Arbelaez, H., Bravo, V., Hernandez, R., Sierra, W., & Venegas, O. (2021). Integral transforms for logharmonic mappings. J. Inequal. Appl., 2021(1), 48.
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(alpha)(z) = integral(z)(0)(f(zeta)/zeta)(alpha)d zeta or Falpha(z) = integral(z)(0)(f '(zeta))(alpha)d zeta appear. In this note we extend the classical problem of finding the values of alpha is an element of C for which either f(alpha) or Falpha 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 SheilSmall in (Clunie and SheilSmall in Ann. Acad. Sci. Fenn., Ser. A I 9:325, 1984) to this new scenario.
