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A Sharp Inequality for Harmonic Diffeomorphisms of the Unit Disk

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Abstract

We extend the classical Schwarz–Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Duren, P.: Harmonic Mappings in the Plane, vol. 156. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  3. Kalaj, D.: On harmonic diffeomorphisms of the unit disc onto a convex domain. Complex Var. Theory Appl. 48(2), 175–187 (2003)

    MathSciNet  MATH  Google Scholar 

  4. Kalaj, D., Marković, M., Mateljević, M.: Carathéodory and Smirnov type theorems for harmonic mappings of the unit disk onto surfaces. Ann. Acad. Sci. Fenn. Math. 38(2), 565–580 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Pavlović, M.: Function Classes on the Unit Disc. An Introduction, vol. 52. De Gruyter, Berlin (2014)

    MATH  Google Scholar 

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Acknowledgements

I am grateful to the referee for useful suggestions and corrections.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put B.B, 81000, Podgorica, Montenegro

    David Kalaj

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  1. David Kalaj
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Correspondence to David Kalaj.

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Kalaj, D. A Sharp Inequality for Harmonic Diffeomorphisms of the Unit Disk. J Geom Anal 29, 392–401 (2019). https://doi.org/10.1007/s12220-018-9996-3

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  • DOI: https://doi.org/10.1007/s12220-018-9996-3

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