Inner Functions in the Hyperbolic Little Bloch Class

摘要

The hyperbolic derivative of an analytic self-map ψ: D → D of the unit disk is given by |ψ'|/(1-|ψ|~2). To explain the terminology, we note that integrating |ψ'|/(1-|ψ|~2) over a rectifiable cure γ in D gives the hyperbolic arclength of ψ(γ). This notion of derivative has been used by Yamashita to study hyperbolic versions of the classical Hardy and Dirichlet spaces; see [Y1] and [Y2]. More re- cently in [MM]ψ and [SZ], hyperbolic derivatives have been shown to be pertinent to the study of composition operator on certain subspaces of H(D), the space of analytic functions on D.