Subordination and $H^ p$ functions

Abstract:

Let $\phi$ and $W$ be inner functions with $\phi (0) = W(0) = 0$. It is shown that if $F$ is an exposed point of the unit ball of ${H^1}$ and \[ F(W({e^{it}}))/F(\phi ({e^{it}})) > 0\] almost everywhere, then $F \circ W = F \circ \phi$. If $f = zF$ such that $F$ and $1/F$ are in ${H^r}$ and ${H^s}$, respectively, where $1/r + 1/s \leq 2$ and $\phi$ is a finite Blaschke product, then a necessary and sufficient condition is provided in order for $f(W({e^{it}}))/f(\phi ({e^{it}}))$ to be positive almost everywhere.