Pre-Schwarzian Norm Estimates for the class of Janowski Starlike Functions

Lemma 1.1. [1, Lemma 2.2] Suppose that $\omega : \mathbb{D} \to \mathbb{D}$ is an analytic function with $\omega(0) = 0$ . If $-1 < A \le 1$ , then $$M(z) := \frac{(|z|^2 - |\omega(z)|^2) + |\omega(z)|(1 - |z|^2)|2 + A\omega(z)|}{|z|(1 - |\omega(z)|^2)|2 + A\omega(z)|} \le 1, \quad z \in \mathbb{D}.$$

Lemma 2.1. For $-1 \le B < A \le 1$ , let (2.1) $$\phi_{A,B}(z) = \frac{2 + Az}{(1 + Az)(1 + Bz)}.$$ Then $\phi_{A,B}(z)$ satisfies the following sharp inequalities. (i) If $AB \geq 0$ then for every $z \in \mathbb{D}$ , we have (2.2) $$|\phi_{A,B}(z)| \le \begin{cases} \phi_{A,B}(-|z|) & \text{for } 0 \le B < A \le 1\\ \phi_{A,B}(|z|) & \text{for } -1 \le B < A \le 0. \end{cases}$$ (ii) If AB < 0 then for every $z \in \mathbb{D}$ , we have (2.3) $$|\phi_{A,B}(z)| \le \begin{cases} \phi_{A,B}(|z|) & \text{for } |z|^2 \le \frac{A+2B}{A^2B} \\ \phi_{A,B}(-|z|) & \text{for } |z|^2 \ge \frac{A+2B}{A^2B}. \end{cases}$$

Theorem 2.1. For $-1 \le B < A \le 1$ , let $f \in \mathcal{S}^*(A, B)$ be of the form (1.1). Then we have the sharp inequality $||T_f|| \le ||T_{K_{A,B}}||$ , where $K_{A,B}$ is given by (1.3). (i) If $AB \geq 0$ , then $$||T_{K_{A,B}}|| = \begin{cases} (A-B)\gamma_1(\alpha_1), & \text{for } 0 \le B < A < 1, \\ (A-B)\gamma_1(\alpha_1), & \text{for } A = 1, B < \frac{1}{3}, \\ 2, & \text{for } A = 1, B \ge \frac{1}{3}, \\ (A-B)\gamma_2(\alpha_2), & \text{for } -1 < B < A \le 0, \\ 2(2+A), & \text{for } B = -1, -1 < A \le 0, \end{cases}$$ where $\gamma_1$ and $\gamma_2$ are given by (2.8) and $\alpha_1$ , $\alpha_2$ are the unique roots in (0,1) of the equations $h_1(x) = 0$ , $h_2(x) = 0$ respectively and $h_1$ , $h_2$ are given by (2.9), (2.10) respectively. (ii) If AB < 0 with $\beta = \sqrt{(A+2B)/A^2B}$ , then $$||T_{K_{A,B}}|| = \begin{cases} (A-B)\gamma_1(\alpha_1), & \text{for } A+2B>0, \\ 2(A-B), & \text{for } A+2B=0, \\ (A-B)\max\{\gamma_2(\alpha_2), \gamma_1(\beta)\}, & \text{for } A+2B<0, \ \beta<1, \\ (A-B)\gamma_2(\alpha_2), & \text{for } A+2B<0, \ \beta\geq1, \ B\neq-1, \\ 2(2+A), & \text{for } A>0, \ B=-1, \end{cases}$$ where $\gamma_1$ and $\gamma_2$ are given by (2.8) and $\alpha_1$ , $\alpha_2$ are the unique roots in (0,1) of the equations $h_1(x) = 0$ , $h_2(x) = 0$ respectively and $h_1$ , $h_2$ are given by (2.9), (2.10) respectively.

Corollary 2.1. [1, Theorem 2] Let $f \in \mathcal{S}^*(\alpha)$ , $0 \le \alpha < 1$ be of the form (1.1). Then the sharp inequality $||T_f|| \le 6 - 4\alpha$ , holds and equality occurs for the function $k_{\alpha}(z) = z/(1-z)^{2(1-\alpha)}$ .

Corollary 2.2. Let $f \in \mathcal{S}^*$ be of the form (1.1). Then the sharp inequality $||T_f|| \le 6$ , holds and equality occurs for the Koebe function $k(z) = z/(1-z)^2$ . Taking A = 1 and B = 0, we get the following result.

Corollary 2.3. Let $f \in \mathcal{S}^*(1,0)$ be of the form (1.1). Then the sharp inequality $||T_f|| \leq 9/4$ , holds and equality occurs for the function $f(z) = ze^z$ . Taking $A = \alpha$ and $B = -\alpha$ , $0 < \alpha < 1$ , we get the following result.

Corollary 2.4. Let $f \in \mathcal{S}^*(\alpha, -\alpha)$ , $0 < \alpha \le 1$ be of the form (1.1). Then the sharp inequality $$||T_f|| \le \begin{cases} 2\alpha(1-x_0^2)(2+\alpha x_0)/(1-\alpha^2 x_0^2), & \text{for } 0 < \alpha < 1, \\ 6, & \text{for } \alpha = 1 \end{cases}$$ holds, where $x_0$ is an unique root in (0,1) of the equation: $$\alpha^{3}x^{4} + (\alpha^{3} - 3\alpha)x^{2} + (4\alpha^{2} - 4)x + \alpha = 0.$$ Moreover, the equality occurs for the function $f(z) = z/(1 - \alpha z)^2$ .