The Gakhov barriers and extremals for the level lines

Abstract

The regular Gakhov class G1 consists of all holomorphic and locally univalent functions f in the unit disk with only one root of the Gakhov equation, which is the maximum of the hyperbolic derivative (conformal radius) of the function f . For the classes H defined by the conditions of Nehari and Becker’s type, as well as by some other inequalities, we have solved the problem of calculation of the Gakhov barrier, i.e., the value ρ(H) = sup{r ≥ 0 : Hr ⊂G1}, where Hr = {fr : f ∈H}, 0 ≤ r ≤ 1, and of an effective description of the Gakhov extremal, i.e., the set of f ’s in H with the level sets fr leaving G1 when r passes through ρ(H). Both possible variants of bifurcation, which provide an exit out of G1 along the level lines, are represented.

Keywords

• gakhov equation
• gakhov set
• hyperbolic derivative
• inner mapping (conformal) radius
• gakhov width
• gakhov barrier
• gakhov extremal