2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74839In this paper we investigate the following problem: when a bounded analytic function $ϕ$ on the unit disk $\mathbb{D}$, fixing 0, is such that $\{ϕ^n : n = 0, 1, 2, . . . \}$ 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 $H^2$.8 pages, 1 figure. See also http://webdelprofesor.ula.ve/nucleotachira/gchacon or http://webdelprofesor.ula.ve/humanidades/grchaconFunctional AnalysisComplex Variables47B33 (primary); 47B38, 47A16 (secondary)text