
Bravo, V., Hernandez, R., Ponnusamy, S., & Venegas, O. (2022). PreSchwarzian and Schwarzian derivatives of logharmonic mappings. Monatsh. fur Math., Early Access.
Abstract: We introduce definitions of preSchwarzian and Schwarzian derivatives for logharmonic mappings, and basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the preSchwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the preSchwarzian derivative is holomorphic.



Hernandez, R. (2011). Prescribing The Preschwarzian In Several Complex Variables. Ann. Acad. Sci. Fenn. Ser. A1Math., 36(1), 331–340.
Abstract: We solve the several complex variables preSchwarzian operator equation [D f (z)](1) D2 f (z) = A(z), z is an element of 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 > f(alpha) transform via the operator equation [D f(alpha) (z)](1) D2 f(alpha) (z) = alpha[D f (z)](1) D2 f (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. Mobius invariance and the multivariables Schwarzian derivative operator of Oda [17] play a central role in this work.



Hernandez, R. (2022). A criterion of univalence in Cn in terms of the Schwarzian derivative. Stud. Univ. BabesBolyai Math., 67(2), 421–430.
Abstract: Using the Loewner Chain Theory, we obtain a new criterion of univalence 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.



Hernandez, R., & Martin, M. J. (2015). Criteria for univalence and quasiconformal extension of harmonic mappings in terms of the Schwarzian derivative. Arch. Math., 104(1), 53–59.
Abstract: We prove that if the Schwarzian norm of a given complexvalued locally univalent harmonic mapping f in the unit disk is small enough, then f is, indeed, globally univalent in the unit disk and can be extended to a quasiconformal mapping in the extended complex plane.



Hernandez, R., & Martin, M. J. (2015). PreSchwarzian and Schwarzian Derivatives of Harmonic Mappings. J. Geom. Anal., 25(1), 64–91.
Abstract: In this paper we introduce a definition of the preSchwarzian and the Schwarzian derivatives of any locally univalent harmonic mapping f in the complex plane without assuming any additional condition on the (second complex) dilatation omega(f) of f. Using the new definition for the Schwarzian derivative of harmonic mappings, we prove theorems analogous to those by Chuaqui, Duren, and Osgood. Also, we obtain a Beckertype criterion for the univalence of harmonic mappings.

