Hohlov Effects for Pre-Schwarzian Derivatives of Functions in the Gakhov Class

Abstract

We find an example of a situation when the exit from Gakhov’s class along some parametrical family of functions is connected with boundary bifurcation of the Gakhov equation. The corresponding condition of hit in Gakhov’s class is described by the construction of the Goryainov-Hohlov type, i.e. this is a subordination condition where the majorant itself is defined by (another) subordination. Next, we introduce and study a new concept of sharpness in the conditions of belonging to Gakhov’s class in the form of subordination of pre-Schwarzian derivatives to starlike functions; this concept is based on the Novikov-Hohlov’s effect in the inverse problems for the potentials and for the analytic functions. Finally, we study the Gakhov equation for the Biernacki-Hohlov operator.