A criterion of univalence in $C^n$ in terms of the Schwarzian derivative

Rodrigo Hernández


By using the using the Loewner Chain Theory, we obtain a new criterion of univalence in $C^n$ in terms of the Schwarzian derivative introduced in [3] by using the arguments in [8]. We as well derive explicitly the formula given in [3] by relating the Schwarzian derivative with the numerical method of approximation of zeros due to Halley.


Univalence criterion, Schwarzian derivative, Loewner chain, Halley method.

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J. Becker, Lo ̈wnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972), 23-43.

I. Graham, G. Kohr, Geometric Function Theory in One and Higher Dimen- sions, Monographs and Textbooks in Pure and Applied Mathematics, 255. Marcel Dekker, Inc., New York, 2003.

R. Herna ́ndez, Schwarzian derivatives and a linearly invariant family in Cn, Pacific J. Math. 228 (2006), no. 2, 201-218.

R. Herna ́ndez, Schwarzian derivatives and some criteria for univalence in Cn, Complex Var. Elliptic Equ. 52 (2007), no. 5, 397-410.

H. Lewy, On the non-vanishing of the Jacobian of a homeomorphism by harmonic gradients, Ann. of Math. (2) 88 (1968), 518-529.

J. Mujica, Complex Analysis in Banach Spaces, North-Holland Mathematics Studies, vol. 120, Amsterdam, 1986, reprinted by Dover, 2010.

Z. Nehari, The Schwarzian derivative and schlicht functions. Bull. Amer. Math. Soc., 55 (1949), 545-551.

T. Oda, On Schwarzian derivatives in several variables (in Japanese). Kokyuroku of R.I.M., Kioto Univ., 226 (1975), 82-85.

J. A. Pfaltzgraff, Subordination chains and univalence of holomorphic mappings in Cn. Math. Ann., 210 (1974), 55-68.

J.A. Pfaltzgraff, Subordination chains and quasiconformal extension of holomorphic maps in Cn, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 1, 13-25.

M. Yoshida, Canonical forms of some system of linear partial differential equa- tions, Proc. Japan Acad., 52 (1976), 473-476.

DOI: http://dx.doi.org/10.24193/subbmath.2022.2.16


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