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 |
| Published: |
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| Summary: | 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 $\mathbf{D}.$ |
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| DOI: | 10.37863/umzh.v75i7.7222 |