Year: 2020
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3092835157