Norm Inequalities for Composition Operators on Hardy and Weighted Bergman Spaces

Any analytic self-map of the open unit disk induces a bounded composition operator on the Hardy space H 2 and on the standard weighted Bergman spaces A 2 β. For a particular self-map, it is reasonable to wonder whether there is any meaningful relationship between the norms of the corresponding operators acting on each of these spaces. In this paper, we demonstrate an inequality which, at least to a certain degree, provides an answer to this question.