Ma-Minda φ-classes studied in this paper:
Abstract
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
Results & Lemmas (2)
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Theorem 3.1 · radius
Theorem 3.1. The following results hold for the classes and. (i) (ii) (iii)
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)
$$R_{S^(\alpha)}(\mathcal{T}_3) = R_{S^_{\alpha}}(\mathcal{T}_3) = (1-\alpha)/(\sqrt{8+2\alpha-\alpha^2})$$
Theorem 3.8 · radius
Theorem 3.8. The following results hold for the class. (i) (ii) (iii)
Theorem 3.8. The following results hold for the class $\mathcal{S}_{SG}^*$ .
(i)
$$R_{\mathcal{S}_{SG}^*}(\mathcal{T}_1) = (3 + 7e - \sqrt{41 + 42e + 17e^2})/(8 + 8e) \approx 0.177213$$
(ii)
$$R_{\mathcal{S}_{SG}^*}(\mathcal{T}_2) = (1 + 3e - \sqrt{5 + 6e + 5e^2})(2 + 2e) \approx 0.204712$$
(iii)
$$R_{\mathcal{S}_{SG}^*}(\mathcal{T}_3) = (\sqrt{1 - 2e + e^2})/(8 + 20e + 8e^2) \approx 0.1559$$
Function classes studied:
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