Journal of Geometric Analysis

, Volume 19, Issue 4, pp 755–771 | Cite as

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

Article
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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 ∫ 0 1 M 1 p (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.

Keywords

Univalent functions Hardy spaces Integral means Arc-length Dirichlet type spaces 

Mathematics Subject Classification (2000)

30C55 30D55 46E15 
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Copyright information

© Mathematica Josephina, Inc. 2009

Authors and Affiliations

  1. 1.Dept. Análisis MatemáticoUniv. MálagaMálagaSpain
  2. 2.Dept. MatemáticasUniv. CórdobaCórdobaSpain

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