Sufficient Conditions and Radius Problems for the Silverman Class

Pub Date : 2024-09-06 DOI:10.1007/s11253-024-02331-w

S. Sivaprasad Kumar, Priyanka Goel

引用次数: 0

Abstract

For 0 < α ≤ 1 and λ > 0, let

\({G}_{\lambda ,\alpha }=\left\{f \in A: \left|\frac{1-\alpha +\alpha zf^{\prime\prime}\left(z\right)/{f}^{{^{\prime}}}\left(z\right)}{z{f}^{{^{\prime}}}\left(z\right)/f\left(z\right)}-\left(1-\alpha \right)\right|< \lambda , z \in {\mathbb{D}}\right\}. (0.1)\)

The general form of the Silverman class was introduced by Tuneski and Irmak [Int. J. Math. Math. Sci., 2006, Article ID 38089 (2006)]. Our differential-inequality formulation is based on several sufficient conditions for this class. Further, we consider a class Ω given by

\(\Omega =\left\{f\in A:\left|z{f{^{\prime}}}^{\left(z\right)}-f\left(z\right)\right|<\frac{1}{2},z\in {\mathbb{D}}\right\}. (0.2)\)

For these two classes, we establish inclusion relations involving some well-known subclasses of S^* and compute radius estimates featuring various pairings of these classes.

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西尔弗曼类的充分条件和半径问题

For 0 < α ≤ 1 and λ > 0, let\({G}_{\lambda ,\alpha }=\left\{f \in A：\left||frac{1-\alpha +\alpha zf^{prime\prime}\left(z\right)/{f}^{^{\prime}}left(z\right)}{z{f}^{^{\prime}}left(z\right)/f\left(z\right)}-left(1-\alpha\right)\right|<；\lambda , z in {\mathbb{D}}\right\}.(0.1)\)The general form of the Silverman class was introduced by Tuneski and Irmak [Int. J. Math. Math. Sci.我们的微分不等式表述基于该类的几个充分条件。此外，我们还考虑了一个类 Ω，该类由以下条件给出：(\Omega =\left\{fin A:\left|z{f{^{\prime}}}^{left(z\right)}-f\left(z\right)\|<\frac{1}{2},z\in {\mathbb{D}}}\right\}.(0.2)\)For these two classes, we establish inclusion relations involving some well-known subclasses of S* and compute radius estimates featuring various pairings of these classes.

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