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Univalent functions in Hardy, Bergman, Bloch and related spaces

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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 Möbius invariant function spaces, such as BMOA, Q p 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 H pα and S pα if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller Möbius invariant subspace of the Bloch space than Q p still containing all univalent Bloch functions.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, 38271, La Laguna, Tenerife, Spain

    Fernando Pérez-González

  2. University of Joensuu Mathematics, P. O. Box 111, 80101, Joensuu, Finland

    Jouni Rättyä

Authors
  1. Fernando Pérez-González
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  2. Jouni Rättyä
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Correspondence to Fernando Pérez-González.

Additional information

This research has been supported in part by the MEC-Spain MTM2005-07347, the Spanish Thematic Network MTM2006-26627-E, and the Academy of Finland 210245.

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Pérez-González, F., Rättyä, J. Univalent functions in Hardy, Bergman, Bloch and related spaces. J Anal Math 105, 125–148 (2008). https://doi.org/10.1007/s11854-008-0032-6

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  • DOI: https://doi.org/10.1007/s11854-008-0032-6

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