
Bravo, V., Hernandez, R., Ponnusamy, S., & Venegas, O. (2022). PreSchwarzian and Schwarzian derivatives of logharmonic mappings. Monatsh. fur Math., 199(4), 733–754.
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.



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 [1] to those mappings. (C) 2017 Elsevier Inc. All rights reserved.



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 Falpha(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 Falpha 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 SheilSmall in (Clunie and SheilSmall in Ann. Acad. Sci. Fenn., Ser. A I 9:325, 1984) to this new scenario.



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 adhoc
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 nonnegligible
group persists in the “calculating” strategy, which is partly explained by the
perception of each of the strategies.



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 SheilSmall in [4]. (C) 2020 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.



Bravo, V., Hernandez, R., & Venegas, O. (2023). TwoPoint Distortion Theorems for Harmonic Mappings. Bull. Malaysian Math. Sci., 46(3), 100.
Abstract: We establish twopoint distortion theorems for sensepreserving 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 twopoint distortion theorems for the cases when h is a normalized convex function and, more generally, when h(D) is a clinearly connected domain.

