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Convex harmonic mappings are not necessarily in $ h^{1/2}$


Authors: Alexandru Aleman and María J. Martín
Journal: Proc. Amer. Math. Soc. 143 (2015), 755-763
MSC (2010): Primary 30C45; Secondary 30H10, 31A05
DOI: https://doi.org/10.1090/S0002-9939-2014-12281-7
Published electronically: October 10, 2014
MathSciNet review: 3283662
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Abstract: We construct convex harmonic mappings in the unit disk that do not belong to the harmonic Hardy space $ h^{1/2}$. This provides a negative answer to a question raised by P. Duren.


References [Enhancements On Off] (What's this?)

  • [1] Yusuf Abu-Muhanna and Glenn Schober, Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), no. 6, 1489-1530. MR 918391 (89e:30011), https://doi.org/10.4153/CJM-1987-071-4
  • [2] Joseph A. Cima and Albert E. Livingston, Integral smoothness properties of some harmonic mappings, Complex Variables Theory Appl. 11 (1989), no. 2, 95-110. MR 1005664 (91f:30031)
  • [3] J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3-25. MR 752388 (85i:30014)
  • [4] Michael John Dorff, Harmonic univalent mappings onto asymmetric vertical strips, Computational methods and function theory 1997 (Nicosia), Ser. Approx. Decompos., vol. 11, World Sci. Publ., River Edge, NJ, 1999, pp. 171-175. MR 1700345 (2000d:30023)
  • [5] Peter L. Duren, Theory of $ H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York, 1970. MR 0268655 (42 #3552)
  • [6] Peter Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. MR 2048384 (2005d:31001)
  • [7] Armen Grigoryan and Maria Nowak, Estimates of integral means of harmonic mappings, Complex Variables Theory Appl. 42 (2000), no. 2, 151-161. MR 1788247 (2001e:30023)
  • [8] Javad Mashreghi, Representation theorems in Hardy spaces, London Mathematical Society Student Texts, vol. 74, Cambridge University Press, Cambridge, 2009. MR 2500010 (2011e:30001)
  • [9] Maria Nowak, Integral means of univalent harmonic maps, Ann. Univ. Mariae Curie-Skłodowska Sect. A 50 (1996), 155-162. MR 1472587 (98f:30019)
  • [10] Metin Öztürk, Univalent harmonic mappings onto half planes, Turkish J. Math. 23 (1999), no. 2, 301-313. MR 1739170 (2001f:30015)
  • [11] Joel H. Shapiro, Linear topological properties of the harmonic Hardy spaces $ h^p$ for $ 0<p<1$, Illinois J. Math. 29 (1985), no. 2, 311-339. MR 784526 (86f:46023)

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Additional Information

Alexandru Aleman
Affiliation: Department of Mathematics, Lund University, P. O. Box 118, S-221 00 Lund, Sweden
Email: aleman@maths.lth.se

María J. Martín
Affiliation: Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
Email: maria.martin@uef.fi

DOI: https://doi.org/10.1090/S0002-9939-2014-12281-7
Keywords: Convex harmonic mappings, Hardy spaces
Received by editor(s): March 4, 2013
Received by editor(s) in revised form: May 21, 2013
Published electronically: October 10, 2014
Additional Notes: This work was supported by Grant MTM2012-37436-C02-02, MINECO, Spain. The second author was also partially supported by “Beca Fundación Caja Madrid. Movilidad de Profesores, Convocatoria 2012/13”, Spain
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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