  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.
