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On the Harmonic Möbius Transformations

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Abstract

It is well-known that two locally univalent analytic functions have equal Schwarzian derivative if and only if each one of them is a composition of the other with a non-constant Möbius transformation. The main goal in this paper is to extend this result to the cases when the functions considered are harmonic. That is, we identify completely the transformations that preserve the (harmonic) Schwarzian derivative of locally univalent harmonic functions.

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References

  1. Abu Muhanna, Y., Ali, R.M., Ponnusamy, S.: The spherical metric and univalent harmonic mappings. Monatsh. Math. 188, 703–716 (2019)

    Article  MathSciNet  Google Scholar 

  2. Ahlfors, L.: Sufficient conditions for quasiconformal extension. Ann. Math. Stud. 79, 23–29 (1974)

    MathSciNet  Google Scholar 

  3. Ahlfors, L.V., Weill, G.: A uniqueness theorem for Beltrami equations. Proc. Am. Math. Soc. 13, 975–978 (1962)

    Article  MathSciNet  Google Scholar 

  4. Becker, J.: Löwnersche differentialgleichung und quasikonform fortsetzbare schlichte functionen. J. Reine Angew. Math. 255, 23–43 (1972)

    MathSciNet  MATH  Google Scholar 

  5. Becker, J., Pommerenke, Ch.: Schlichtheitskriterien und Jordangebiete. J. Reine Angew. Math. 354, 74–94 (1984)

    MathSciNet  MATH  Google Scholar 

  6. Becker, J., Pommerenke, Ch.: Locally univalent functions and the Bloch and Dirichlet norm. Comput. Methods Funct. Theory 16, 43–52 (2016)

    Article  MathSciNet  Google Scholar 

  7. Chen, S., Ponnusamy, S.: John disks and \(K\)-quasiconformal harmonic mappings. J. Geom. Anal. 27, 1468–1488 (2017)

    Article  MathSciNet  Google Scholar 

  8. Chuaqui, M., Osgood, B., Pommerenke, Ch.: John domains, quasidisks and the Nehari class. J. Reine Angew. Math. 471, 77–114 (1996)

    MathSciNet  MATH  Google Scholar 

  9. Chuaqui, M., Duren, P., Osgood, B.: The Schwarzian derivative for harmonic mappings. J. Anal. Math. 91, 329–351 (2003)

    Article  MathSciNet  Google Scholar 

  10. Chuaqui, M., Hernández, R., Martín, M.J.: Order of affine and linear invariant families of harmonic mappings. Math. Ann. 367, 1099–1122 (2017)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  12. Duren, P.: Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics 156. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  13. Gehring, F.W., Pommerenke, Ch.: On the Nehari univalence criterion and quasicircles. Comment. Math. Helvetici 59, 226–242 (1984)

    Article  MathSciNet  Google Scholar 

  14. Graham, I., Kohr, G.: Geometric Function Theory in one and Higher Dimensions, Monographs and Textbooks in Pure and Applied Mathematics 255. Marcel Dekker Inc., New York (2003)

    Google Scholar 

  15. Hag, K., Hag, P.: John disks and the pre-Schwarzian derivative. Ann. Acad. Sci. Fenn. Math. 26, 205–224 (2001)

    MathSciNet  MATH  Google Scholar 

  16. Hernández, R., Martín, M.J.: Quasiconformal extensions of harmonic mappings. Ann. Acad. Sci. Fenn. Ser. A. I Math. 38, 617–630 (2013)

    Article  MathSciNet  Google Scholar 

  17. Hernández, R., Martín, M.J.: Pre-Schwarzian and Schwarzian derivatives of harmonic mappings. J. Geom. Anal. 25, 64–91 (2015)

    Article  MathSciNet  Google Scholar 

  18. Hernández, R., Martín, M.J.: Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative. Arch. Math. (Basel) 105, 53–59 (2015)

    Article  MathSciNet  Google Scholar 

  19. Huusko, J.-M., Martín, M.J.: Criteria for bounded valence of harmonic mappings. Comput. Methods Funct. Theory 17, 603–612 (2017)

    Article  MathSciNet  Google Scholar 

  20. Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)

    Article  MathSciNet  Google Scholar 

  21. Nehari, Z.: The Schwarzian derivative and Schlicht functions. Bull. Am. Math. Soc. 55, 545–551 (1949)

    Article  MathSciNet  Google Scholar 

  22. Sheil-Small, T.: Constants for planar harmonic mappings. J. Lond. Math. Soc. 42, 237–248 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We would like to thank the referee for the careful reading of the previous version of this paper and for the useful suggestions to improve the exposition.

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Authors and Affiliations

  1. Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile

    Rodrigo Hernández

  2. Departamento de Análisis Matemático, Universidad de La Laguna, Av. Astrofísico Francisco Sánchez, s/n. 38271, La Laguna, Sta. Cruz de Tenerife, Spain

    María J. Martín

Authors
  1. Rodrigo Hernández
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  2. María J. Martín
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Correspondence to Rodrigo Hernández.

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Dedicated to Professor Fernando Pérez-González on the occasion of his retirement.

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The first author is supported by Grant Fondecyt \(\#\)1190756, Chile. The second author is supported by Spanish research projects PID2019-106093GB-I00 and PID2019-106870GB-I00 (MICINN).

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Hernández, R., Martín, M.J. On the Harmonic Möbius Transformations. J Geom Anal 32, 18 (2022). https://doi.org/10.1007/s12220-021-00809-8

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