The $n$-valent convexity of Frasin integral operators
UDC 517.5 Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$We consider the Frasin integral operator as follows:\begin{gather*}\label{e1...
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| Date: | 2021 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2021
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/88 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.5
Let $ f_{i},$ $i\in\lbrace 1,2,\ldots, k\rbrace,$ is an analytic function on the unit disk in the complex plane of the form$f_{i}(z) = z^{n} + a_{i,n+1}z^{n+1} + \ldots, n\in\mathbb{N} = \lbrace 1,2,\ldots\rbrace.$We consider the Frasin integral operator as follows:\begin{gather*}\label{e1.3}G_{n}(z)=\int\limits_{0}^{z} n\xi^{(n-1)}\bigg(\dfrac{f'_{1}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{1}}\cdots\bigg(\dfrac{f'_{k}(\xi)}{n\xi^{n-1}}\bigg)^{\alpha_{k}}d\xi.\end{gather*}In this paper, we obtain a sufficient condition under which this integral operator is $n$-valent convex and get other interesting results.
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| DOI: | 10.37863/umzh.v73i2.88 |