Radius of Starlikeness Through Subordination

Theorem 1.1. The following sharp radius results hold for the class $S_{\alpha,\beta}^*$ : (i) For $-1 \le B < A \le 1$ , the $\mathcal{S}^*[A, B]$ radius $$R_{\mathcal{S}^*[A,B]} = \min\{1, \ (A-B)/\left(|A+B-2\beta B| + 2(1-\beta)\right)\}.$$ - (ii) For $0 \le \gamma < 1$ , $\gamma > \beta$ , the $S^(\gamma)$ radius $R_{S^(\gamma)} = (1 \gamma)/(1 + \gamma 2\beta)$ . - (iii) The $S_L$ radius $R_{S_L} = (\sqrt{2} 1)/(\sqrt{2} + 1 2\beta)$ . - (iv) The $S_e$ radius $R_{S_e} = (e-1)/(e+1-2\beta)$ . The idea of the proof is to use inclusion results for the class $\mathcal{S}_{\alpha,\beta}$ with the class of starlike functions of order $\beta$ . Singh and Gupta [27, Corollary 4.1] have shown that $\mathcal{S}_{\alpha,\beta}^ \subseteq \mathcal{S}^*(\beta)$ . In order to use this inclusion, we first find the various radii for the class of starlike functions of order $\beta$ in the following:

Lemma 1.2. The following sharp radius results hold for the class $S^*(\beta)$ : (i) For $-1 \le B < A \le 1$ , the $\mathcal{S}^*[A, B]$ radius $$R_{\mathcal{S}^*[A,B]} = \min\{1, \ (A-B)/\left(|A+B-2\beta B| + 2(1-\beta)\right)\}.$$ (ii) For $0 \le \gamma < 1$ , $\gamma > \beta$ , the $S^(\gamma)$ radius $R_{S^(\gamma)} = (1 - \gamma)/(1 + \gamma - 2\beta)$ . (iii) The $S_L$ radius $R_{S_L} = (\sqrt{2} - 1)/(\sqrt{2} + 1 - 2\beta)$ . (iv) The $S_e$ radius $R_{S_e} = (e - 1)/(e + 1 - 2\beta)$ . (iv) The $$S_e^*$$ radius $R_{S^*} = (e-1)/(e+1-2\beta)$ . Theorem 1.1 follows from this lemma except for the sharpness. To find the extremal function $\tilde{f}$ for the class $\mathcal{S}_{\alpha,\beta}^*$ , write $\tilde{f}$ as $$\tilde{f}(z) = z + \sum_{n=2}^{\infty} a_n z^n$$ and determine the coefficients $a_n$ from $$\frac{z\tilde{f}'(z)}{\tilde{f}(z)}\left(1+\alpha\frac{z\tilde{f}''(z)}{f_1'(z)}\right) = \varphi_p(z) \tag{1.3}$$ where $\varphi_p$ is given by (1.1). Writing $$C = 2(2\alpha - \beta) - 4\alpha\beta, \ D = 2(\alpha + \beta) + 2\alpha\beta(2\beta - 3) - 1,$$ the equation (1.3) readily gives $$a_2 = \frac{C+2}{1+2\alpha} = 2(1-\beta)$$ $$a_n = \frac{\left(C+2(n-1)+2\alpha(n-1)(n-2)\right)}{(1+n\alpha)(n-1)} a_{n-1} + \frac{\left(D-(n-2)-\alpha(n-2)(n-3)\right)}{(1+n\alpha)(n-1)} a_{n-2}.$$ Calculating the coefficients $a_n$ from the above recurrence relation, we see that the extremal function $\hat{f}$ is the generalised Koebe's function given by $$\tilde{f}(z) = \frac{z}{(1-z)^{2-2\beta}}. (1.4)$$ Interestingly, it is the extremal of the class $S^*(\beta)$ and hence the sharpness of our theorem follows trivially. It is also well-known that a convex function is starlike of order 1/2 and so the class K of convex function is contained in the class $S^*(1/2)$ of starlike functions of order 1/2. This inclusion and Lemma 1.2 together readily yields the following radii results for the class of convex functions:

Corollary 1.3. The following sharp radius results hold for the class K: (i) For $-1 \le B < A \le 1$ , the $S^*[A, B]$ radius $$R_{\mathcal{S}^*[A,B]} = \min\{1, (A-B)/(1+|A|)\}.$$ - (ii) For $0 \le \gamma < 1$ , $\gamma > 1/2$ , the $S^(\gamma)$ radius $R_{S^(\gamma)} = (1 \gamma)/\gamma$ . - (iii) The $S_L$ radius $R_{S_L} = 1 1/\sqrt{2} \approx 0.2929$ . - (iv) The $S_e$ radius $R_{S_e} = 1 1/e \approx 0.6321$ . The method of convolution can also be applied to find radius problems of various classes. Corollary 1.3 (ii) requires the largest number $\rho$ such that the function $l_{\rho}$ : $\mathbb{D} \to \mathbb{C}$ is a starlike of order $\gamma \geq 1/2$ , where $f_{\rho}(z) = f(z) * l_{\rho}(z)$ . Here l(z) = z/(1-z)is the convolution identity and the functions $f_{\rho}, l_{\rho}: \mathbb{D} \to \mathbb{C}$ are defined respectively by $f_{\rho}(z) = f(\rho z)/\rho$ and $l_{\rho}(z) = z/(1-\rho z)$ . This is equivalent to find the number $\rho$ such that $\text{Re}(\rho z/(1-\rho z)) > \gamma - 1$ . It follows by simple computation that $\rho = (1-\gamma)/\gamma$ , since the real part of the function $(\rho z/(1-\rho z))$ attains minimum at z = -1.