Integral transforms for logharmonic mappings

Hugo Arbeláez, Víctor Bravo, Rodrigo Hernández, Willy Sierra, Osvaldo Venegas

Research output: Contribution to journalArticlepeer-review

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 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.

Original languageEnglish
Article number48
JournalJournal of Inequalities and Applications
Volume2021
Issue number1
DOIs
StatePublished - 2021

Keywords

  • Integral transform
  • Logharmonic mappings
  • Shear construction
  • Univalent mappings

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