Bravo, V., Hernandez, R., Ponnusamy, S., & Venegas, O. (2022). Pre-Schwarzian and Schwarzian derivatives of logharmonic mappings. Monatsh. fur Math., 199(4), 733–754.
Abstract: We introduce definitions of pre-Schwarzian 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 pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when the pre-Schwarzian derivative is holomorphic.
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Bravo, V., Hernandez, R., & Venegas, O. (2023). Two-Point Distortion Theorems for Harmonic Mappings. Bull. Malaysian Math. Sci., 46(3), 100.
Abstract: We establish two-point distortion theorems for sense-preserving planar harmonic map -pings f = h + g in the unit disk D which satisfy harmonic versions of the univalence criteria due to Becker and Nehari. In addition, we also find two-point distortion theo-rems for the cases when h is a normalized convex function and, more generally, when h(D) is a c-linearly connected domain.
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Hernandez, R. (2022). A criterion of univalence in C-n in terms of the Schwarzian derivative. Stud. Univ. Babes-Bolyai Math., 67(2), 421–430.
Abstract: Using the Loewner Chain Theory, we obtain a new criterion of univalence in C-n 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.
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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 complex-valued 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.
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