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The Journal of Analysis

, Volume 26, Issue 1, pp 135–150 | Cite as

Area distortion under some orientation preserving harmonic maps

  • Yusuf Abu Muhanna
  • Issam Louhichi
Original Research Paper

Abstract

We study the distortion of area under the class of orientation preserving harmonic maps into the unit disk and the class of harmonic univalent maps onto the unit disk.

Keywords

Hyperbolic metric Pseudo-hyperbolic metric Harmonic univalent map 

Mathematics Subject Classification

Primary 30C62 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare no conflict of interest. The authors certify that they have NO affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.

References

  1. 1.
    Astala, K. 1994. Area distortion for quasiconformal mappings. Acta Mathematica 173: 37–60.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    . Beardon and D. Minda, The hyperbolic metric and geometric function theory. In: Proceedings of the International Workshop on Quasiconformal Mappings and their Applications (IWQCMA05).Google Scholar
  3. 3.
    Eremenko, A., and D.H. Hamilton. 1995. On the area distortion of quasiconformal mappings. Proceeding of the AMS 123 (9): 2793–2797.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    J. Garnett, Bounded analytic functions, Springer, 2007.Google Scholar
  5. 5.
    Gehring, F.W., and E. Reich. 1966. Area distortion under quasiconformal mappings. Annales Academiae Scientiarum Fennicae 388: 1–14.MathSciNetzbMATHGoogle Scholar
  6. 6.
    Heinz, E. 1959. On one-to-one harmonic mappings. Pacific Journal of Mathematics 9: 101105.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Lewy, H. 1936. On the non-vanishing of the Jacobian in certain one-to-one mappings. Bulletin of the American Mathematical Society 42: 689–692.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Abu-Muhanna, Y. 2010. Convex and harmonic Bohr’s inequalities. Complex Variables and Elliptic functions Journal 55 (11): 1071–1078.CrossRefzbMATHGoogle Scholar
  9. 9.
    Abu-Muhanna, Y. 1999. On harmonic univalent functions. Complex Variables Journal 139 (4): 341–348.MathSciNetzbMATHGoogle Scholar
  10. 10.
    Duren, P. 1983. Univalent functions. Heidelberg: Springer.zbMATHGoogle Scholar
  11. 11.
    Duren, P., and G. Schober. 1989. Linear extremal problems for harmonic mappings of the disk. Proceedings of the American Mathematical Society 105 (4): 967–973.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hedenmalm, H., B. Korenblum, and K. Zhu. 2000. Theory of Bergman Spaces. New York: Springer.CrossRefzbMATHGoogle Scholar
  13. 13.
    Heinonen, J. 2001. Lectures on analysis on metric spaces. Heidelberg: Springer.CrossRefzbMATHGoogle Scholar
  14. 14.
    Koh, N.-T., and L.V. Kovalev. 2011. Area contraction for harmonic automorphisms of the disk. Bulletin of the London Mathematical Society 43: 91–96.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kalaj, D., and M. Vuorinen. 2012. On harmonic functions and the Schwarz lemma. Proceedings of the American Mathematical Society 140 (1): 161–165.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, College of Arts and SciencesAmerican University of SharjahSharjahUnited Arab Emirates

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