The Gakhov barriers and extremals for the level lines

A. V. Kazantsev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 160 (2018) no. 4, pp. 750-761

Résumé

The regular Gakhov class $\mathcal{G}_1$ 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 $\mathcal{H}$ 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 $\rho(\mathcal{H}) = \sup \{r\ge 0: \mathcal{H}_r\subset \mathcal{G}_1\}$, where $\mathcal{H}_r = \{f_r: f\in \mathcal{H}\}$, $0\le r\le 1$, and of an effective description of the Gakhov extremal, i.e., the set of $f$'s in $\mathcal{H}$ with the level sets $f_r$ leaving $\mathcal{G}_1$ when $r$ passes through $\rho(\mathcal{H})$. Both possible variants of bifurcation, which provide an exit out of $\mathcal{G}_1$ along the level lines, are represented.

Détail

Keywords: Gakhov equation, Gakhov set, hyperbolic derivative, inner mapping (conformal) radius, Gakhov width, Gakhov barrier, Gakhov extremal.