Univalence criteria for locally univalent analytic functions
UDC 517.5 Suppose that  $p(z)=1+z\phi''(z)/\phi'(z),$ where   $\phi(z)$ is a locally univalent analytic function in the unit disk $\mathbf{D}$  with $\phi(0)=\phi'(1)-1=0.$  We establish the lower an...
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| Date: | 2023 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7222 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1865793814866690048 |
|---|---|
| author | Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan |
| author_facet | Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan |
| author_institution_txt_mv | [
{
"author": "Zhenyong Hu",
"institution": "School of Mathematics and Statistics, Nanjing University of Science and Technology, China"
},
{
"author": "Jinhua Fan",
"institution": "School of Mathematics and Statistics, Nanjing University of Science and Technology, China"
},
{
"author": "Xiaoyuan Wang",
"institution": "School of Mathematics and Statistics, Nanjing University of Science and Technology, China"
}
] |
| author_sort | Hu, Zhenyong |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T15:57:32Z |
| description | UDC 517.5
Suppose that  $p(z)=1+z\phi''(z)/\phi'(z),$ where   $\phi(z)$ is a locally univalent analytic function in the unit disk $\mathbf{D}$  with $\phi(0)=\phi'(1)-1=0.$  We establish the lower and upper bounds for the best constants $\sigma_{0}$ and $\sigma_{1}$ such that  $e^{-\sigma_{0}/2}<|p(z)|<e^{\sigma_{0}/2}$ and  $|p(w)/p(z)|<e^{\sigma_{1}}$ for $z,w\in\mathbf{D},$  respectively, imply the univalence of $\phi(z)$  in $\mathbf{D}.$ |
| doi_str_mv | 10.37863/umzh.v75i7.7222 |
| first_indexed | 2026-03-24T03:31:51Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7222 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:51Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-72222023-08-15T15:57:32Z Univalence criteria for locally univalent analytic functions Univalence criteria for locally univalent analytic functions Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan locally univalent analytic functions, John constant, univalence criterion. 30C55, 30C50. UDC 517.5 Suppose that  $p(z)=1+z\phi''(z)/\phi'(z),$ where   $\phi(z)$ is a locally univalent analytic function in the unit disk $\mathbf{D}$  with $\phi(0)=\phi'(1)-1=0.$  We establish the lower and upper bounds for the best constants $\sigma_{0}$ and $\sigma_{1}$ such that  $e^{-\sigma_{0}/2}<|p(z)|<e^{\sigma_{0}/2}$ and  $|p(w)/p(z)|<e^{\sigma_{1}}$ for $z,w\in\mathbf{D},$  respectively, imply the univalence of $\phi(z)$  in $\mathbf{D}.$ УДК 517.5 Критерії однозначності для локально однозначних аналітичних функцій Припустимо, що $p(z)=1+z\phi''(z)/\phi'(z),$ де $\phi(z)$ – локально однозначна аналітична функція  в одиничному диску $\mathbf{D}$ з $\phi(0)=\phi'(1)-1=0.$  Отримано нижню та верхню оцінки для найкращих сталих $\sigma_{0}$ та $\sigma_{1},$ таких що $e^{-\sigma_{0}/2}<|p(z)|<e^{\sigma_{0}/2}$ і $|p(w)/p(z)|<e^{\sigma_{1}}$ для $z, w\in\mathbf{D}$ відповідно означають однозначність $\phi(z)$ в $\mathbf{D}. $ Institute of Mathematics, NAS of Ukraine 2023-07-25 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7222 10.37863/umzh.v75i7.7222 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 7 (2023); 987 - 994 Український математичний журнал; Том 75 № 7 (2023); 987 - 994 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7222/9756 Copyright (c) 2023 Zhenyong Hu, Jinhua Fan, Xiaoyuan Wang |
| spellingShingle | Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan Hu, Zhenyong Fan, Jinhua Wang, Xiaoyuan Univalence criteria for locally univalent analytic functions |
| title | Univalence criteria for locally univalent analytic functions |
| title_alt | Univalence criteria for locally univalent analytic functions |
| title_full | Univalence criteria for locally univalent analytic functions |
| title_fullStr | Univalence criteria for locally univalent analytic functions |
| title_full_unstemmed | Univalence criteria for locally univalent analytic functions |
| title_short | Univalence criteria for locally univalent analytic functions |
| title_sort | univalence criteria for locally univalent analytic functions |
| topic_facet | locally univalent analytic functions John constant univalence criterion. 30C55 30C50. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7222 |
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