Shalu Yadav, Kanika Sharma, V. Ravichandran · 2021 📄 arXiv →
A starlike univalent function $f$ is characterized by the function $zf'(z)/f(z)$; several subclasses of these functions were studied in the past by restricting the function $zf'(z)/f(z)$ to take values in a region $Ω$ on the right-half plane, or, equivalently, by requiring the function $zf'(z)/f(z)$ to be subordinate to the corresponding mapping of the unit disk $\mathbb{D}$ to the region $Ω$. The mappings $w_1(z):=z+\sqrt{1+z^2}, w_2(z):=\sqrt{1+z}$ and $w_3(z):=e^z$ maps the unit disk $\math
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Theorem 3.1. The following results hold for the classes $S^(\alpha)$ and $S^_{\alpha}$ .
(i)
$$R_{S^(\alpha)}(\mathcal{T}_1) = R_{S^_{\alpha}}(\mathcal{T}_1) = (7 - 2\alpha - \sqrt{17 + 4\alpha + 4\alpha^2})/8$$
(ii)
$$R_{S^(\alpha)}(\mathcal{T}_2) = R_{S^_{\alpha}}(\mathcal{T}_2) = ((3 - \alpha - \sqrt{5 - 2\alpha + \alpha^2}))/2$$
(iii)
$$R_{S^(\alpha)}(\mathcal{T}_3) = R_{S^_{\alpha}}(\mathcal{T}_3) = (1-\alpha)/(\sqrt{8+2\alpha-\alpha^2})$$We use first-party site metrics to improve navigation: page visits, internal clicks, search result counts (not search terms), and approximate active time. A random tab ID groups events for seven days; GPC/DNT turns this off. The separate visitor count uses a self-hosted pseudonymous fingerprint and also respects GPC/DNT.