TY - JOUR

T1 - Integral transforms for logharmonic mappings

AU - Arbeláez, Hugo

AU - Bravo, Víctor

AU - Hernández, Rodrigo

AU - Sierra, Willy

AU - Venegas, Osvaldo

N1 - Publisher Copyright:
© 2021, The Author(s).

PY - 2021

Y1 - 2021

N2 - 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 fα(z)=∫0z(f(ζ)/ζ)αdζ or Fα(z)=∫0z(f′(ζ))αdζ appear. In this note we extend the classical problem of finding the values of α∈ C for which either fα or Fα are univalent, whenever f belongs to some subclasses of univalent mappings in 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.

AB - 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 fα(z)=∫0z(f(ζ)/ζ)αdζ or Fα(z)=∫0z(f′(ζ))αdζ appear. In this note we extend the classical problem of finding the values of α∈ C for which either fα or Fα are univalent, whenever f belongs to some subclasses of univalent mappings in 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.

KW - Integral transform

KW - Logharmonic mappings

KW - Shear construction

KW - Univalent mappings

UR - http://www.scopus.com/inward/record.url?scp=85102167191&partnerID=8YFLogxK

U2 - 10.1186/s13660-021-02578-y

DO - 10.1186/s13660-021-02578-y

M3 - Article

AN - SCOPUS:85102167191

VL - 2021

JO - Journal of Inequalities and Applications

JF - Journal of Inequalities and Applications

SN - 1025-5834

IS - 1

M1 - 48

ER -