On the Gakhov equation for the Biernacki operator

The paper establishes the region in the parameter plane such that the image of any starlike function with the zero root of the Gakhov equation under the mapping by the Biernacki operator corresponding to the parameter of this region is found in the Gakhov class consisting of the functions, for which this root is unique. It is demonstrated that the above region cannot be extended without loss of the uniqueness of the root for the image of at least one starlike function. For the image of the whole class of starlike functions having the zero root, the Gakhov width is calculated and the effective description is given for the set of trajectories of the exit out of the Gakhov class along the level lines.