Gakhov set in the Merkes theorem on convex combinations

The Merkes theorem deduces the starlikeness of any convex combination $f_\lambda$ of identity mapping and a holomorphic convex function $f$ in the unit disk with $f''(0)=0$. Under the same conditions, all of the functions $f_\lambda$ (except the mappings onto a strip when $\lambda=1$) are proved to belong also to the Gakhov set characterized by the property of (no more than) uniqueness of the root of the Gakhov equation. These results allow for the analogies for the exterior of the unit disk. The behavior of convex combinations is studied on the functions of the Alexander classes. For the exhaustion of each such class by the “level curves”, the “stopping moment” is found which corresponds to the exit out of the Gakhov set, and all of the trajectories of such an exit are described.