Geodesic
On the exit out of the Gakhov set controlled by the subordination conditions
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 1, pp. 31-43

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A Gakhov set $\mathcal G$ is the class of all holomorphic and locally univalent functions in the unit disk, which have no more than one root of the Gakhov equation. For the series of the well-known subclasses of univalent functions having the zero root of the Gakhov equation, an effective desription is given for the set of all trajectories of the exit out of $\mathcal G$; such an exit takes place when the parameter value corresponds to the sharp constant in the appropriate uniqueness condition of the root. It is shown that the exit out of $\mathcal G$ may occur due to the bifurcations of the two following types only: 1) the maximum at zero splits into two maxima and the saddle; 2) the non-zero semisaddle appears and then divides into the maximum and the saddle.
Keywords: hyperbolic derivative, conformal (inner mapping) radius, bifurcations of critical points, Gakhov set, class of starlike functions
Mots-clés : subordination conditions. 
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     author = {A. V. Kazantsev},
     title = {On the exit out of the {Gakhov} set controlled by the subordination conditions},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {31--43},
     publisher = {mathdoc},
     volume = {156},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2014_156_1_a3/}
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A. V. Kazantsev. On the exit out of the Gakhov set controlled by the subordination conditions. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 1, pp. 31-43. http://geodesic.mathdoc.fr/item/UZKU_2014_156_1_a3/