A convergence question in $H^{p}$

Abstract:

Let $\phi \in {H^p}$ (unit disc), $0 < p < \infty$. and let ${\phi _r}(z) = \phi (rz),r < 1$ . If $\phi$ contains a nontrivial inner factor, it is known that $\phi /{\phi _r}$ is unbounded in ${H^p}$-norm. We prove that if $\phi$ is analytic on the closed disc and has no zeros on the open disc, then $\phi /{\phi _r} \to 1$ in ${H^p}$, as $r \to 1$. The same conclusion follows if $1/\phi \in {H^\infty }$. We construct an outer function $\phi$ which is continuous on the closed disc, analytic for $z \ne 1$, and such that $\phi /{\phi _r}$ is unbounded in every ${H^p}$.