
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.



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.



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. (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.



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.

