
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.



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.



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.



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.



Efraimidis, I., Gaona, J., Hernandez, R., & Venegas, O. (2017). On harmonic Blochtype mappings. Complex Var. Elliptic Equ., 62(8), 1081–1092.
Abstract: Let f be a complexvalued harmonicmapping defined in the unit disk D. We introduce the following notion: we say that f is a Blochtype function if its Jacobian satisfies This gives rise to a new class of functions which generalizes and contains the wellknown analytic Bloch space. We give estimates for the schlicht radius, the growth and the coefficients of functions in this class. We establish an analogue of the theorem which, roughly speaking, states that for. analytic log. is Bloch if and only if. is univalent.



Hernandez, R., & Venegas, O. (2019). Distortion Theorems Associated with Schwarzian Derivative for Harmonic Mappings. Complex Anal. Oper. Theory, 13(4), 1783–1793.
Abstract: Let f be a complexvalued harmonic mapping defined in the unit disc D. The theorems of Chuaqui and Osgood (J Lond Math Soc 2:289298, 1993), which assert that the bounds of the size of the hyperbolic norm of the Schwarzian derivative for an analytic function f imply certain bounds for distortion and growth of f, are extended to the harmonic case.

