Leopold Koczan

Domains of Univalence for Typically-real Odd Functions

The univalence problems in the class of typically-real functions were considered by Golusin [G. Golusin (1950). On typically-real functions, (Russian). Mat. Sb. , 27 (69), 201-218]. He proved that the domain of local and global univalence for typically-real functions coincide. In this note we have proved that this is not the case for typically-real odd functions. Moreover, we have observed that for this class there can be infinitely many domains of univalence. We have determined some of them.

Covering problems for functions n-fold symmetric and convex in the direction of the real axis

Abstract Let F denote the class of all functions univalent in the unit disk and convex in the direction of the real axis. In the paper we discuss the functions of the class F which are n -fold symmetric, where n is positive even integer. For the class of such functions we find the Koebe set as well as the covering set, i.e. ⋂ f ∈ F f ( Δ ) and ⋃ f ∈ F f ( Δ ) . Moreover, the Koebe constant and the covering constant are obtained.