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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.
Arbelaez, H., Bravo, V., Hernandez, R., Sierra, W., & Venegas, O. (2020). A new approach for the univalence of certain integral of harmonic mappings. Indag. Math.-New Ser., 31(4), 525–535.
Abstract: The principal goal of this paper is to extend the classical problem of finding the values of alpha is an element of C for which either (f) over cap (alpha) (z) = integral(z)(0) (f (zeta)/zeta)(alpha) d zeta or f(alpha) (z) = integral(z)(0)(f' (zeta))(alpha)d zeta are univalent, whenever f belongs to some subclasses of univalent mappings in D, to the case of harmonic mappings, by considering the shear construction introduced by Clunie and Sheil-Small in . (C) 2020 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
Gaona, J., Hernández, R., Guevara, F., & Bravo, V. (2022). Influence of a Function’s Coefficients and Feedback of the Mathematical Work When Reading a Graph in an Online Assessment System. Int. J. Emerg. Technol. Learn., 17(20), 77–98.
Abstract: This paper shows the results of an experiment applied to 170
students from two Chilean universities who solve a task about reading a graph
of an affine function in an online assessment environment where the parameters
(coefficients of the graphed affine function) are randomly defined from an ad-hoc
algorithm, with automatic correction and automatic feedback. We distinguish two
versions: one of them with integer coefficients and the other one with decimal
coefficients in the affine function. We observed that the nature of the coefficients
impacts the mathematical work used by the students, where we again focus on
two of them: by direct estimation from the graph or by calculating the equation of
the line. On the other hand, feedback oriented towards the “estimation” strategy
influences the mathematical work used by the students, even though a non-negligible
group persists in the “calculating” strategy, which is partly explained by the
perception of each of the strategies.
Keywords: digital technology; distance education and online learning; mathematical activity; post-secondary education; task design
Arbelaez, H., Bravo, V., Hernandez, R., Sierra, W., & Venegas, O. (2021). Integral transforms for logharmonic mappings. J. Inequal. Appl., 2021(1), 48.
Abstract: Bieberbach's conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type f(alpha)(z) = integral(z)(0)(f(zeta)/zeta)(alpha)d zeta or F-alpha(z) = integral(z)(0)(f '(zeta))(alpha)d zeta appear. In this note we extend the classical problem of finding the values of alpha is an element of C for which either f(alpha) or F-alpha are univalent, whenever f belongs to some subclasses of univalent mappings in D, to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3-25, 1984) to this new scenario.
Bravo, V., Hernandez, R., & Venegas, O. (2017). On the univalence of certain integral for harmonic mappings. J. Math. Anal. Appl., 455(1), 381–388.
Abstract: We generalize the problem of univalence of the integral of f'(z)(alpha) when f is univalent to the complex harmonic mappings. To do this, we extend the univalence criterion by Ahlfors in  to those mappings. (C) 2017 Elsevier Inc. All rights reserved.
Keywords: Harmonic mapping; Univalent functions; Integral transform
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.
Keywords: Pre-Schwarzian and Schwarzian derivatives; Harmonic and logharmonic mappings; Univalence criterion