Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function
UDC 517.5 Investigation of the theory of complex functions  is one of the most fascinating aspects of theory of complex analytic functions of one variable.  It has a huge impact on all areas of mathematics.  Many mathematical concepts...
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| Datum: | 2023 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
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Institute of Mathematics, NAS of Ukraine
2023
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7214 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512629515091968 |
|---|---|
| author | Şahin, Hasan Yildiz, İsmet Şahin, Hasan Yildiz, İsmet |
| author_facet | Şahin, Hasan Yildiz, İsmet Şahin, Hasan Yildiz, İsmet |
| author_sort | Şahin, Hasan |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T15:57:32Z |
| description | UDC 517.5
Investigation of the theory of complex functions  is one of the most fascinating aspects of theory of complex analytic functions of one variable.  It has a huge impact on all areas of mathematics.  Many mathematical concepts are explained when viewed through the theory  of complex functions. Let $f(z)\in A,$ $f(z)=z+\sum_{n\geq 2}^{\infty }a_{n}z^{n} , $  be an analytic function in the open unit disc  $U=\left\{z\colon |z|<1,\ z\in \mathbb{C}\right\}$ normalized by $f(0)=0$ and $f'(0)=1.$  For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where $r$ is a positive integer of order $2^{-r} $ $\left(0<2^{-r} \le \dfrac{1}{2}\right).$  By using  subordination, we propose a criterion for $f(z)\in S^* [a^{r},b^{r}].$ The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study  their regional transformations.  In addition, the properties of convexity for $f(z)\in A$ are examined. |
| doi_str_mv | 10.37863/umzh.v75i7.7214 |
| first_indexed | 2026-03-24T03:31:50Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7214 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:50Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-72142023-08-15T15:57:32Z Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function Şahin, Hasan Yildiz, İsmet Şahin, Hasan Yildiz, İsmet Analytic function convexity unit disk Analytic Function Univalent Function Convexity UDC 517.5 Investigation of the theory of complex functions  is one of the most fascinating aspects of theory of complex analytic functions of one variable.  It has a huge impact on all areas of mathematics.  Many mathematical concepts are explained when viewed through the theory  of complex functions. Let $f(z)\in A,$ $f(z)=z+\sum_{n\geq 2}^{\infty }a_{n}z^{n} , $  be an analytic function in the open unit disc  $U=\left\{z\colon |z|<1,\ z\in \mathbb{C}\right\}$ normalized by $f(0)=0$ and $f'(0)=1.$  For close-to-convex and starlike functions, new and different conditions are obtained by using subordination properties, where $r$ is a positive integer of order $2^{-r} $ $\left(0<2^{-r} \le \dfrac{1}{2}\right).$  By using  subordination, we propose a criterion for $f(z)\in S^* [a^{r},b^{r}].$ The relations for starlike and close-to-convex functions are investigated under certain conditions according to their subordination properties. At the same time, we analyze the convexity of some analytic functions and study  their regional transformations.  In addition, the properties of convexity for $f(z)\in A$ are examined. УДК 517.5 Визначення деяких властивостей зіркоподібних i близьких до опуклих функцій за підпорядкованими умовами з опуклістю певної аналітичної функції Дослідження з теорії комплексних функцій є одним із найцікавіших аспектів теорії комплексних аналітичних функцій однієї змінної.  Вони мають величезний вплив на всі області математики.  Багато математичних понять пояснюються з точки зору теорії комплексних функцій.  Нехай $f(z)\in A,$ $f(z)=z+\sum_{n\geq 2}^{\infty }a_{n}z^{n},$ –  аналітична функція на відкритому одиничному диску $U=\left\{z\colon |z|<1,\ z\in \mathbb{C}\right\}$, що нормована таким чином: $f(0)=0,$  $f'(0)=1.$  Для зірчастих функцій та функцій, що близькі до опуклих, за допомогою властивостей підпорядкування  отримано нові та відмінні умови, де $r$ – натуральне число порядку $2^{-r}$ $\left(0<2^{-r} \le \dfrac{1}{2}\right).$  Використовуючи підпорядкованість, ми пропонуємо деякий критерій для $f(z)\in S^* [a^{r},b^{r}].$  Досліджено співвідношення між зірчастими функціями та функціями, що близькі до опуклих,  за певних умов згідно з властивостями підпорядкування.  Також досліджено опуклість деяких аналітичних функцій і вивчено їх локальні перетворення.  Крім того, досліджено властивості опуклості для $f(z)\in A$. Institute of Mathematics, NAS of Ukraine 2023-07-25 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7214 10.37863/umzh.v75i7.7214 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 7 (2023); 995 - 1008 Український математичний журнал; Том 75 № 7 (2023); 995 - 1008 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7214/9757 Copyright (c) 2023 Hasan Şahin |
| spellingShingle | Şahin, Hasan Yildiz, İsmet Şahin, Hasan Yildiz, İsmet Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_alt | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_full | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_fullStr | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_full_unstemmed | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_short | Determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| title_sort | determination of some properties of starlike and close-to-convex functions according to subordinate conditions with convexity of a certain analytic function |
| topic_facet | Analytic function convexity unit disk Analytic Function Univalent Function Convexity |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7214 |
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