On Composition Operators on N+(?)

Let N(?) denote the class of analytic functions fin a domain ?, contained in the complex numbers C, such that log(1+| f |) has a harmonic majorant. The subclass N+(?) of N(?) consists of all f such that log(1+| f |) has a quasi-bounded harmonic majorant. Let ? be a non-constant analytic function from ? into itself Define the composition operator C?, on N(?) by C?f=fo?, V f € N(?). Then C?, maps N+(?) into itself. Here we characterize the invertibility of C? when ? is finitely connected with boundary ? consisting of disjoint analytic simple closed curves and we give a necessary condition for the density of the range of C?, in N+(?). Moreover, we consider linear isometries on N+(?) and their relation to C?.