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We characterize all isometries among the composition operators acting on the Bloch space in terms of the hyperbolic derivative and cluster set of the symbol, and display a class of nontrivial examples.

Abstract. We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov, and other authors.

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We study the action of fractional differentiation and integration on weighted Bergman spaces and also the Taylor coefficients of functions in certain subclasses of these spaces. We then derive several criteria for the multipliers between such spaces, complementing and extending various recent results. Univalent Bergman functions are also considered.