Rodrigo Hernández

Rodrigo Hernández
Universidad Adolfo Ibáñez · Facultad de Ingeniería y Ciencias

PhD
I am working in Geometric Function Theory in SCV, Harmonic Mappings, and Holomorphic Functions

About

52
Publications
7,309
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307
Citations
Citations since 2017
31 Research Items
230 Citations
201720182019202020212022202301020304050
201720182019202020212022202301020304050
201720182019202020212022202301020304050
201720182019202020212022202301020304050
Introduction
Mathematical interest are: Geometric function theory, Complex anlysis, Planar harmonic mappings, Several Complex variables.
Additional affiliations
January 2015 - December 2017
Bok Editores
Position
  • Managing Director
Description
  • Editorial
January 2008 - present
u-planner
Position
  • Academic VP
Description
  • High Performance in Higher Education Software based on our own mathematical algorithms and AI for data analysis for Higher Education Institutions.
March 2005 - present
Universidad Adolfo Ibáñez
Position
  • Professor (Associate)
Education
March 2000 - December 2004
March 1995 - June 1999

Publications

Publications (52)
Article
Full-text available
The family F_λ of orientation preserving convex harmonic functions f = h + \overline{g} in the unit disk D (normalized in the standard way) that satisfy that for some λ ∈ ∂D, h (z) + g (z) = 1 (1 + λz)(1 + λz) , z ∈ D , and their rotations play an important role among those functions that are harmonic, preserve the orientation, and map the unit dis...
Preprint
Full-text available
A pre-Schwarzian and a Schwarzian derivative for locally univalent pluriharmonic mappings in ${\mathbb C}^n$ are introduced. Basic properties such as the chain rule, multiplicative invariance and affine invariance are proved for these operators. It is shown that the pre-Schwarzian is stable only with respect to rotations of the identity. A characte...
Preprint
Full-text available
The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order \(alpha\) in terms of the value of \(f''(0)\), in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita...
Preprint
Full-text available
The main purpose of this paper is to obtain sharp bounds of the norm of Schwarzian derivative for convex mappings of order $alpha$ in terms of the value of $f''(0)$, in particular, when this quantity is equal to zero. In addition, we obtain sharp bounds for distortion and growth for this mappings and we generalized the results obtained by Suita and...
Article
Full-text available
This paper shows the results of an experiment applied to 170 students from two Chilean universities who solve a task about reading a graph of an affine function in an online assessment environment where the parameters (coefficients of the graphed affine function) are randomly defined from an ad-hoc algorithm, with automatic correction and automatic...
Chapter
Full-text available
La teoría de funciones univalentes es un tópico clásico dentro de la teoría geométrica de funciones, la cual da cuenta de la relación entre aspectos geométricos y analíticos de ellas. Desde los inicios del siglo pasado y siendo en la década de los sesenta un campo muy prolífico para muchos matemáticos. El tema principal de este cursillo es introduc...
Preprint
Full-text available
We study classes of locally biholomorphic mappings defined in the $\P$ that have bounded Schwarzian operator in the Bergman metric. We establish important properties of specific solutions of the associated system of differential equations and show a geometric connection between the order of the classes and a covering property. We show for modified...
Preprint
Full-text available
We establish two-point distortion theorems for sense-preserving planar harmonic mappings $f=h+\overline{g}$ which satisfies the univalence criteria in the unit disc such that, Becker's and Nehari`s harmonic version. In addition, we find the sharp two-point distortion theorem when $h$ is a convex function, and normalized mappings such that $h(\D)$ i...
Presentation
Full-text available
Motivation Funciones armónicas complejas Pre-Schwarziana y Schwarziana para f = h + g. Teoremas "Two-point distortion" para funciones armónicas complejas Rodrigo Hernández Universidad Adolfo Ibáñez, Viña del Mar, Chile Trabajo junto con V. Bravo y O. Venegas July 6, 2022 Rodrigo Hernández Two-point distortion theorems Motivation Funciones armónicas...
Conference Paper
Full-text available
Consideremos f una función analítica univalente en el disco unidad D. Por ejemplo f una función de Möbius, satisface que $|f(a) - f(b)| = |a - b|\sqrt{|f'(a)||f'(b)|}$ Nos preguntamos qué tan grande es |f(a) - f(b)| para $a$ y $b$ en el disco unitario, pero en términos de la distancia euclidiana o hiperbólica.
Article
Full-text available
We introduce definitions of pre-Schwarzian 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 pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the...
Article
Full-text available
By using the using the Loewner Chain Theory, we obtain a new criterion of univalence in C n in terms of the Schwarzian derivative introduced in [3] by using the arguments in [8]. We as well derive explicitly the formula given in [3] by relating the Schwarzian derivative with the numerical method of approximation of zeros due to Halley. Mathematics...
Article
Full-text available
It is well-known 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 non-constant Möbius 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 tra...
Article
In this paper, we define both the upper and lower order of a sense-preserving 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 equ...
Preprint
Full-text available
This paper shows the results of an experiment applied to 170 students from two Chilean universities who solve a task about reading a graph of an affine function in an online assessment environment where the parameters (coefficients of the graphed affine function) are randomly defined from an ad-hoc algorithm, with automatic correction and automatic...
Preprint
Full-text available
We provide two new formulas for quasiconformal extension to $\overline{\mathbb{C}}$ for harmonic mappings defined in the unit disk and having sufficiently small Schwarzian derivative. Both are generalizations of the Ahlfors-Weill extension for holomorphic functions.
Article
Full-text available
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 α (z) = z 0 (f (ζ)/ζ) α dζ or F α (z) = z 0 (f ′ (ζ)) α...
Article
A pre-Schwarzian 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 pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is g...
Preprint
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 $f_\alpha(z)=\int_0^z(f(\zeta)/\zeta)^\alpha d\zeta$ or $F...
Conference Paper
Full-text available
La Teoría Geométrica de Funciones intenta dar una explicación analítica de ciertos aspectos geométricos de las funciones tales como, crecimiento, distorsión, univalencia, convexidad, etc. En esta charla abordaremos algunos de estos aspectos en funciones analíticas, harmónicas y algo más.
Preprint
Full-text available
We introduce a new definition of pre-Schwarzian derivative 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 pre-Schwarzian is stable only with respect to rotations of the identity. A characterization is given for the case when t...
Preprint
Full-text available
We introduce a new definition of pre-Schwarzian derivative 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 pre-Schwarzain is stable only with respect to rotations of the identity. A characterization is given for the case when t...
Article
Full-text available
The main purpose of this paper is to study the concept of normal function in the context of harmonic mappings from the unit disk $\mathbb{D}$ to the complex plane. In particular, we obtain necessary conditions for that a function $f$ to be normal.
Preprint
Full-text available
The principal goal of this paper is to extend the classical problem of find the values of $\alpha\in \C$ for which the mappings, either $F_\alpha(z)=\int_0^z(f(\zeta)/\zeta)^\alpha d\zeta$ or $f_\alpha(z)=\int_0^z(f'(\zeta))^\alpha d\zeta$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\D$, but in the case of harmo...
Article
Full-text available
The principal goal of this paper is to extend the classical problem of find the values of α ∈ C for which the mappings, either F α (z) = z 0 (f (ζ)/ζ) α dζ or f α (z) = z 0 (f (ζ)) α dζ are univalent, whenever f belongs to some subclasses of univalent mappings in D, but in the case of harmonic mappings, considering the shear construction introduced...
Article
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Let f be a complex-valued harmonic mapping defined in the unit disc D\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {D}}$$\end{document}. The theorems of Chua...
Conference Paper
Full-text available
The family of (normalized) complex-valued harmonic mappings in the unit disk that map D onto a convex domain is much wider than its analytic counterpart. Many fundamental questions related to its structure (for instance, the sharp version of the growth theorem) are still unresolved. Perhaps this lack of knowledge is somehow motivated by the fact th...
Article
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We characterize in various ways the weighted composition transformations which preserve the class P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {P}$$\end{do...
Article
Full-text available
It is well-known that two locally univalent analytic functions ϕ and ψ have equal Schwarzian derivative if and only if there exists a non-constant Möbius transformation T such that ϕ = T \circ ψ. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schwarzian derivative.
Article
It is well-known that two locally univalent analytic functions $\varphi$ and $\psi$ have equal Schwarzian derivative if and only if there exists a non-constant M\"obius transformation $T$ such that $\varphi=T\circ \psi$. In this paper, we identify completely the relationship between two locally univalent harmonic mappings with equal (harmonic) Schw...
Article
Full-text available
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 Sheil-Small, 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 \(\mathcal {S}_H\) of u...
Article
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We generalize the problem of univalence of the integral of f ′ (z) α when f is univalent to the complex harmonic mappings. To do this, we extend the univalence criterion by Ahlfors in [1] to those mappings.
Article
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Let f be a complex-valued harmonic mapping defined in the unit disk . We introduce the following notion: we say that f is a Bloch-type function if its Jacobian satisfies This gives rise to a new class of functions which generalizes and contains the well-known analytic Bloch space. We give estimates for the schlicht radius, the growth and the coeff...
Article
Full-text available
We give a generalization of Ahlfors quasiconformal criterion in terms of pre-Schwarzian derivative for sense-preserving harmonic mappings and we use that extend the problem of the univalence of $\int_0^z(f'(\zeta�))^\alpha\;�d\zeta$� for sense-preserving harmonic mapping $f = h + \overline{g}$.
Article
Full-text available
We derive an Ahlfors-Weill type extension for a class of holomorphic mappings defined in the ball script 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...
Article
Full-text available
We prove that if the Schwarzian norm of a given complex-valued locally univalent harmonic mapping $f$ in the unit disk is small enough, then $f$ is, indeed, globally univalent and can be extended to a quasiconformal mapping in the extended complex plane.
Article
Full-text available
In this paper we obtain certain sufficient conditions for the univalence of pluriharmonic mappings defined in the unit ball \(\mathbb{B}^n \) of ℂ n . The results are generalizations of conditions of Chuaqui and Hernández that relate the univalence of planar harmonic mappings with linearly connected domains, and show how such domains can play a rol...
Article
Full-text available
Given any sense preserving harmonic mapping f=h+ḡ in the unit disk, we prove that for all |λ|=1 the functions fλ=h+λḡ are univalent (resp. close-to-convex, starlike, or convex) if and only if the analytic functions Fλ=h+λg are univalent (resp. close-to-convex, starlike, or convex) for all such λ. We also obtain certain necessary geometric condition...
Article
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Let f be a sense-preserving harmonic mapping in the unit disk. We give a sufficient condition in terms of the pre-Schwarzian derivative of f to ensure that it can be extended to a quasiconformal map in the complex plane.
Article
Full-text available
In this paper we introduce a definition of the pre-Schwarzian 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 analogou...
Article
Full-text available
We study the (trace) norm of a linearly invariant family in the ball in $\C$. By adapting an approach that in one variable yields optimal results, we are able to derive an upper bound for the norm of the family in terms of the Schwarzian norm and the dimension $n$.
Article
Full-text available
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(1<|z|) log 1/1|z| |f' (z)| <∞, and the analytic Besov spaces Bp,1 ≤ p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention t...
Conference Paper
Full-text available
Resumen En este trabajo damos una definicion de derivada schwarziana para funciones armonicas que preservan la orientacion y sin ninguna hipotesis adicional sobre la dilatacion compleja w. La manera de derivar esta fórmula esta basada en el hecho de que debe caracterizar a las transformaciones de Möbius armónicas, usando la idea de la mejor aproxim...
Conference Paper
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Article
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We solve the several complex variables preSchwarzian operator equation $[Df(z)]^{-1}D^2f(z)=A(z)$, $z\in \C^n$, where $A(z)$ is a bilinear operator and $f$ is a $\C^n$ valued locally biholomorphic function on a domain in $\C^n$. Then one can define a several variables $f\to f_\alpha$ transform via the operator equation $[Df_\alpha(z)]^{-1}D^2f_\alp...
Article
Full-text available
We study a sufficient condition for univalence in the polydisk in terms of the size of the norm of the Schwarzian operator. Examples show that our result is close to optimal in dimension two. This paper extends work by the second author concerning similar criteria in the ball.
Conference Paper
Full-text available
Article
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Oda gave a definition for Schwarzian derivatives in several complex variables [Oda, T., 1974, On Schwarzian derivatives in several variables (in Japanese). Kokyuroku Research Institute for Mathematical Sciences, Kioto University, 226, 82–85.], which we used in [Hernández, R., Schwarzian derivatives and linear invariant families in . Journal of Math...
Article
Full-text available
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...
Article
Full-text available
We investigate the relationship between the univalence of f and of h in the decomposition f = h + (g) over bar of a serise-preserving 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...

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Projects

Projects (5)
Project
Determinar las cotas precisas para la derivada schwarziana para mapeos alpha-convexos.
Project
Extended the classical results in geometric function theory of one complex variable a several complex variables