Integral transforms for logharmonic mappings
Integral transforms for logharmonic mappings
Authors
Arbelaez, Hugo
Bravo, Victor
Hernandez, Rodrigo
Sierra, Willy
Venegas, Osvaldo
Bravo, Victor
Hernandez, Rodrigo
Sierra, Willy
Venegas, Osvaldo
Authors
Date
Datos de publicaciĆ³n:
10.1186/s13660-021-02578-y
Keywords
Collections
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 F-alpha(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 F-alpha 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.