  The Krzy\.z conjecture concerns the largest values of the Taylor coefficients
of a non-vanishing analytic function bounded by one in modulus in the unit
disk. It has been open since 1968 even though information on the structure of
extremal functions is available. The purpose of this paper is to collect
various conditions that the coefficients of an extremal function (and various
other quantities associated with it) should satisfy if the conjecture is true
and to show that each one of these properties is equivalent to the conjecture
itself. This may provide several possible starting points for future attempts
at solving the problem.
