Consider $H^2(\mathbb D),$ the Hardy space over the unit disk. We know that if $\phi\in H^\infty(\mathbb D),$ the space of bounded holomorphic functions in the unit disk, then $\phi^n$ belongs to $H^2(\mathbb D)$ for each $n\geq1$ where $\phi^n(z)=(\phi(z))^n$.
Now we ask a converse question. Suppose $\phi$ is a function in $H^2(\mathbb D)$ such that $\phi^n$ is also in $H^2(\mathbb D)$ for each $n.$ Then does it imply that $\phi$ belongs to $H^\infty(\mathbb D)$.
Any help will be appreciated.