On the decay of singular inner functions

It is known that, if $S(z)$ is a non-constant, singular inner function defined on the unit disk, then ${\min }_{\mathrm{\mid }z\mathrm{\mid }\le r}\mathrm{\mid }S(z)\mathrm{\mid }\to 0$ as $r\to 1^{-}$. We show that the convergence may be arbitrarily slow.