Sufficient conditions and radius problems for the Silverman class
UDC 517.5 For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}\colon \left|\frac{1-\alpha+\alpha zf''(z)/f'(z)}{z f'(z)/f(z)}-(1-\alpha)\right|<\lambda,\ z\in\mat...
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| Datum: | 2024 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2024
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7317 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.5
For $0<\alpha\leq1$ and $\lambda>0,$ let \begin{equation}\label{1} G_{\lambda,\alpha}=\left\{f\in\mathcal{A}\colon \left|\frac{1-\alpha+\alpha zf''(z)/f'(z)}{z f'(z)/f(z)}-(1-\alpha)\right|<\lambda,\ z\in\mathbb{D}\right\}.\tag{0.1}\end{equation} The general form of the Silverman class introduced by Tuneski and Irmak [Int. J. Math. and Math. Sci., {\bf 2006}, Article~ID 38089 (2006)].  Our differential inequality formulation lays out several sufficient conditions for this class.  Further, we consider a class $\Omega$ given by \begin{equation}\label{omega}\Omega=\left\{f\in\mathcal{A}\colon  |zf'(z)-f(z)|<\frac{1}{2},\ z\in\mathbb{D}\right\}.\tag{0.2}\end{equation} For these two classes, we establish inclusion relations involving some well-known subclasses of $\mathcal{S}^*$ and compute radius estimates featuring various pairings of these classes. |
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| DOI: | 10.3842/umzh.v76i3.7317 |