Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space A(alpha)(psi) coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function psi satisfies the so-called Delta(2)-condition. In addition we prove that this condition characterizes those A(alpha)(psi) on which every composition operator is bounded or order bounded into the Orlicz space L-alpha(psi) This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when psi satisfies the Delta(2)-condition, a composition operator is compact on A(alpha)(psi) if and only if it is order bounded into the so-called Morse-Transue space M-alpha(psi). Our results stand in the unit ball of C-N.