Composition operators between Nevanlinna classes and Bergman spaces with weights

We investigate composition operators between spaces of analytic functions on the unit disk A in the complex plane. The spaces we consider are the weighted Nevanlinna class N-alpha, which consists of all analytic functions f on Delta such that f(Delta)log(+) \f (z)\ (1-\z\(2))(alpha) dx dy <infinity, and the corresponding weighted Bergman spaces A(alpha)(rho), -1 < a < infinity, 0 < p < infinity. Let X be any of the spaces A(alpha)(rho), and N-alpha and y any of the spaces A(beta)(q), N-beta, beta > -1, 0 < q < infinity.We characterize, in function theoretic terms, when the composition operator C-phi : f --> f o phi induced by an analytic function phi : Delta --> Delta defines an operator X --> Y which is continuous, respectively compact, respectively order bounded.