arXiv subjects

O. Venegas

Publications and source records attributed to O. Venegas.

4 recordsLinked to original sources

arXiv2023

A Characterization of concave mappings

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 domain is either bounded or unbounded. In the case of a bounded convex domain, we investigate the pivotal role played by the Schwarzian derivative and the order of the functions in understanding the behavior and properties of these mappings.

math.CV↗

arXiv2020

Integral transform for Logharmonic mappings

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 $f_α(z)=\int_0^z(f(ζ)/ζ)^αdζ$ or $F_α(z)=\int_0^z(f'(ζ))^αdζ$ appear. In this notes we extend the classical problem of finding the values of $α\in\mathbb{C}$ for which either $f_α$ or $F_α$ are univalent, whenever $f$ belongs to some subclasses of univalent mappings in $\mathbb D$, to the case of logharmonic mappings, by considering the extension of the \textit{shear construction} introduced by Clunie and Sheil-Small in \cite{CSS} to this new scenario.

math.CV↗

arXiv2020

Pre-Schwarzian derivative for Logharmonic mapppings

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 the pre-Schwarzian derivative is holomorphic.

math.CV↗

arXiv2016

On harmonic Bloch-type mappings

Let $f$ be a complex-valued harmonic mapping defined in the unit disk $\mathbb D$. We introduce the following notion: we say that $f$ is a Bloch-type function if its Jacobian satisfies $$ \sup_{z\in\mathbb D}(1-|z|^2)\sqrt{|J_f(z)|}<\infty. $$ 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 coefficients of functions in this class. We establish an analogue of the theorem which states that an analytic $φ$ is Bloch if and only if there exists $c>0$ and a univalent $ψ$ such that $φ= c \log ψ'$.

math.CV↗