Univalent functions in Hardy, Bergman, Bloch and related spaces

Abstract

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several Mobius invariant function spaces, such as BMOA, Qp or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space \( \mathcal{D}_{p + \alpha }^p \) are the same as the univalent functions in Hαp and Sαp if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller Mobius invariant subspace of the Bloch space than Qp still containing all univalent Bloch functions.