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 $M \ge (3+\sqrt{5})/4 \approx 1.309$ ;

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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  $M \ge (3+\sqrt{5})/4 \approx 1.309$ ;

Corollary 3.2. $\mathcal{S}^*[1-\alpha,0] \subset \mathcal{S}^*_{car}$ for all $\alpha \in [1/2,1)$ and $\mathcal{S}^*[\alpha,-\alpha] \subset \mathcal{S}^*_{car}$ for all $\alpha \in (0,1/3]$ . In particular, if $f \in \mathcal{A}$ satisfies $$\left|\frac{zf'(z)}{f(z)} - 1\right| < \frac{1}{2} \quad or \quad \left|\frac{zf'(z)/f(z) - 1}{zf'(z)/f(z) + 1}\right| < \frac{1}{3}$$ for all $z \in \mathbb{D}$ , then $f \in \mathcal{S}_{car}^*$ .

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Corollary 3.2. $\mathcal{S}^[1-\alpha,0] \subset \mathcal{S}^_{car}$  for all  $\alpha \in [1/2,1)$  and  $\mathcal{S}^[\alpha,-\alpha] \subset \mathcal{S}^_{car}$  for all  $\alpha \in (0,1/3]$ . In particular, if  $f \in \mathcal{A}$  satisfies

$$\left|\frac{zf'(z)}{f(z)} - 1\right| < \frac{1}{2} \quad or \quad \left|\frac{zf'(z)/f(z) - 1}{zf'(z)/f(z) + 1}\right| < \frac{1}{3}$$

for all  $z \in \mathbb{D}$ , then  $f \in \mathcal{S}_{car}^*$ .

Corollary 3.3. (i) If $f \in A$ is given by (1.3) such that $$\sum_{n=2}^{\infty} (2n-1)|a_n| \le 1$$ then $f \in \mathcal{S}_{car}$ . (ii) A function $f_n(z) = z + a_n z^n$ (n = 2, 3, ...) belongs to the class $\mathcal{S}_{car}$ if and only if $|a_n| \leq 1/(2n-1)$ .

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.  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 $R_3 = \min\{1, 3/(2A - 5B)\}$ .

Corollary 4.2. (i) For $0 \le \alpha < 1$ , the $\mathcal{S}^*_{car}$ -radius for the class $\mathcal{S}^*(\alpha)$ is $1/(3-4\alpha)$ if $0 \le \alpha \le 1/4$ and $3/(7-4\alpha)$ if $1/4 < \alpha < 1$ . (ii) For $0 \le \alpha < 1$ , the $\mathcal{S}^*_{car}$ -radius for the class $\mathcal{S}^*[1-\alpha,0]$ is $1/(2(1-\alpha))$ if $0 < \alpha < 1/2$ and 1 if $1/2 \le \alpha < 1$ . (iii) For $0 < \alpha \le 1$ , the $\mathcal{S}^*_{car}$ -radius for the class $\mathcal{S}^*[\alpha,-\alpha]$ is 1 if $0 < \alpha \le 1/3$ and $1/(3\alpha)$ if $1/3 < \alpha \le 1$ . (iv) For M > 1/2, the $\mathcal{S}^*_{car}$ -radius for the class $\mathcal{S}^*[1,-(M-1)/M]$ is M/(3M-1).

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Corollary 4.2. (i) For  $0 \le \alpha < 1$ , the  $\mathcal{S}^_{car}$ -radius for the class  $\mathcal{S}^(\alpha)$  is  $1/(3-4\alpha)$  if  $0 \le \alpha \le 1/4$  and  $3/(7-4\alpha)$  if  $1/4 < \alpha < 1$ . (ii) For  $0 \le \alpha < 1$ , the  $\mathcal{S}^_{car}$ -radius for the class  $\mathcal{S}^[1-\alpha,0]$  is  $1/(2(1-\alpha))$  if  $0 < \alpha < 1/2$  and 1 if  $1/2 \le \alpha < 1$ . (iii) For  $0 < \alpha \le 1$ , the  $\mathcal{S}^_{car}$ -radius for the class  $\mathcal{S}^[\alpha,-\alpha]$  is 1 if  $0 < \alpha \le 1/3$  and  $1/(3\alpha)$  if  $1/3 < \alpha \le 1$ . (iv) For M > 1/2, the  $\mathcal{S}^_{car}$ -radius for the class  $\mathcal{S}^[1,-(M-1)/M]$  is M/(3M-1).

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

$$\operatorname{Re}\left(\frac{zf'(z)}{f(z)}\right) > \min_{|z|=r} \operatorname{Re}(\phi_{car}(z)) = 1 - r + \frac{r^2}{2} > \alpha$$

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.

![](_page_17_Figure_2.jpeg)

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  $\phi_{car}(\mathbb{D}_{s_6})$ .

![](_page_17_Figure_9.jpeg)

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

(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

(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.

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).

![](_page_19_Figure_11.jpeg)

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.

Corollary 6.2. Let $f, g \in S$ . Then $h_{\rho}(z) = (f g)(\rho z)/\rho \in S^*_{car}$ for $0 < \rho \le (4 - \sqrt{13})/3 \approx 0.1314829$ . The radius is sharp. Proof. By the proof of Theorem 6.1(d), the $\mathcal{S}_{car}$ -radius of the class $\mathcal{F}_3^{z/(1-z)^2}$ is $\rho_0 = (4-\sqrt{13})/3$ and this radius is sharp for the function $f_0(z) = z(1+z)/(1-z)^3$ . Since $f, g \in \mathcal{S}$ , the functions F and G defined by zF'(z) = f(z) and zG'(z) = g(z) belongs to the class $\mathcal{K}$ . Consequently $F G \in \mathcal{K}$ . By using Lemma 2.3, $(f_0 F G)(\rho z)/\rho \in \mathcal{S}_{car}$ for $0 < \rho \le \min\{1/2, \rho_0\} = \rho_0$ . Since $f g = f_0 F * G$ , the result follows.