Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian
It was established that if the left-hand side of the Gakhov equation is bounded by the constant 2, then this equation has exactly one root in the unit disk, where the constant is sharp and the root is not necessarily zero. We revealed two aspects arising with regard to this connection. The first asp...
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Kazan Federal University
2019-12-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
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| Online Access: | https://kpfu.ru/uz-eng-phm-2019-4-4.html |
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| author | A.V. Kazantsev |
| author_facet | A.V. Kazantsev |
| author_sort | A.V. Kazantsev |
| collection | DOAJ |
| description | It was established that if the left-hand side of the Gakhov equation is bounded by the constant 2, then this equation has exactly one root in the unit disk, where the constant is sharp and the root is not necessarily zero. We revealed two aspects arising with regard to this connection. The first aspect concerns the pre-Schwarzian immersion of the Gakhov class into the space of bounded holomorphic functions. It was shown that the width of this immersion is equal to 2; the full description was done for the intersection of the boundary of the immersion with the ball of the radius 2 centered at the origin. The second aspect is connected with maintenance of the uniqueness of the root when the linear or fractional linear actions on the pre-Schwarzian with multiplying by the unit disk variable are bounded. Some uniqueness conditions were constructed in the form of S.N. Kudryashov’s univalence criteria. |
| format | Article |
| id | doaj-art-f655395608fa4280ab8f7f9b6742f95e |
| institution | Kabale University |
| issn | 2541-7746 2500-2198 |
| language | English |
| publishDate | 2019-12-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj-art-f655395608fa4280ab8f7f9b6742f95e2025-01-02T17:56:26ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982019-12-01161452653510.26907/2541-7746.2019.4.526-535Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-SchwarzianA.V. Kazantsev0Kazan Federal University, Kazan, 420008 RussiaIt was established that if the left-hand side of the Gakhov equation is bounded by the constant 2, then this equation has exactly one root in the unit disk, where the constant is sharp and the root is not necessarily zero. We revealed two aspects arising with regard to this connection. The first aspect concerns the pre-Schwarzian immersion of the Gakhov class into the space of bounded holomorphic functions. It was shown that the width of this immersion is equal to 2; the full description was done for the intersection of the boundary of the immersion with the ball of the radius 2 centered at the origin. The second aspect is connected with maintenance of the uniqueness of the root when the linear or fractional linear actions on the pre-Schwarzian with multiplying by the unit disk variable are bounded. Some uniqueness conditions were constructed in the form of S.N. Kudryashov’s univalence criteria.https://kpfu.ru/uz-eng-phm-2019-4-4.htmlgakhov equationconformal radiuspre-schwarzian |
| spellingShingle | A.V. Kazantsev Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian Учёные записки Казанского университета: Серия Физико-математические науки gakhov equation conformal radius pre-schwarzian |
| title | Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian |
| title_full | Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian |
| title_fullStr | Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian |
| title_full_unstemmed | Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian |
| title_short | Root uniqueness of the Gakhov equation in the classes of functions with the bounded pre-Schwarzian |
| title_sort | root uniqueness of the gakhov equation in the classes of functions with the bounded pre schwarzian |
| topic | gakhov equation conformal radius pre-schwarzian |
| url | https://kpfu.ru/uz-eng-phm-2019-4-4.html |
| work_keys_str_mv | AT avkazantsev rootuniquenessofthegakhovequationintheclassesoffunctionswiththeboundedpreschwarzian |