On the exit of the Gakhov set along the family of Avkhadiev's classes
Professor F.G. Avkhadiev has played a crucial role in the formation of the finite-valence theory for the classes of holomorphic functions with bounded distortion. We call them the Avkhadiev classes, and their elements are called the Avkhadiev functions. In this paper, we have studied the connections...
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Kazan Federal University
2017-09-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| author | A.V. Kazantsev |
| author_facet | A.V. Kazantsev |
| author_sort | A.V. Kazantsev |
| collection | DOAJ |
| description | Professor F.G. Avkhadiev has played a crucial role in the formation of the finite-valence theory for the classes of holomorphic functions with bounded distortion. We call them the Avkhadiev classes, and their elements are called the Avkhadiev functions. In this paper, we have studied the connections of the above classes with the Gakhov set G consisting of all holomorphic and locally univalent functions f in the unit disk D with (no more than) the unique root of the Gakhov equation in D. In particular, for the one-parameter series of the Avkhadiev classes constructing on the rays α ln f', α ≥ 0, where |f'(ς)| ͼ (e–π/2, eπ/2), ς ͼ D, and f''(0)=0, we have shown that the Gakhov barrier (the exit value of the parameter out of G) of the given series coincides with its Avkhadiev barrier (the exit value of the parameter out of the univalence class), and we have found the extremal family of the Avkhadiev functions. This family is characterized by the coincidence of its individual exit value out of G and the Gakhov barrier for the whole series. |
| format | Article |
| id | doaj-art-79ddd64b813c4f2b921bae7e2509b514 |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2017-09-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-79ddd64b813c4f2b921bae7e2509b5142025-01-02T20:31:36ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982017-09-011593318326On the exit of the Gakhov set along the family of Avkhadiev's classesA.V. Kazantsev 0Kazan Federal University, Kazan, 420008 RussiaProfessor F.G. Avkhadiev has played a crucial role in the formation of the finite-valence theory for the classes of holomorphic functions with bounded distortion. We call them the Avkhadiev classes, and their elements are called the Avkhadiev functions. In this paper, we have studied the connections of the above classes with the Gakhov set G consisting of all holomorphic and locally univalent functions f in the unit disk D with (no more than) the unique root of the Gakhov equation in D. In particular, for the one-parameter series of the Avkhadiev classes constructing on the rays α ln f', α ≥ 0, where |f'(ς)| ͼ (e–π/2, eπ/2), ς ͼ D, and f''(0)=0, we have shown that the Gakhov barrier (the exit value of the parameter out of G) of the given series coincides with its Avkhadiev barrier (the exit value of the parameter out of the univalence class), and we have found the extremal family of the Avkhadiev functions. This family is characterized by the coincidence of its individual exit value out of G and the Gakhov barrier for the whole series.https://kpfu.ru/on-the-exit-of-the-gakhov-set-along-the-family-of_332923.htmlgakhov setgakhov equationgakhov widthinner mapping (conformal) radiushyperbolic derivativeadmissible functionalavkhadiev classesgakhov barrieravkhadiev barrier |
| spellingShingle | A.V. Kazantsev On the exit of the Gakhov set along the family of Avkhadiev's classes Учёные записки Казанского университета: Серия Физико-математические науки gakhov set gakhov equation gakhov width inner mapping (conformal) radius hyperbolic derivative admissible functional avkhadiev classes gakhov barrier avkhadiev barrier |
| title | On the exit of the Gakhov set along the family of Avkhadiev's classes |
| title_full | On the exit of the Gakhov set along the family of Avkhadiev's classes |
| title_fullStr | On the exit of the Gakhov set along the family of Avkhadiev's classes |
| title_full_unstemmed | On the exit of the Gakhov set along the family of Avkhadiev's classes |
| title_short | On the exit of the Gakhov set along the family of Avkhadiev's classes |
| title_sort | on the exit of the gakhov set along the family of avkhadiev s classes |
| topic | gakhov set gakhov equation gakhov width inner mapping (conformal) radius hyperbolic derivative admissible functional avkhadiev classes gakhov barrier avkhadiev barrier |
| url | https://kpfu.ru/on-the-exit-of-the-gakhov-set-along-the-family-of_332923.html |
| work_keys_str_mv | AT avkazantsev ontheexitofthegakhovsetalongthefamilyofavkhadievsclasses |