  In this paper we investigate the following problem: when a bounded analytic
function $\phi$ on the unit disk $\mathbb{D}$, fixing 0, is such that $\{\phi^n
: n = 0, 1, 2, . . . \}$ is orthogonal in $\mathbb{D}$?, and consider the
problem of characterizing the univalent, full self-maps of $\mathbb{D}$ in
terms of the norm of the composition operator induced. The first problem is
analogous to a celebrated question asked by W. Rudin on the Hardy space setting
that was answered recently ([3] and [15]). The second problem is analogous to a
problem investigated by J. Shapiro in [14] about characterization of inner
functions in the setting of $H^2$.
