arXiv subjects

Hugo Arbeláez

Publications and source records attributed to Hugo Arbeláez.

6 recordsLinked to original sources

arXiv2026

Omitted values for some subclasses of univalent mappings

We study the range of $\operatorname{Re}\{a_2 f(z)\}$ for normalized analytic functions $f$ in the unit disk belonging to several classes of conformal mappings. As our main contribution, we introduce the class $CC_α$ of completely convex mappings of order $α$, defined by a uniform two-point starlikeness condition, and we estimate the range of $\operatorname{Re}\{a_2f(z)\}$ in terms of $α$, for all $f\in CC_α$ and $z\in \mathbb{D}$, generalizing the classical result of Fournier--Ma--Ruscheweyh, which is recovered for $α=0$. We also determine omitted value sets for convex functions of order $α$, spherically convex mappings, uniformly starlike functions, and Nehari classes $\mathcal{N}_t$. The proofs rely primarily on the Schwarz--Pick lemma applied to auxiliary functions constructed from the two-point kernel $zf'(z)/(f(z)-f(x))$.

math.CV↗

arXiv2024

Properties of Besov and $Q_p$ spaces in terms of the Schwarzian derivative of harmonic mappings

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 $\mathcal{QT}_p$, based on the Jacobian operator, and begin a study of these.

math.CV↗

arXiv2024

On an Internal Characterization of Horocyclically Convex Domains in the Unit Disk

A proper subdomain $G$ of the unit disk $\mathbb{D}$ is horocyclically convex (horo-convex) if, for every $ω\in \mathbb{D}\cap \partial G$, there exists a horodisk $H$ such that $ω\in \partial H$ and $G\cap H=\emptyset$. In this paper we give an internal characterization of these domains, namely, that $G$ is horo-convex if and only if any two points can be joined inside $G$ by a $C^1$ curve composed with finitely many Jordan arcs with hyperbolic curvature in $(-2,2)$. We also give a lower bound for the hyperbolic metric of horo-convex regions and some consequences.

math.CV↗

arXiv2019

Level sets of the Hyperbolic Derivative for analytic self-maps of the unit disk

Let the function $φ$ be holomorphic in the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$ and let $φ(\mathbb{D})\subset \mathbb{D}$. We study the level sets and the critical points of the hyperbolic derivative of $φ$, $$|D_φ(z)|:=\frac{(1-|z|^2)|φ'(z)|}{1-|φ(z)|^2}.$$ In particular, we show how the Schwarzian derivative of $φ$ reveals the nature of the critical points.

math.CV↗

arXiv2019

A new approach for the univalence of certain integral of harmonic mappings

The principal goal of this paper is to extend the classical problem of find the values of $α\in \C$ for which the mappings, either $F_α(z)=\int_0^z(f(ζ)/ζ)^αdζ$ or $f_α(z)=\int_0^z(f'(ζ))^αdζ$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\D$, but in the case of harmonic mappings, considering the \textit{shear construction} introduced by Clunie and Sheil-Small in \cite{CSS}.

math.CV↗

arXiv2018

Normal harmonic mappings

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.

math.CV↗