Hyperbolic geodesics, Krzyz's conjecture and beyond

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.