The Schwarzian Derivative of Harmonic Mappings in the Plane

Abstract

In this paper, the authors introduce a definition of the Schwarzian derivative of any locally univalent harmonic mapping defined on a simply connected domain in the complex plane. Using the new definition, the authors prove that any harmonic mapping f which maps the unit disk onto a convex domain has Schwarzian norm ∥S _f∥ ≤ 6. Furthermore, any locally univalent harmonic mapping f which maps the unit disk onto an arbitrary regular n-gon has Schwarzian norm $\left\| {{S_f}} \right\| \leq {8 \over 3}$.

Keywords

Schwarzian derivative / Schwarzian norm / Harmonic mapping

References

[1] Chuaqui M, Duren P, Osgood B. The Schwarzian derivative for harmonic mappings. J. Anal. Math., 2003, 91: 329-351

[2] Chuaqui M, Duren P, Osgood B. Schwarzian derivative criteria for valence of analytic and harmonic mappings. Math. Proc. Camb. Philos. Soc., 2007, 143: 473-486

[3] Chuaqui M, Duren P, Osgood B. Schwarzian derivative and uniform local univalence. Comput. Methods Funct. Theory, 2008, 8: 21-34

[4] Clunie J, Sheil-Small T. Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. AI Math., 1984, 9: 3-25

[5] Duren P. Univalent Functions, 1983, Heidelberg, New York: Springer-Verlag

[6] Duren P. Harmonic Mappings in the Plane, 2004, Cambridge: Cambridge University Press

[7] Hernández R, Martín M. Pre-Schwarzian and Schwarzian derivative of harmonic mappings. J. Geom. Anal., 2015, 25: 64-91

[8] Kraus W. Über den zusammenhang einiger characteristiken eines einfach zusammenhängenden bereiches mit der kreisabbildung. Mitt. Math. Sem. Giessen, 1932, 21: 1-28

[9] Nehari Z. The Schwarzian derivative and schlicht functions. Bull. Am. Math. Soc., 1949, 55: 545-551

[10] Nehari Z. Univalence criteria depending on the Schwarzian derivative. Illinois J. Math., 1979, 23: 345-351

[11] Osgood B, Stowe D. The Schwarzian derivative and conformal mappings of Riemannian mainfolds. Duke. Math. J., 1992, 67: 57-99

[12] Pommerenke Ch. Linear-invariante familien analytischer funktionen I. Math. Annalen, 1964, 155: 108-154

[13] Sheil-Small T. Constants for planar harmonic mappings. J. London Math. Soc., 1990, 42: 237-248