n \ge 1$,withequalityonlyforthefunction$f(z) = \exp [(z^n - 1)/(z^n + 1)]$ 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.