Radius Problems for Certain Classes of Analytic Functions

Theorem 2.1. For $0 < \lambda \le \pi/2$ , let $f \in \mathcal{S}_{\lambda e}^*$ . Then the pre-Schwarzian norm satisfies the following sharp inequality <span id="page-4-1"></span> $$||P_f|| \le \frac{(1-\alpha^2)(e^{\lambda\alpha} + \lambda\alpha - 1)}{\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation $$(2.1) 1 + r^2 - 2\lambda r^3 - e^{\lambda r} (1 - \lambda r + r^2 + \lambda r^3) = 0.$$ For the particular value $\lambda = 1$ , we get the sharp estimate of the pre-Schwarzian norm for functions in $\mathcal{S}^*(e^z)$ . <span id="page-4-0"></span>Corollary 2.1. For any $f \in S^*(e^z)$ , the pre-Schwarzian norm satisfies the following sharp inequality $$||P_f|| \le \frac{(1-\alpha^2)(e^\alpha + \alpha - 1)}{\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation $$1 + r^2 - 2r^3 - e^r(1 - r + r^2 + r^3) = 0.$$ The next theorem, gives the sharp estimate of the pre-Schwarzian norm for functions in the class $S^*(q_c)$ .

Theorem 2.2. For $0 < c \le 1$ , let $f \in \mathcal{S}^*(q_c)$ . Then the pre-Schwarzian norm satisfies the following sharp inequality $$||P_f|| \le \frac{c(1-\alpha^2)}{2(1-c\alpha)} + \frac{(1-\alpha^2)(1-\sqrt{1-c\alpha})}{\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation <span id="page-4-3"></span> $$(2.2) -2 + 4cr - (2+c^2)r^2 + 2cr^3 - c^2r^4 + (1-cr)^{3/2}(2-cr+2r^2-3cr^3) = 0.$$ For the particular value c=1, the class $\mathcal{S}^(q_c)$ reduce to the class $\mathcal{S}^(q_1)=:$ $\mathcal{S}^(\sqrt{1+z})$ . The sharp estimate of the pre-Schwarzian norm for functions in $\mathcal{S}^(\sqrt{1+z})$ is given by the following result. Corollary 2.2. Let $f \in \mathcal{S}^*(\sqrt{1+z})$ be of the form (1.1). Then the pre-Schwarzian norm satisfies the following sharp inequality $$||P_f|| \le \frac{(1+\alpha)(2-\alpha-2(1-\alpha)^{3/2})}{2\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation $$-2 - r^2 + (2 + r + 3r^2)\sqrt{1 - r} = 0.$$ The next two theorems, we provide the sharp estimate of the pre-Schwarzian norm of the Alexander transformation for functions in the class $\mathcal{S}_{\lambda e}$ and $\mathcal{S}^(q_c)$ .

Theorem 2.3. For any $f \in \mathcal{S}_{\lambda e}^*$ and $g \in \mathcal{C}_{\lambda e}$ , the pre-Schwarzian norm satisfies the following sharp inequality $$||P_g|| = ||P_{J[f]}|| \le \frac{(1 - \alpha^2)(e^{\lambda \alpha} - 1)}{\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation <span id="page-5-2"></span><span id="page-5-1"></span>(2.3) $$\lambda r e^{\lambda r} (1 - r^2) - (1 + r^2)(e^{\lambda r} - 1) = 0.$$

Theorem 2.4. For any $f \in \mathcal{S}^*(q_c)$ and $g \in \mathcal{C}(q_c)$ , the pre-Schwarzian norm satisfies the following sharp inequality $$||P_g|| = ||P_{J[f]}|| \le \frac{(1 - \alpha^2)(1 - \sqrt{1 - c\alpha})}{\alpha},$$ where $\alpha$ is the unique root in (0,1) of the equation <span id="page-5-3"></span>(2.4) $$2(1+r^2)(1-\sqrt{1-cr})-c(r+3r^3)=0.$$

Lemma 3.1. For a fixed c with $0 < c \le 1$ , let $$k_1(s) = (1 - cs)^2(-8 + 12cs - 3c^2s^2 - 4cs^3 + 3c^2s^4),$$ $$k_2(s) = -4\sqrt{1 - cs}(-2 + 6cs - 6c^2s^2 + cs^3 + c^3s^3),$$ $$k(s) = \frac{k_1(s) + k_2(s)}{(1 - cs)^{7/2}},$$ where 0 < s < 1. Then k(s) < 0 for all $s \in (0,1)$ and for each fixed $c \in (0,1]$ .