An-Najah Staff

Let N(Ω) denote the class of analytic functions fin a domain Ω, contained in the complex numbers C, such that log(1+| f |) has a harmonic majorant. The subclass N⁺ (Ω) of N(Ω) consists of all f such that log(1+| f |) has a quasi-bounded harmonic majorant. Let Ф be a non-constant analytic function from Ω into itself Define the composition operator C_ф , on N(Ω) by C_Фf=foФ , ∀ f € N(Ω). Then C_ф , maps N⁺ (Ω) into itself. Here we characterize the invertibility of C_Ф  when Ω 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_Ф, in N⁺(Ω). Moreover, we consider linear isometries on N⁺ (Ω) and their relation to C_Ф.