On the exit of the Gakhov set along the family of Avkhadiev's classes

Professor F. G. Avkhadiev has played a crucial role in the formation of the finite-valence theory for the classes of holomorphic functions with bounded distortion. We call them the Avkhadiev classes, and their elements are called the Avkhadiev functions. In this paper, we have studied the connections of the above classes with the Gakhov set $\mathcal G$ consisting of all holomorphic and locally univalent functions $f$ in the unit disk $\mathbb D$ with (no more than) the unique root of the Gakhov equation in $\mathbb D$. In particular, for the one-parameter series of the Avkhadiev classes constructing on the rays $\alpha\ln f'$, $\alpha\ge0$, where $|f'(\zeta)|\in(e^{-\pi/2},e^{\pi/2})$, $\zeta\in\mathbb D$, and $f''(0)=0$, we have shown that the Gakhov barrier (the exit value of the parameter out of $\mathcal G$) of the given series coinsides with its Avkhadiev barrier (the exit value of the parameter out of the univalence class), and we have found the extremal family of the Avkhadiev functions. This family is characterized by the coincidence of its individual exit value out of $\mathcal G$ and the Gakhov barrier for the whole series.