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.

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., & 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., & 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.

Hernandez, R. R. (2006). Schwarzian derivatives and a linearly invariant family in c(n). Pac. J. Math., 228(2), 201–218.
Abstract: We use Oda's definition of the Schwarzian derivative for locally univalent holomorphic maps F in several complex variables to define a Schwarzian derivative operator phi F. We use the Bergman metric to define a norm 11,vertical bar vertical bar phi F vertical bar vertical bar for this operator, which in the ball is invariant under composition with automorphisms. We study the linearly invariant family Falpha = {F : Bn > Cn vertical bar F(0) = 0, DF(0) = Id, vertical bar vertical bar phi F vertical bar vertical bar <= alpha}, estimating its order and norm order.
