# Rodrigo HernándezUniversidad Adolfo Ibáñez · Facultad de Ingeniería y Ciencias

Rodrigo Hernández

PhD

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

## About

62

Publications

9,193

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386

Citations

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

Education

March 2000 - December 2004

March 1995 - June 1999

## Publications

Publications (62)

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...

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...

The purpose of this paper is to establish new characterizations of concave functions f defined in 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{d...

In this paper we give a characterization of $\log J_f$ belongs to $\widetilde{\mathcal{B}}_p$ or $\widetilde{\mathcal{Q}}_p$ spaces for any locally univalent sense-preserving harmonic mappings $f$ defined in the unit disk, using the Schwarzian derivative of $f$ and Carleson meseaure. In addition, we introduce the classes $\mathcal{BT}_p$ and $\math...

We study the convolution of functions of the form
\[
f_\alpha (z) := \dfrac{\left( \frac{1 + z}{1 - z} \right)^\alpha - 1}{2 \alpha},
\]
which map the open unit disk of the complex plane onto polygons of 2 edges when $\alpha\in(0,1)$. Inspired by a work of Cima, we study the limits of convolutions of finitely many $f_\al$ and the convolution of arb...

En este manuscrito medimos los estilos de pensamiento matemático, visual, analítico e integrado, los estilos de pensamiento interno y externo, las creencias en matemáticas aplicado, teórico, formal, competitivo y rígido, como también la auto percepción de eficacia en tareas matemáticas de la vida cotidiana, en estudiantes de primer año de una unive...

It is well-known that lens maps are convex mappings defined in the unit disc to itself. In this brief note, we show that these mappings are convex of order α > 0, and starlike of order β > 0, and establish the precise orders in terms of opening angle of the lens map.

We study the convolution of functions of the form $f_α(z) := \frac{((1+z)/(1−z))^α − 1}{2α} , which map the open unit disk of the complex plane onto polygons of 2 edges when α ∈ (0, 1). We extend results by Cima by studying limits of convolutions of finitely many fα and by considering the convolution of arbitrary unbounded convex mappings. The anal...

This study focuses on Concave mappings, a class of univalent functions that exhibit a unique property: they map the unit disk onto a domain whose complement is convex. The main objective of this work is to characterize these mappings in terms of the real part of the expression $1 +zf''(z)/f'(z)$, considering scenarios where the omitted convex domai...

We study classes of locally biholomorphic mappings defined in the polydisk $ℙ^n$
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 fo...

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 Ahlfors-Weill extension for holomorphic functions.

We establish two-point distortion theorems for sense-preserving planar harmonic mappings \(f=h+\overline{g}\) in the unit disk \({\mathbb D}\) which satisfy harmonic versions of the univalence criteria due to Becker and Nehari. In addition, we also find two-point distortion theorems for the cases when h is a normalized convex function and, more gen...

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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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.

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...

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...

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...

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.

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...

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...

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.

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...

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...

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...

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...

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...

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.

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...

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...

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.

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...

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 $f = h + \overline{g}$.

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...

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.

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...

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...

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.

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...

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 $n$.

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...

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...

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...

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.

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...

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...

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...