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.

Arbelaez, H., Hernandez, R., & Sierra, W. (2019). Normal harmonic mappings. Mon.heft. Math., 190(3), 425–439.
Abstract: The main purpose of this paper is to study the concept of normal function in the context of harmonic mappings from the unit disk D to the complex plane. In particular, we obtain necessary conditions for a function f to be normal.

Arbelaez, H., Hernandez, R., & Sierra, W. (2022). Lower and upper order of harmonic mappings. J. Math. Anal. Appl., 507(2), 125837.
Abstract: In this paper, we define both the upper and lower order of a sensepreserving harmonic mapping in D. We generalize to the harmonic case some known results about holomorphic functions with positive lower order and we show some consequences of a function having finite upper order. In addition, we improve a related result in the case when there is equality in a known distortion theorem for harmonic mappings with finite upper order. Some examples are provided to illustrate the developed theory. (C) 2021 Elsevier Inc. All rights reserved.

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.

Chuaqui, M., & Hernandez, R. (2007). Univalent harmonic mappings and linearly connected domains. J. Math. Anal. Appl., 332(2), 1189–1194.
Abstract: We investigate the relationship between the univalence of f and of h in the decomposition f = h + (g) over bar of a serisepreserving harmonic mapping defined in the unit disk D subset of C. Among other results, we determine the holomorphic univalent maps It for which there exists c > 0 such that every harmonic mapping of the form f = h + (g) over bar with vertical bar g'vertical bar < c vertical bar h'vertical bar is univalent. The notion of a linearly connected domain appears in our study in a relevant way. (c) 2006 Elsevier Inc. All rights reserved.

Efraimidis, I., FerradaSalas, A., Hernandez, R., & Vargas, R. (2021). Schwarzian derivatives for pluriharmonic mappings. J. Math. Anal. Appl., 495(1), 124716.
Abstract: A preSchwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in Cn are introduced. 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. Furthermore, it is shown that if the Schwarzian derivative of a pluriharmonic mapping vanishes then the analytic part of this mapping is a Mobius transformation. Some observations are made related to the dilatation of pluriharmonic mappings and to the dilatation of their affine transformations, revealing differences between the theories in the plane and in higher dimensions. An example is given that rules out the possibility for a shear construction theorem to hold in Cn, for n >= 2. (C) 2020 Elsevier Inc. All rights reserved.

Efraimidis, I., Hernandez, R., & Martin, M. J. (2023). AhlforsWeill extensions for harmonic mappings. J. Math. Anal. Appl., 523(2), 127053.
Abstract: We provide two new formulas for quasiconformal extension to C for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the AhlforsWeill extension for holomorphic functions.(c) 2023 Elsevier Inc. All rights reserved.

FerradaSalas, A., Hernandez, R., & Martin, M. J. (2017). On Convex Combinations Of Convex Harmonic Mappings. Bull. Aust. Math. Soc., 96(2), 256–262.
Abstract: The family Flambda of orientationpreserving harmonic functions f = h + (g) over bar in the unit disc D (normalised in the standard way) satisfying h' (z) + g' (z) = 1/(1 + lambda z)(1 + (lambda) over barz), z is an element of D, for some lambda is an element of partial derivative D, along with their rotations, play an important role among those functions that are harmonic and orientationpreserving and map the unit disc onto a convex domain. The main theorem in this paper generalises results in recent literature by showing that convex combinations of functions in Flambda are convex.

Hernandez, R., & Martin, M. J. (2013). Quasiconformal Extension Of Harmonic Mappings In The Plane. Ann. Acad. Sci. Fenn. Ser. A1Math., 38(2), 617–630.
Abstract: Let f be a sensepreserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the preSchwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane.

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.

Hernandez, R., & Martin, M. J. (2022). On the Harmonic Mobius Transformations. J. Geom. Anal., 32(1), 18.
Abstract: It is wellknown that two locally univalent analytic functions have equal Schwarzian derivative if and only if each one of them is a composition of the other with a nonconstant Mobius transformation. The main goal in this paper is to extend this result to the cases when the functions considered are harmonic. That is, we identify completely the transformations that preserve the (harmonic) Schwarzian derivative of locally univalent harmonic functions.

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.
