The Gakhov barriers and extremals for the level lines

A.V. Kazantsev

Kazan Federal University, Kazan, 420008 Russia

E-mail: avkazantsev63@gmail.com

Received March 22, 2018

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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 : H_r ⊂G₁}, where H_r = {f_r : 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 f_r 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

Acknowledgments. The study was supported by the Russian Foundation for Basic Research and the Government of the Republic of Tatarstan (project no. 18-41-160017).

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For citation: Kazantsev A.V. The Gakhov barriers and extremals for the level lines. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2018, vol. 160, no. 4, pp. 750–761. (In Russian)

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