Compact composition operators on $H^2$ and Hardy-Orlicz spaces

We compare the compactness of composition operators on $H^2$ and on Orlicz-Hardy spaces $H^\Psi$. We show in particular that exists an Orlicz function $\Psi$ such that $H^{3+\eps} \subseteq H^\Psi \subseteq H^3$ for every $\eps >0$, and a composition operator $C_\phi$ which is compact on $H^3$ and on $H^{3+\eps}$, but not compact on $H^\Psi$.