A criterion of univalence in Cⁿ in terms of the Schwarzian derivative
DOI:
https://doi.org/10.24193/subbmath.2022.2.16Keywords:
Univalence criterion, Schwarzian derivative, Loewner chain, Halley method.Abstract
Using the Loewner Chain Theory, we obtain a new criterion of uni- valence in Cⁿ in terms of the Schwarzian derivative for locally biholomorphic mappings. We as well derive explicitly the formula of this Schwarzian derivative using the numerical method of approximation of zeros due by Halley.
Mathematics Subject Classification (2010): 32A10, 32W50, 32H02.
Received 14 January 2022; Accepted 1 February 2022.
References
Becker, J., Lownersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math., 255(1972), 23-43.
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.
Hernandez, R., Schwarzian derivatives and a linearly invariant family in Cn, Pacific J. Math., 228(2006), no. 2, 201-218.
Hernandez, R., Schwarzian derivatives and some criteria for univalence in Cn, Complex Var. Elliptic Equ., 52(2007), no. 5, 397-410.
Lewy, H., On the non-vanishing of the Jacobian of a homeomorphism by harmonic gradients, Ann. of Math. (2), 88(1968), 518-529.
Mujica, J., Complex Analysis in Banach Spaces, North-Holland Mathematics Studies, vol. 120, Amsterdam, 1986, Reprinted by Dover, 2010.
Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc., 55(1949), 545-551.
Oda, T., On Schwarzian derivatives in several variables, (Japanese), Kokyuroku of R.I.M., Kioto Univ., 226(1975), 82-85.
Pfaltzgraff, J.A., Subordination chains and univalence of holomorphic mappings in Cn, Math. Ann., 210(1974), 55-68.
Pfaltzgraff, J.A., Subordination chains and quasiconformal extension of holomorphic maps in Cn, Ann. Acad. Sci. Fenn. Ser. A I Math., 1(1975), no. 1, 13-25.
Yoshida, M., Canonical forms of some system of linear partial differential equations, Proc. Japan Acad., 52(1976), 473-476.
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