Abstract:
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.
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