A complete description of the modulus of an outer function on the
real line is well known. Indeed, this characterization is considered
as one of the classical results of the theory of Hardy spaces.
However, a satisfactory characterization of the argument of an outer
function on the real line is not available yet. In this paper, we
define some classes of real functions which can serve as the
argument of an outer function. In particular, for any $0 < p \leq
\infty$, an increasing bi-Lipschitz function is the argument of an
outer function in $H^p(\mathbb{R})$.