Inclusion relations and radius problems for a subclass of starlike functions

Lemma 2.1. If 0 < r < 1, then the function $\phi_{car}$ defined by (1.1) satisfies $$\min_{|z|=r} \operatorname{Re}(\phi_{car}(z)) = \begin{cases} 1 - r + r^2/2, & 0 < r \le 1/2\\ (3 - 2r^2)/4, & 1/2 \le r < 1 \end{cases}$$ and $$\max_{|z|=r} \text{Re}(\phi_{car}(z)) = 1 + r + \frac{r^2}{2} = \phi_{car}(r).$$

Lemma 2.2. For 1/2 < a < 5/2, let $r_a$ be given by $$r_a = \begin{cases} (2a-1)/2, & 1/2 < a \le 3/2\\ (5-2a)/2, & 3/2 \le a < 5/2 \end{cases}$$ and $R_a$ be given by $$R_a = \begin{cases} (5 - 2a)/2, & 1/2 < a \le 7/6\\ \sqrt{(2a - 1)^3/(8(a - 1))}, & 7/6 \le a < 5/2. \end{cases}$$ Then $$\{w : |w - a| < r_a\} \subseteq \Omega_{car} \subseteq \{w : |w - a| < R_a\}$$

Lemma 2.3. Let $f \in \mathcal{S}_{car}$ and $g \in \mathcal{K}$ . Then the function $(f g)(\rho z)/\rho \in \mathcal{S}_{car}^*$ for $0 < \rho \le 1/2$ .

Theorem 3.1. The class $S_{car}^*$ satisfies the following relationships: - (i) $S_{car}^ \subset S^(\alpha)$ for $0 \le \alpha \le 1/4$ ; (ii) $S_{car}^ \subset SS^(\beta)$ for $\beta_0 \le \beta \le 1$ , where $\beta_0 = (2/\pi) \tan^{-1}(3\sqrt{3/5}) \approx 0.743253$ ; - (iii) $k \mathcal{S}^ \subset \mathcal{S}^_{car}$ for $k \geq 5/3$ ; - (iv) $\mathcal{S}_e^(\alpha) \subset \mathcal{S}_{car}$ for $\alpha_0 \leq \alpha < 1$ , where $\alpha_0 = (e-2)/(2(e-1)) \approx 0.209011$ ; - (v) $\mathcal{S}_L^(\alpha) \subset \mathcal{S}_{car}$ for $1/2 \le \alpha < 1$ ; - (vi) $S^(\sqrt{1+cz}) \subset S^_{car}$ for $0 < c \le 3/4$ ; - (vii) $\mathcal{S}^[A, B] \subset \mathcal{S}^_{car}$ , where $-1 < \overline{B} < A \le 1$ if one of the following conditions holds: (a) $1 B^2 < 2(1 AB) \le 3(1 B^2)$ and $2A \le 1 + B$ , (b) $$3(1-B^2) \le 2(1-AB) < 5(1-B^2)$$ and $2A \le 3+5B$ ; (viii) $S_{car}^ \subset S^[1, -(M-1)/M]$ for $M \ge (3+\sqrt{5})/4 \approx 1.309$ ;

Theorem 3.4. Let $f \in \mathcal{S}_{car}$ . Then the second partial sum $f_2$ is starlike in |z| < 1/2 and convex in |z| < 1/4. Also, $f_2(\rho z)/\rho \in \mathcal{S}_{car}$ for $0 < \rho \le 1/3$ . All the constants are sharp.

Theorem 3.5. Let $f \in A$ be given by (1.3) and $f_2$ denotes its second partial sum. Then we have the following: - (1) If $f \in \mathcal{K}$ , then $f_2(\rho z)/\rho \in \mathcal{S}_{car}^*$ for $0 < \rho \le 1/3$ ; - (2) If $f \in \mathcal{S}$ (or $\mathcal{S}^*$ or $\mathcal{C}$ ), then $f_2(\rho z)/\rho \in \mathcal{S}^-_{car}$ for $0 < \rho \le 1/6$ . Proof. Since $f_2(\rho z)/\rho = z + a_2\rho z^2$ , therefore if $3|a_2|\rho \leq 1$ , then $f_2(\rho z)/\rho \in \mathcal{S}^*_{car}$ by Corollary 3.3. If $f \in \mathcal{K}$ , then $|a_2| \leq 1$ so that $3|a_2|\rho \leq 1$ for $0 < \rho \leq 1/3$ . The result is sharp for the half-plane mapping l(z) = z/(1-z). Similarly, if $f \in \mathcal{S}$ , then $|a_2| \leq 2$ which gives $3|a_2|\rho \leq 1$ for $0 < \rho \leq 1/6$ and in this case, Koebe function $k(z) = z/(1-z)^2$ verifies the sharpness of the result. <span id="page-8-0"></span> <span id="page-8-1"></span>FIGURE 3. $\mathcal{S}_{car}^*$ -radius is unity. Before closing this section, let us establish the inclusion relation of the class $\mathcal{S}_{car}$ with the classes $\mathcal{S}_{SG}^ := \mathcal{S}^(2/(1+e^{-z}))$ , $\mathcal{S}_{cosh}^ := \mathcal{S}^(\cosh z)$ and $\mathcal{S}_R^ := \mathcal{S}^*(\psi_R)$ where (3.2) $$\psi_R(z) = 1 + \frac{z}{k} \left( \frac{k+z}{k-z} \right), \quad k = \sqrt{2} + 1.$$ These classes were introduced by Goel and Kumar [12], Alotaibi et al. [5] and Kumar and Ravichandran [20] respectively. Figure 3 shows that the image domains $|\log(w/(2-w))| < 1$ , $|\log(w+\sqrt{w^2-1})| < 1$ and $\psi_R(\mathbb{D})$ of the functions $2/(1+e^{-z})$ , $\cosh z$ and $\psi_R$ respectively under the unit disk lie inside $\Omega_{car}$ . As a result $\mathcal{S}_{SG}$ , $\mathcal{S}_{cosh}$ and $\mathcal{S}_R$ are subclasses of $\mathcal{S}_{car}$ . Consequently the $\mathcal{S}_{car}$ -radius of the classes $\mathcal{S}_{SG}$ , $\mathcal{S}_{cosh}$ and $\mathcal{S}_R$ is unity. The radii constants associated with other subclasses of starlike functions are investigated in the next section. 4. $$\mathcal{S}_{car}^*$$ -Radius Recently, several other subclasses of starlike functions have been defined using the unified class introduced by Ma and Minda [25] based on some interesting curves in the Euclidean plane. The classes $\mathcal{S}_C^ := \mathcal{S}^(1+4z/3+2z^2/3), \, \mathcal{S}_{lim}^ := \mathcal{S}^(1+\sqrt{2}z+z^2/2), \, \mathcal{S}_{ne}^ := \mathcal{S}^(1+z-z^3/3)$ and $\mathcal{S}_{RL}^ := \mathcal{S}^(\sqrt{2}-(\sqrt{2}-1)((1-z)/(1+2(\sqrt{2}-1)z))^{1/2})$ are the classes of starlike functions associated with cardioid $(9u^2+9v^2-18u+5)^2-16(9u^2+9v^2-6u+1)=0$ , limacon $(4u^2+4v^2-8u-5)^2+8(4u^2+4v^2-12u-3)=0$ , nephroid $((u-1)^2+v^2-4/9)^3-4v^2/3=0$ and left-half of the shifted lemniscate of Bernoulli $|(w-\sqrt{2})^2-1|=1$ respectively. Other classes associated with univalent functions include $\mathcal{S}_{(u)}^ := \mathcal{S}^(z+\sqrt{1+z^2})$ and $\mathcal{S}_{sin}^ := \mathcal{S}^(1+\sin z)$ . These classes have been introduced and studied in [7,11,29,33,38,39,46-48]. In this section, we will determine $\mathcal{S}_{car}$ -radius for all the classes discussed here and in the previous section. Firstly, let us determine the $\mathcal{S}_{car}$ -radius of the class $\mathcal{S}^*[A,B]$ , where $-1 \leq B < A \leq 1$ .

Theorem 4.1. Let $-1 \le B < A \le 1$ . If $B \ge 0$ , then the $\mathcal{S}_{car}$ -radius for the class $\mathcal{S}^[A, B]$ is $R_2 = \min\{1, 1/(2A - B)\}$ . If B < 0, then the $\mathcal{S}_{car}$ -radius for the class $\mathcal{S}^[A, B]$ is $R_2$ if $R_2 \le R_1$ and $R_3$ if $R_2 > R_1$ , where $R_1 = 1/\sqrt{B(3B - 2A)}$ and $R_3 = \min\{1, 3/(2A - 5B)\}$ .

Theorem 4.3. The $S_{car}^*$ -radius for various subclasses of starlike functions is given as follows: | S.No. | Class | Radius | | | | |--------------|------------------------------|--------------------------------------------------------------------------------------------------------|--|--|--| | (a) | $\mathcal{S}^*(\sqrt{1+cz})$ | $r_1 = \frac{3}{4c} \left(\frac{3}{4} < c \le 1\right)$ | | | | | (b) | $\mathcal{S}_L^*(\alpha)$ | $r_2 = \frac{3 - 4\alpha}{4(1 - \alpha)^2} \left(0 \le \alpha < \frac{1}{2}\right)$ | | | | | (c) | $\mathcal{S}_e^*(\alpha)$ | $r_3 = \log\left(\frac{2(1-\alpha)}{1-2\alpha}\right) \left(0 \le \alpha < \frac{e-2}{2(e-1)}\right)$ | | | | | (d) | $\mathcal{S}^*_{RL}$ | $r_4 = \frac{1}{82}(39 + 17\sqrt{2}) \approx 0.7688$ | | | | | (e) | $\mathcal{S}_C^*$ | $r_5 = \frac{1}{2} \approx 0.5$ | | | | | (f) | $\mathcal{S}^*_{lim}$ | $r_6 = \sqrt{2} - 1 \approx 0.414$ | | | | | g | $\mathcal{S}^*_{\mathbb{Q}}$ | $r_7 = \frac{3}{4} \approx 0.75$ | | | | | (h) | $\mathcal{S}^*_{sin}$ | $r_8 = \sin^{-1}\left(\frac{1}{2}\right) \approx 0.523598$ | | | | | ( <i>i</i> ) | $\mathcal{S}_{ne}^*$ | $r_9 \approx 0.557875$ | | | | Here $r_9$ is the smallest positive real root of the equation $2r^3 - 6r + 3 = 0$ in (0,1). All bounds are sharp.

Theorem 4.4. (i) The $S_{car}$ -radius of the class $\mathcal{BL}(\alpha)$ , $0 \le \alpha < 1$ is $1/(1+\sqrt{1+\alpha})$ . (ii) The $S_{car}$ -radius of the class $\mathcal{M}(\beta)$ , $\beta > 1$ is $1/(4\beta - 3)$ .

Theorem 5.1. For the class $S_{car}^*$ , the following radii constants are sharp: (a) For $0 \le \alpha < 1$ , the $S^*(\alpha)$ -radius is $s_1 := s_1(\alpha)$ where $$s_1(\alpha) = \begin{cases} 1, & 0 \le \alpha \le 1/4, \\ \sqrt{\frac{3 - 4\alpha}{2}}, & 1/4 < \alpha \le 5/8, \\ 1 - \sqrt{2\alpha - 1}, & 5/8 < \alpha < 1. \end{cases}$$ - (b) For $0 \le \alpha < 1$ , the $S_L^*(\alpha)$ -radius is $s_2 := s_2(\alpha) = -1 + ((2\sqrt{2} 1) 2(\sqrt{2} 1)\alpha))^{1/2}$ . In particular, $s_2(0) = -1 + (2\sqrt{2} - 1)^{1/2}$ . - (c) The $S_{RL}^*$ -radius is $s_3 := -1 + (1 + 2(-\gamma + \sqrt{\gamma})^{1/2})^{1/2} \approx 0.253734$ , where $\gamma = 2\sqrt{2} 2$ . - (d) The $S_R^*$ -radius is $s_4 := 1 (4\sqrt{2} 5)^{1/2} \approx 0.189535$ . - (e) The $S_{sin}^*$ -radius is $s_5 := -1 + \sqrt{1 + 2\sin(1)} \approx 0.637969$ . - (f) The $S_{cosh}^*$ -radius is $s_6 := -1 + \sqrt{-1 + 2\cosh(1)} \approx 0.444355$ . - (g) The $S_{ne}^*$ -radius is $s_7 := (\sqrt{21} 3)/3 \approx 0.527525$ . - (h) The $S_{SG}^*$ -radius is $$s_8 := -1 + \sqrt{1 + \frac{2(e-1)}{e+1}} \approx 0.387168$$ (i) For $0 \le \alpha < 1$ , the $S^*[1 - \alpha, 0]$ -radius is $s_9 := -1 + \sqrt{3 - 2\alpha}$ . (j) For $0 < \alpha \le 1$ , the $S^*[\alpha, -\alpha]$ -radius is $s_{10} := s_{10}(\alpha)$ where $$s_{10}(\alpha) := \begin{cases} w_{\alpha}, & 0 < \alpha \le \alpha^, \\ 1, & \alpha^ \le \alpha \le 1. \end{cases}$$ where $$w_{\alpha} = \frac{2\alpha}{\sqrt{1 - \alpha^2}} \sqrt{\frac{2}{\sqrt{1 + 3\alpha^2}} - 1}$$ and $\alpha^* = ((5 + 2\sqrt{13})/27)^{1/2} \approx 0.672505$ . (k) For M > 1/2, the $S^*[1, -(M-1)/M]$ -radius is $s_{11} := s_{11}(M)$ where $$s_{11}(M) = \begin{cases} a_M, & 1/2 \le M \le M^, \\ b_M, & M^ \le M < (3 + \sqrt{5})/4, \\ 1, & M \ge (3 + \sqrt{5})/4, \end{cases}$$ where $a_M = -1 + \sqrt{M-1}$ , $$b_M = \sqrt{2\sqrt{2}M\sqrt{\frac{M-1}{2M-1}} - 2(M-1)}$$ and $M^* \approx 1.1423$ is the root of the equation $a_M = b_M$ . - (l) The $\mathcal{S}_C^*$ -radius is $s_{12} := 1$ . - (m) For $\beta > 1$ , $\mathcal{M}(\beta)$ -radius is $s_{13} := s_{13}(\beta)$ where $$s_{13}(\beta) = \begin{cases} \sqrt{2\beta - 1} - 1, & 1 < \beta \le 5/2, \\ 1, & \beta \ge 5/2. \end{cases}$$ Proof. Since $f \in \mathcal{S}_{car}^*$ , $zf'(z)/f(z) \prec \phi_{car}(z)$ where $\phi_{car}$ is given by (1.1) and $\phi_{car}(\mathbb{D}) = \Omega_{car}$ . Let |z| = r. Let $f_{car}$ denotes the function given by (1.2). For (a), by Theorem 3.1(i), $f \in \mathcal{S}^*(\alpha)$ for $0 \le \alpha \le 1/4$ . Let $1/4 < \alpha < 1$ . For $0 < r \le 1/2$ , Lemma 2.1 gives <span id="page-16-0"></span> $$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > \min_{|z|=r} \operatorname{Re}(\phi_{car}(z)) = 1 - r + \frac{r^2}{2} > \alpha$$ if $r < 1 - \sqrt{2\alpha - 1}$ , where $5/8 < \alpha < 1$ . Similarly for the case when $1/2 \le r < 1$ , it is easily seen that $\text{Re}(zf'(z)/f(z)) > (3-2r^2)/4 > \alpha$ provided $r < \sqrt{(3-4\alpha)/2}$ , where $1/4 < \alpha \le 5/8$ . The result is sharp for the function $f_{car}$ . For $f \in \mathcal{S}^*_{car}$ , $zf'(z)/f(z) \prec 1 + z + z^2/2$ so that $$\left|\frac{zf'(z)}{f(z)} - 1\right| \le r + \frac{r^2}{2}.$$ For (b), the disk (5.1) lies inside the domain $|((w-\alpha)/(1-\alpha))^2-1|<1$ provided $r+r^2/2 \leq (\sqrt{2}-1)(1-\alpha)$ by [19, Lemma 2.3, p. 238]. This implies $r\leq -1+((2\sqrt{2}-1)-2(\sqrt{2}-1)\alpha))^{1/2}:=s_2(\alpha)$ . This bound is best possible for the function $f_{car}$ since $zf'_{car}(z)/f_{car}(z)=\alpha+(1-\alpha)\sqrt{2}$ at $z=s_2$ . In (c), the disk (5.1) lies inside the domain $|(w-\sqrt{2})^2-1|<1$ if $r+r^2/2\leq ((2\sqrt{2}-2)^{1/2}-(2\sqrt{2}-2))^{1/2}$ by [29, Lemma 3.2]. This simplifies to $r\leq s_3\approx 0.2537371$ . The sharpness of the radii $s_2(0)$ and $s_3$ is depicted in Figure 6. <span id="page-17-0"></span> FIGURE 6. Sharpness of $\mathcal{S}_{L}^{}$ , $\mathcal{S}_{RL}^{}$ , $\mathcal{S}_{ne}^{}$ and $\mathcal{S}_{SG}^{}$ radii for the class $\mathcal{S}_{car}^{*}$ . For proving (d), observe that a necessary condition for the subordination $\phi_{car}(z) \prec \psi_R(z)$ to hold in $\mathbb{D}_r$ is $$2(\sqrt{2}-1) \le \psi_R(-1) \le \psi_R(-r) \le \phi_{car}(-r) = 1 - r + \frac{r^2}{2}$$ where $\psi_R$ is given by (3.2). This is possible if $r \leq 1 - (4\sqrt{2} - 5)^{1/2} := s_4$ . In fact, $\phi_{car}(\mathbb{D}_{s_4}) \subset \psi_R(\mathbb{D})$ (see Figure 7(a)). Thus the $\mathcal{S}_R$ -radius of the class $\mathcal{S}_{car}$ is at least $s_4$ . This bound cannot be further improved as seen by considering the function $f_{car}$ . Similarly, for part (e), the relation $$1 + r + \frac{r^2}{2} = \phi_{car}(r) \le \psi_s(r) \le \psi_s(1) = 1 + \sin(1)$$ is necessary for the subordination $\phi_{car}(z) \prec \psi_s(z)$ to satisfy in $\mathbb{D}_r$ , where $\psi_s$ is defined in Theorem 4.3(h). This gives $r \leq -1 + \sqrt{1 + 2\sin(1)} := s_5$ . Also, $\phi_{car}(\mathbb{D}_{s_5}) \subset \psi_s(\mathbb{D})$ by Figure 7(b) so that $\mathcal{S}^_{sin}$ -radius is at least $s_5$ . The bound is sharp for the function $f_{car}$ . The same procedure can be applied in part (f) to show that $\mathcal{S}^_{cosh}$ -radius is $s_6 := -1 + \sqrt{-1 + 2\cosh(1)}$ by considering the inequality $1 + r + r^2/2 \leq \cosh(1)$ and Figure 7(c) illustrating that the image domain of the function $\cosh z$ under the unit disk contains $\phi_{car}(\mathbb{D}_{s_6})$ . <span id="page-17-1"></span> FIGURE 7. Inclusions associated with a rational, trigonometric and hyperbolic function. In order to prove (g), note that the disk (5.1) lies inside the domain $\psi_{ne}(\mathbb{D})$ where $\psi_{ne}$ is defined in Theorem 4.3(i) if $r + r^2/2 \leq 2/3$ (see [47]). This simplifies to $r \leq (\sqrt{21} - 3)/3 := s_7$ . The result is sharp for the function $f_{car}(z) = z \exp(z + z^2/4)$ as $z f'_{car}(z)/f_{car}(z) = 5/3 = \psi_{ne}(1)$ at $z = s_7$ . In the similar fashion, the part (h) can proved by applying [12, Lemma 2.2] to show that the disk (5.1) lies in the domain $\psi_{SG}(\mathbb{D})$ if $r + r^2/2 \leq (e - 1)/(e + 1)$ , where $\psi_{SH}(z) = 2/(1 + e^{-z})$ . The last inequality gives the desired bound $s_8 := -1 + (1 + 2((e - 1)/(e + 1)))^{1/2}$ and $z f'_{car}(z)/f_{car}(z)$ equals $2e/(1 + e) = \psi_{SH}(1)$ at $z = s_8$ . The sharpness of constants $s_7$ and $s_8$ is also depicted graphically in Figure 6. For (i), the disk (5.1) lies inside the domain $|w-1| < 1-\alpha$ provided $r + r^2/2 \le 1-\alpha$ which gives $r \le -1 + \sqrt{3-2\alpha} := s_9$ and at the point $z = s_9$ , $zf'_{car}(z)/f_{car}(z) = 2-\alpha$ . In part (j), we will show that $\phi_{car}(z) \in \{w : |(w-1)/(w+1)| < \alpha\}$ for all $z \in \mathbb{D}_{s_{10}}$ . For $z = re^{it}$ , we have $$|\phi_{car}(z) - 1|^2 = r^2 + r^3 \cos t + \frac{r^4}{4}$$ and $$|\phi_{car}(z) + 1|^2 = 4 + 4r\cos t + r^2 + 2r^2\cos(2t) + r^3\cos t + \frac{r^4}{4}.$$ If we set $x = \cos t$ , then the problem now reduces to show that the function $$g(x,r) := \alpha^{2} |\phi_{car}(z) + 1|^{2} - |\phi_{car}(z) - 1|^{2}$$ $$= 4\alpha^{2} + 4\alpha^{2}rx - \alpha^{2}r^{2} + 4\alpha^{2}r^{2}x^{2} + \alpha^{2}r^{3}x + \frac{\alpha^{2}r^{4}}{4} - r^{2} - r^{3}x - \frac{r^{4}}{4}$$ is positive for all $x \in [-1, 1]$ and $0 < r < s_{10}$ . Note that <span id="page-18-0"></span>(5.2) $$h(x,r) := \frac{\partial}{\partial x} g(x,r) = -r^3 + 4r\alpha^2 + r^3\alpha^2 + 8r^2x\alpha^2$$ and $\frac{\partial^2}{\partial x^2} g(x,r) = 8r^2\alpha^2$ . Consider the following four observations: - (1) $h(x_0, r) = 0$ where $x_0 = (r^2(1 \alpha^2) 4\alpha^2)/(8\alpha^2 r)$ . By (5.2), g(x, r) has a relative minimum at $x_0$ . - (2) The quantity $$g(-1,r) = \frac{1}{4}(2r(1-\alpha) - (1-\alpha)r^2 + 4\alpha)((1+\alpha)r^2 + 4\alpha - 2r(1+\alpha))$$ is positive if $r \le 1 - \sqrt{(1-3\alpha)/(1+\alpha)} := u_{\alpha}$ , $0 < \alpha \le 1/3$ and g(-1,r) > 0 for all r if $\alpha \ge 1/3$ . (3) The quantity $$g(1,r) = \frac{1}{4}(4\alpha - 2r(1-\alpha) - (1-\alpha)r^2)(4\alpha + 2r(1+\alpha) + (1+\alpha)r^2)$$ is positive if $r \leq \sqrt{(1+3\alpha)/(1-\alpha)} - 1 := v_{\alpha}$ , $0 < \alpha \leq 3/7$ and g(1,r) > 0 for every r if $\alpha \geq 3/7$ . (4) $g(x_0, r) > 0$ provided $r \le w_\alpha$ , $0 < \alpha \le \alpha$ and $g(x_0, r) > 0$ for all r and $\alpha \ge \alpha$ , where $w_\alpha$ and $\alpha^*$ are given in the statement of the theorem. Note that $w_\alpha \le u_\alpha$ for all $\alpha \in (0, 1/3]$ and $w_\alpha \le v_\alpha$ for all $\alpha \in (0, 3/7]$ . These observations lead us to the desired result with sharpness depicted by the function $f_{car}$ . For proving (k), we need to prove that $\phi_{car} \in \{w : |w - M| < M\}$ for all $|z| < r_{11}$ . Let $z = re^{it}$ , $x = \cos t$ . It suffices to show that the function $$p(x,r) = M^2 - |\phi_{car}(z) - M|^2$$ $$= 2M - 1 - Mr^2 - r^3x - \frac{r^4}{4} - 2(1 - M)rx - 2(1 - M)r^2x^2$$ is positive for all $x \in [-1, 1]$ and $0 < r \le s_{11}$ . Note that p(-1, r) > 0 for all r, p(1, r) > 0 for all $r \in (0, a_M]$ with $1/2 < M \le 5/4$ and p(1, r) > 0 for all $r \in (0, 1]$ if M > 5/4. Also, we have <span id="page-19-0"></span>(5.3) $$\frac{\partial}{\partial x}p(x,r) = 2(M-1)r - r^3 + 4(M-1)r^2x$$ and $\frac{\partial^2}{\partial x^2}p(x,r) = 4(M-1)r^2$ . If $1/2 < M \le 1$ , then (5.3) shows that p(x,r) is a decreasing function of x so that $p(x,r) \ge p(1,r) > 0$ as $r \le a_M$ . If M > 1, then $(\partial/\partial x)p(x,r)$ vanishes at $x_0 = (r^2 - 2(M-1))/(4(M-1)r)$ which is a point of relative minima and $$p(x_0, r) = \frac{4M^2(3 - 2r^2) + (r^2 - 2)^2 - 2M(r^2 - 2)(r^2 - 4)}{8(M - 1)} > 0$$ provided $0 < r < b_M$ and $1 < M < (3 + \sqrt{5})/4$ . Let $M$ be the root of the equation $a_M = b_M$ . Similar calculation carried out in [38, Theorem 3.1] shows that if $1 < M \le M$ , then p(x,r) > 0 for $r \le a_M$ and if $M^* \le M < (3 + \sqrt{5})/4$ , then p(x,r) > 0 for $r \le b_M$ . If $M \ge (3 + \sqrt{5})/4$ , then the result follows by Theorem 3.1(viii). Here $a_M$ and $b_M$ are defined in the statement of the theorem. <span id="page-19-1"></span>Figure 8 shows that the inclusion $\phi_{car}(\mathbb{D}) \subset \psi_C(\mathbb{D})$ holds where $\psi_C$ is defined in Theorem 4.3(e). Hence $\mathcal{S}_{car}^ \subset \mathcal{S}_C$ and this proves (l).  FIGURE 8. $\mathcal{S}_{C}^{}$ -radius for the class $\mathcal{S}_{car}^{}$ is unity. For proving (m), firstly we will show that $\mathcal{S}^_{car} \subset \mathcal{M}(\beta)$ if $\beta \geq 5/2$ . To see this, note that if $f \in \mathcal{S}^_{car}$ , Lemma 2.1 gives $\text{Re}(zf'(z)/f(z)) < \max\{\text{Re}(\phi_{car}(z)) : |z| = 1\} = 5/2$ . Thus the $\mathcal{M}(\beta)$ -radius of the class $\mathcal{S}^*_{car}$ is 1 if $\beta \in [5/2, \infty)$ . Again, Lemma 2.1 implies that $\text{Re}(zf'(z)/f(z)) < 1 + r + r^2/2 < \beta$ provided $r < -1 + \sqrt{2\beta - 1} := s_{12}$ where $1 < \beta < 5/2$ . The function $f_{car}$ satisfies $zf'_{car}(z)/f_{car}(z) = \beta$ at $z = s_{12}$ .

Theorem 6.1. The $S_{car}^*$ -radius of the classes $\mathcal{F}_i^{\chi}$ (i = 1, 2, 3) is given by the following table: | S.No. | χ | $R_1^{\chi}$ | $R_2^{\chi}$ | $R_3^{\chi}$ | |-------|---------------------|-------------------------|-------------------------|-------------------------| | (a) | z | $\frac{1}{4+\sqrt{17}}$ | $\frac{1}{3+2\sqrt{3}}$ | $\frac{1}{2+\sqrt{5}}$ | | (b) | $\frac{z}{1+z}$ | $5 - 2\sqrt{6}$ | $\sqrt{17}-4$ | $3-2\sqrt{2}$ | | (c) | $\frac{z}{1-z^2}$ | $r_1 \approx 0.11667$ | $r_2 \approx 0.14326$ | $r_3 \approx 0.20213$ | | (d) | $\frac{z}{(1-z)^2}$ | $\frac{6-\sqrt{33}}{3}$ | $5 - 2\sqrt{6}$ | $\frac{4-\sqrt{13}}{3}$ | | (e) | $z + \frac{z^2}{2}$ | $s_1 \approx 0.10924$ | $s_2 \approx 0.13414$ | $s_3 \approx 0.19028$ | Here $r_1$ , $r_2$ , $r_3$ are the smallest positive real root of the equations $3r^4 - 8r^3 - 4r^2 - 8r + 1 = 0$ , $r^4 - 6r^3 - 6r^2 - 6r + 1 = 0$ and $3r^4 - 4r^3 - 4r^2 - 4r + 1 = 0$ respectively in (0,1). Also $s_1$ , $s_2$ , $s_3$ are the smallest positive real root of the equations $3r^3 + 6r^2 - 19r + 2 = 0$ , $5r^3 - 15r + 2 = 0$ and $3r^3 + 2r^2 - 11r + 2 = 0$ respectively in (0,1). All the estimates are sharp.