The Gakhov barriers and extremals for the level lines
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 Bec...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2018-12-01
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| Series: | Учёные записки Казанского университета. Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/the-gakhov-barriers-and-extremals-for-the-level-403967.html |
| _version_ | 1811233164105875456 |
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| author | A.V. Kazantsev |
| author_facet | A.V. Kazantsev |
| author_sort | A.V. Kazantsev |
| collection | DOAJ |
| description | 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 : Hr ⊂G1}, where Hr = {fr : 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 fr 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. |
| first_indexed | 2024-04-12T11:15:59Z |
| format | Article |
| id | doaj.art-9bf30c14126c4c4ca24c23687ef300af |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2024-04-12T11:15:59Z |
| publishDate | 2018-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета. Серия Физико-математические науки |
| spelling | doaj.art-9bf30c14126c4c4ca24c23687ef300af2022-12-22T03:35:29ZengKazan Federal UniversityУчёные записки Казанского университета. Серия Физико-математические науки2541-77462500-21982018-12-011604750761The Gakhov barriers and extremals for the level linesA.V. Kazantsev0Kazan Federal University, Kazan, 420008 RussiaThe 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 : Hr ⊂G1}, where Hr = {fr : 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 fr 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.https://kpfu.ru/the-gakhov-barriers-and-extremals-for-the-level-403967.htmlgakhov equationgakhov sethyperbolic derivativeinner mapping (conformal) radiusgakhov widthgakhov barriergakhov extremal |
| spellingShingle | A.V. Kazantsev The Gakhov barriers and extremals for the level lines Учёные записки Казанского университета. Серия Физико-математические науки gakhov equation gakhov set hyperbolic derivative inner mapping (conformal) radius gakhov width gakhov barrier gakhov extremal |
| title | The Gakhov barriers and extremals for the level lines |
| title_full | The Gakhov barriers and extremals for the level lines |
| title_fullStr | The Gakhov barriers and extremals for the level lines |
| title_full_unstemmed | The Gakhov barriers and extremals for the level lines |
| title_short | The Gakhov barriers and extremals for the level lines |
| title_sort | gakhov barriers and extremals for the level lines |
| topic | gakhov equation gakhov set hyperbolic derivative inner mapping (conformal) radius gakhov width gakhov barrier gakhov extremal |
| url | https://kpfu.ru/the-gakhov-barriers-and-extremals-for-the-level-403967.html |
| work_keys_str_mv | AT avkazantsev thegakhovbarriersandextremalsforthelevellines AT avkazantsev gakhovbarriersandextremalsforthelevellines |