Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 36, 2011, 331-340

PRESCRIBING THE PRESCHWARZIAN IN SEVERAL COMPLEX VARIABLES

Rodrigo Hernández

Universidad Adolfo Ibáñez, Facultad de Ciencias y Tecnología
Av. Balmaceda 1625 Recreo, Viña del Mar, Chile; rodrigo.hernandez 'at' uai.cl

Abstract. We solve the several complex variables preSchwarzian operator equation [Df(z)]-1D2f(z) = A(z), z \in Cn, where A(z) is a bilinear operator and f is a Cn valued locally biholomorphic function on a domain in Cn. Then one can define a several variables f \to f\alpha transform via the operator equation [Df\alpha(z)]-1D2f\alpha(z) = \alpha[Df(z)]-1D2f(z), and thereby, study properties of f\alpha. This is a natural generalization of the one variable operator f\alpha(z) in [6] and the study of its univalence properties, e.g., the work of Royster [23] and many others. Möbius invariance and the multivariables Schwarzian derivative operator of Oda [17] play a central role in this work.

2000 Mathematics Subject Classification: Primary 32H02, 32W99; Secondary 32A10.

Key words: PreSchwarzian derivative, holomorphic mapping, univalence.

Reference to this article: R. Hernández: Prescribing the preSchwarzian in several complex variables. Ann. Acad. Sci. Fenn. Math. 36 (2011), 331-340.

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doi:10.5186/aasfm.2011.3621

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