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.

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.

Arevalo, I., Hernandez, R., Martin, M. J., & Vukotic, D. (2018). On weighted compositions preserving the Caratheodory class. Mon.heft. Math., 187(3), 459–477.
Abstract: We characterize in various ways the weighted composition transformations which preserve the class P of normalized analytic functions in the disk with positive real part. We analyze the meaning of the criteria obtained for various special cases of symbols and identify the fixed points of such transformations.

Bravo, V., Hernandez, R., Ponnusamy, S., & Venegas, O. (2022). PreSchwarzian and Schwarzian derivatives of logharmonic mappings. Monatsh. fur Math., Early Access.
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.

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.

Chuaqui, M., & Hernandez, R. (2013). The order of a linearly invariant family in Cn. J. Math. Anal. Appl., 398(1), 372–379.
Abstract: We study the (trace) order of the linearly invariant family in the ball Bn defined by parallel to SF parallel to <= alpha, where F : Bn > Cn is locally biholomorphic and SF is the Schwarzian operator. By adapting Pommerenke's approach, we establish a characteristic equation for the extremal mapping that yields an upper bound for the order of the family in terms of alpha and the dimension n. Lower bounds for the order are established in similar terms by means of examples. (C) 2012 Elsevier Inc. All rights reserved.

Chuaqui, M., & Hernandez, R. (2015). AhlforsWeill extensions in several complex variables. J. Reine Angew. Math., 698, 161–179.
Abstract: We derive an AhlforsWeill type extension for a class of holomorphic mappings defined in the ball Bn, generalizing the formula for Nehari mappings in the disk. The class of mappings holomorphic in the ball is defined in terms of the Schwarzian operator. Convexity relative to the Bergman metric plays an essential role, as well as the concept of a weakly linearly convex domain. The extension outside the ball takes values in the projective dual to Cn, that is, in the set of complex hyperplanes.

Chuaqui, M., Hamada, H., Hernandez, R., & Kohr, G. (2014). Pluriharmonic mappings and linearly connected domains in Cn. Isr. J. Math., 200(1), 489–506.
Abstract: In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball of Cn . The results are generalizations of conditions of Chuaqui and Hernandez that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a role in questions regarding injectivity in higher dimensions. In addition, we extend recent work of Hernandez and Martin on a shear type construction for planar harmonic mappings, by adapting the concept of stable univalence to pluriharmonic mappings of the unit ball into Cn .

Chuaqui, M., Hernandez, R., & Martin, M. J. (2017). Affine and linear invariant families of harmonic mappings. Math. Ann., 367(34), 1099–1122.
Abstract: We study the order of affine and linear invariant families of planar harmonic mappings in the unit disk. By using the famous shear construction of Clunie and SheilSmall, we construct a function to determine the order of the family of mappings with bounded Schwarzian norm. The result shows that finding the order of the class SH of univalent harmonic mappings can be formulated as a question about Schwarzian norm and, in particular, our result shows consistency between the conjectured order of SH and the Schwarzian norm of the harmonic Koebe function.

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., 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.

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.

Galanopoulos, P., Girela, D., & Hernandez, R. (2011). Univalent Functions, VMOA and Related Spaces. J. Geom. Anal., 21(3), 665–682.
Abstract: This paper is concerned mainly with the logarithmic Bloch space B(log) which consists of those functions f which are analytic in the unit disc D and satisfy sup(z<1)(1z) log 1/1z f' (z)<infinity, and the analytic Besov spaces Bp, 1 <= p < infinity. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of: A bounded univalent function in U(p>1) B(P) Bp but not in the logarithmic Bloch space. A bounded univalent function in B(log) but not in any of the Besov spaces B(p) with p < 2. We also prove that the situation changes for certain subclasses of univalent functions. Namely, we prove that the convex univalent functions in D which belong to any of the spaces B(0), VMOA, B(p) (1 <= p <= infinity), Blog, or some other related spaces are the same, the bounded ones. We also consider the question of when the logarithm of the derivative, log g', of a univalent function g belongs to Besov spaces. We prove that no condition on the growth of the Schwarzian derivative Sg of g guarantees log g' is an element of B(p). On the other hand, we prove that the condition integral(D) (1z(2))(2p2) Sg(z)(p) d A(z)<infinity implies that log g' is an element of B(p) and that this condition is sharp. We also study the question of finding geometric conditions on the image domain g(D) which imply that log g' lies in Bp. First, we observe that the condition of g( D) being a convex Jordan domain does not imply this. On the other hand, we extend results of Pommerenke and Warschawski, obtaining for every p is an element of (1, infinity), a sharp condition on the smoothness of a Jordan curve Gamma which implies that if g is a conformal mapping from D onto the inner domain of Gamma, then log g' is an element of B(p).

Hernandez, R. (2011). Prescribing The Preschwarzian In Several Complex Variables. Ann. Acad. Sci. Fenn. Ser. A1Math., 36(1), 331–340.
Abstract: We solve the several complex variables preSchwarzian operator equation [D f (z)](1) D2 f (z) = A(z), z is an element of Cn, where A(z) is a bilinear operator and f is a Cn valued locally biholomorphic function on a domain in Cn. Then one can define a several variables f > f(alpha) transform via the operator equation [D f(alpha) (z)](1) D2 f(alpha) (z) = alpha[D f (z)](1) D2 f (z), and thereby, study properties of f alpha. This is a natural generalization of the one variable operator f(alpha) (z) in [6] and the study of its univalence properties, e.g., the work of Royster [23] and many others. Mobius invariance and the multivariables Schwarzian derivative operator of Oda [17] play a central role in this work.

Hernandez, R. (2022). A criterion of univalence in Cn in terms of the Schwarzian derivative. Stud. Univ. BabesBolyai Math., 67(2), 421–430.
Abstract: Using the Loewner Chain Theory, we obtain a new criterion of univalence in Cn 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.

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. (2013). Stable geometric properties of analytic and harmonic functions. Math. Proc. Camb. Philos. Soc., 155(2), 343–359.
Abstract: Given any sense preserving harmonic mapping f = h + (g) over bar in the unit disk, we prove that for all vertical bar lambda vertical bar = 1 the functions f(lambda) = h + lambda(g) over bar are univalent (resp. closetoconvex, starlike, or convex) if and only if the analytic functions Flambda = h + lambda g are univalent (resp. closetoconvex, starlike, or convex) for all such lambda. We also obtain certain necessary geometric conditions on h in order that the functions f(lambda) belong to the families mentioned above. In particular, we see that if f(lambda) are univalent for all lambda on the unit circle, then h is univalent.
