Integral transforms for logharmonic mappings

• Main Authors: Hugo Arbeláez, Víctor Bravo, Rodrigo Hernández, Willy Sierra, Osvaldo Venegas
• Format: Article
• Language: English
• Published: SpringerOpen 2021-03-01
• Series: Journal of Inequalities and Applications
• Subjects: Integral transform; Logharmonic mappings; Shear construction; Univalent mappings
• Online Access: https://doi.org/10.1186/s13660-021-02578-y

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 ) = ∫ 0 z ( f ( ζ ) / ζ ) α d ζ $f_{\alpha }(z)=\int _{0}^{z}(f(\zeta )/\zeta )^{\alpha }\,d\zeta $ or F α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ $F_{\alpha }(z)=\int _{0}^{z}(f'(\zeta ))^{\alpha }\,d\zeta $ appear. In this note we extend the classical problem of finding the values of α ∈ C $\alpha \in \mathbb{C}$ for which either f α $f_{\alpha }$ or F α $F_{\alpha }$ are univalent, whenever f belongs to some subclasses of univalent mappings in D $\mathbb{D}$ , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario.