This article is cited in 10 papers

On the exit out of the Gakhov set controlled by the subordination conditions

A. V. Kazantsev

Institute of Computer Mathematics and Information Technologies, Kazan (Volga Region) Federal University, Kazan, Russia

Abstract: 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, subordination conditions.

UDC: 517.54

Received: 11.01.2013