Harmonic hereditary convexity

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  1. 1.

    Chuaqui, M., Duren, P., Osgood, B.: Ellipses, near ellipses, and harmonic Möbius transformations. Proc. Amer. Math. Soc. 133, 2705–2710 (2005)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

    Duren, P.: Univalent functions. Springer-Verlag, New York (1983)

    Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  6. 6.

    Iwaniec, T., Koh, N., Kovalev, L.V., Onninen, J.: Existence of energy-minimal diffeomorphisms between doubly connected domains. Invent. Math. 186, 667–707 (2011)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Iwaniec, T., Kovalev, L.V., Onninen, J.: The harmonic mapping problem and affine capacity. Proc. Roy. Soc. Edinburgh Sect. A 141, 1017–1030 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Kalaj, D.: Energy-minimal diffeomorphisms between doubly connected Riemann surfaces. Calc. Var. Partial Differential Equations 51, 465–494 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Koh, N.: Hereditary convexity for harmonic homeomorphisms. Indiana University Mathematics Journal 64, 231–243 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Koh, N.: Hereditary circularity for energy minimal diffeomorphisms. Conform. Geom. Dyn. 21, 369–377 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Koh, N.: Harmonic mappings with hereditary starlikeness. J. Math. Anal. Appl. 457(1), 273–286 (2018)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Koh, N., Kovalev, L.V.: Area contraction for harmonic automorphisms of the disk. Bull. London Math. Soc. 43, 91–96 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Lehto, O., Virtanen, K.: Quasiconformal Mappings in the Plane. Springer-Verlag, Berlin (1973)

    Google Scholar 

  14. 14.

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

    MathSciNet  Article  Google Scholar 

  15. 15.

    Nagpal, S., Ravichandran, V.: Fully starlike and fully convex harmonic mappings of order \(\alpha\). Ann. Polon. Math. 108, 85–107 (2013)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Radó, T.: Zu einem Satze von S. Bernstein über Minimalflächen im Großen. Math. Z. 26, 559–565 (1927)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ruscheweyh, S., Salinas, L.C.: On the preservation of direction-convexity and the Goodman-Saff conjecture. Ann. Acad. Sci. Fenn. Math. 14, 63–73 (1989)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Schiff, J.L.: Normal families. Springer-Verlag, New York (1993)

    Google Scholar 

  19. 19.

    Study, E.: Vorlesungen über ausgewählte Gegenstände der Geometrie. Heft II, Teubner, Leipzig (1913)

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Koh, NT. Harmonic hereditary convexity. Complex Anal Synerg 7, 9 (2021). https://doi.org/10.1007/s40627-021-00074-z

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