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Univalent Functions in Hardy Spaces in Terms of the Growth of Arc-Length

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Abstract

Pommerenke (1962) proved that for f univalent in the unit disk and 0<p<2, f∈H p if and only if ∫ 10 M p1 (r,f′)dr<∞. In this paper, we prove that the result continues to be true for p slightly larger than 2, but is false for large p. Also, it turns out that the result is true for all p>0 when f is restricted to the class of close-to-convex functions. Finally, we discuss the membership of univalent functions in some related spaces of Dirichlet type.

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Authors and Affiliations

  1. Dept. Análisis Matemático, Univ. Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Cristóbal González

  2. Dept. Matemáticas, Univ. Córdoba, Edif. Einstein, Campus de Rabanales, 14014, Córdoba, Spain

    José Ángel Peláez

Authors
  1. Cristóbal González
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  2. José Ángel Peláez
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Correspondence to Cristóbal González.

Additional information

Communicated by Marco Peloso.

Both authors partially supported by the Spanish (Grants MTM2007-60854 and MTM2006-26627) and regional Andalusian (Grants FQM210 and P06-FQM01504) Governments.

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González, C., Peláez, J.Á. Univalent Functions in Hardy Spaces in Terms of the Growth of Arc-Length. J Geom Anal 19, 755–771 (2009). https://doi.org/10.1007/s12220-009-9092-9

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