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Boundary Behavior of Univalent Harmonic Mappings

A Survey of Recent Boundary Behavior Results of Univalent Harmonic Mappings

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Current Topics in Pure and Computational Complex Analysis

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Abstract

This chapter is a survey of the boundary behavior properties of univalent harmonic mappings of the unit disk \(\mathbb{D}\) over the last two decades. Particular emphasis has been given to the boundary behavior of univalent harmonic mappings “onto” \(\mathbb{D}\) in the sense of Hengartner and Schober.

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References

  1. Abu-Muhanna, Y., Lyzzaik, A.: The boundary behavior of harmonic univalent maps. Pac. J. Math. 141, 1–20 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beurling, A.: Sur les ensembles exceptionels. Acta Math. 72, 1–13.

    Google Scholar 

  3. Bshouty, D., Hengartner, W.: Boundary values versus dilatations of harmonic mappings. J. Anal. Math. 72, 141–164 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bshouty, D., Lyzzaik, A.: Uniqueness of harmonic mappings into bounded strictly starlike domains with rectifiable boundary. Comput. Method Funct. Theory 8(2), 433–446 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bshouty, D., Lyzzaik, A.: Problems and conjectures in planar harmonic mappings. J. Anal. 18, 698–1 (2010)

    MathSciNet  Google Scholar 

  6. Bshouty, D., Hengartner, W., Hossian, O.: Harmonic typically real mappings. Math. Proc. Camb. Philos. Soc. 19, 673–680 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bshouty, D., Lyzzaik, A., Weitsman, A.: Uniqueness of harmonic mappings with Blaschke dilatations. J. Geom. Anal. 17(1), 41–47 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bshouty, D., Lyzzaik, A., Weitsman, A.: On the boundary behaviour of univalent harmonic mappings onto convex domains. Comput. Method Funct. Theory 8(1–2), 261–275 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bshouty, D., Lyzzaik, A., Weitsman, A.: On the boundary behaviour of univalent harmonic mappings. Ann. Acad. Sci. Fenn. Math. 37, 135–147 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A. I. 9(2), 3–25 (1984)

    MATH  MathSciNet  Google Scholar 

  11. Duren, P.: Harmonic mappings in the plane. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  12. Duren, P., Khavinson, D.: Boundary correspondence and dilatation of harmonic mappings. Complex Var. Theory Appl. 33(1–4), 105–112 (1997)

    Article  MathSciNet  Google Scholar 

  13. Garnett, J., Marshall, D.: Harmonic measure. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  14. Frostman, O.: Potential d’equilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Meddel. Lunds Univ. Mat. Sem., Vol. 3 (1935),1-118.

    Google Scholar 

  15. Gergen, J., Dressel, F.: Mapping by p-regular functions. Duke Math. J. 18, 185–210 (1951)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gergen, J., Dressel, F.: Uniqueness for by p-regular mapping. Duke Math. J. 19, 435–444 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hengartner, W., Schober, G.: On the boundary behavior of orientation-preserving harmonic mappings. Complex Var. Theory Appl. 5, 197–208 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hengartner, W., Schober, G.: Harmonic mappings with given dilatation. J. Lon. Math. Soc. 33(2), 473–483 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  19. Heinz, E.: On one-to-one harmonic mappings. Pac. J. Math. 9(1), 101–105 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kalaj, D.: O harmonic diffeomorphisms of the unit disc onto a convex domain. Complex Var. Theory Appl. 48, 175–187 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Laugesen, R.: Planar harmonic maps with inner and Balschke dilatations. J. London Math. Soc. 56(2), 37–48 (1997)

    Article  MathSciNet  Google Scholar 

  22. Lyzzaik, A.: Local properties of light harmonic mappings. Can. J. Math. 44, 135–153 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Martio, O.: On harmonic quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math. 425, 1–10 (1968)

    MathSciNet  Google Scholar 

  24. Nowak, M.: Integral means of univalent harmonic maps. Ann. Univ. Mariae Curie-Sklodwska 50, 155–162 (1996)

    MATH  Google Scholar 

  25. Protas, D.: On the accumulation of the zeros of a Blaschke product at a boundary point. Proc. Am. Math. Soc. 34, 489–496 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  26. Sheil-Small, T.: On the fourier series of a step function. Mich. Math. J. 36, 459–475 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  27. Tsuji, M.: Potential Theory in Modern Function Theory, 2nd Edn. Chelsea Publishing Company, White River (1975)

    MATH  Google Scholar 

  28. Weitsman, A.: On the dilatation of univalent harmonic mappings. Proc. Am. Math. Soc. 126(2), 447–452 (1998)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Technion, 32000, Haifa, Israel

    Daoud Bshouty

  2. Department of Mathematics, American University of Beirut, Beirut, Lebanon

    Abdallah Lyzzaik

Authors
  1. Daoud Bshouty
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  2. Abdallah Lyzzaik
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Corresponding author

Correspondence to Daoud Bshouty .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Walchand College of Engineering, Sangli, Maharashtra, India

    Santosh Joshi

  2. Department of Mathematics, Brigham Young University, Provo, Utah, USA

    Michael Dorff

  3. Department of Mathematics, University of Kalyani, Kalyani, West Bengal, India

    Indrajit Lahiri

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Bshouty, D., Lyzzaik, A. (2014). Boundary Behavior of Univalent Harmonic Mappings. In: Joshi, S., Dorff, M., Lahiri, I. (eds) Current Topics in Pure and Computational Complex Analysis. Trends in Mathematics. Birkhäuser, New Delhi. https://doi.org/10.1007/978-81-322-2113-5_1

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