On A Special Type Of Ma-Minda Function

Theorem 1.4. (Distortion Theorem for $C(\Phi)$ ). Suppose $f \in C(\Phi)$ and $|z_0| = r < 1$ . Then $$d'_{\Phi}(r) \le |f'(z_0)| \le d'_{\Phi}(-r).$$ Equality holds for some non zero $z_0$ if and only if f is a rotation of $d_{\Phi}$ , given in (1.5).

Theorem 1.7. (Distortion Theorem for $S^(\Phi)$ ). Suppose $f \in S^(\Phi)$ and $|z_0| = r < 1$ . Then $$t'_{\Phi}(r) < |f'(z_0)| < t'_{\Phi}(-r).$$ Equality holds for some non zero $z_0$ if and only if f is a rotation of $t_{\Phi}$ , given in (1.6). We now introduce the following classes involving the special type of Ma-Minda function $\psi$ : $$\mathcal{S}_l^* := \left\{ f \in \mathcal{S} : \frac{zf'(z)}{f(z)} \prec 1 - \log(1+z) \right\} \text{ and } \mathcal{C}_l := \left\{ f \in \mathcal{S} : 1 + \frac{zf''(z)}{f'(z)} \prec 1 - \log(1+z) \right\}.$$ By the structural formula (1.6), we get a function $f \in \mathcal{S}_l^*$ if and only if there exists an analytic function q, satisfying $q(z) \prec \psi(z)$ such that $$f(z) = z \exp\left(\int_0^z \frac{q(t) - 1}{t} dt\right). \tag{1.8}$$ Now, we give some examples of the functions in the class $\mathcal{S}_{l}^{*}$ . For this, let us assume $$\psi_1(z) = 1 - \frac{z}{6}$$ , $\psi_2(z) = \frac{4-z}{4+z}$ , $\psi_3(z) = 1 - z \sin \frac{z}{4}$ and $\psi_4(z) = \frac{8-2z}{8-z}$ . A geometrical observation leads to $\psi_i(z) \subset \psi(z)$ (i = 1, 2, 3, 4). Thus $\psi_i(z) \prec \psi(z)$ . Now, the functions $f_i's$ belonging to the class $\mathcal{S}_l^*$ corresponding to each of the functions $\psi_i's$ are determined by the structural formula (1.8) as follows: $$f_1(z) = z \exp\left(\frac{-z}{6}\right), f_2(z) = \frac{16z}{(4+z)^2}, f_3(z) = z \exp\left(4\left(1+\cos\frac{z}{4}\right)\right) \text{ and } f_4(z) = z - \frac{z^2}{8}.$$ In particular, for $q(z) = \psi(z) = 1 - \log(1+z)$ , the corresponding function obtained as follows: $$f_0(z) = z \exp\left(\int_0^z \frac{-\log(1+t)}{t} dt\right) = z - z^2 + \frac{3}{4}z^3 - \frac{19}{36}z^4 + \frac{107}{288}z^5 + \cdots,$$ (1.9) acts as an extremal function in many cases for $\mathcal{S}_{l}^{*}$ . Remark 1.8. The distortion and growth theorems for $C_l$ and $S_l$ can be obtained from that of $C(\Phi)$ and $S^(\Phi)$ , given in Theorem 1.5. Here, we establish inclusion results, radius problems, majorization result and estimation of the Bloch function norm for the functions in the class $\mathcal{S}_l^*$ . In the coefficient bound section, we consider the class: $$\mathcal{A}(g,h,\phi) =: \mathcal{M}_{g,h}(\phi) = \left\{ f \in \mathcal{A} : \frac{(fg)(z)}{(fh)(z)} \prec \phi(z), \, \phi \in \mathcal{M} \right\},\tag{1.10}$$ where Taylor series expansion of g, h is given by (1.1) and $g_n, h_n > 0$ with $g_n - h_n > 0$ . This class is defined in [18] and authors have obtained Fekete-Szegö bound for the same. We determine the bounds of fourth coefficient $|a_4|$ , second Hankel determinant $|a_2a_4 - a_3^2|$ and the quantity $|a_2a_3 - a_4|$ for the functions in the class $\mathcal{M}_{g,h}(\phi)$ . The importance of this class lies in unification of various subclasses of $\mathcal{S}$ , discussed in detail in the coefficient section. Some of our results reduce to many earlier known results of Lee et al. [13], Mishra et al. [17] and Singh [26]. In view of (1.10), we also consider the class $\mathcal{M}_{g,h}(\Phi)$ for $\Phi$ in $\mathscr{M}^{\circ}$ . Now, we introduce the class: $$\mathcal{M}_{\alpha}(\Phi) = \left\{ f \in \mathcal{A} : \frac{zf'(z) + \alpha z^2 f''(z)}{\alpha z f'(z) + (1 - \alpha)f(z)} \prec \Phi(z), \ (0 \le \alpha \le 1) \right\}.$$ Note that when $g(z) = (z(1+(2\alpha-1)z))/(1-z)^3$ and $h(z) = (z(1+(\alpha-1)z))/(1-z)^2$ , we have $\mathcal{M}_{g,h}(\Phi) =: \mathcal{M}_{\alpha}(\Phi)$ . Further, the power series expansion of g and h, respectively yield $$g_2 = 2(1+\alpha), g_3 = 3(1+2\alpha), g_4 = 4(1+3\alpha)...$$ and $h_2 = 1+\alpha, h_3 = 1+2\alpha, h_4 = 1+3\alpha...$ (1.11) By setting $\mathcal{M}_{\alpha}(\psi) =: \mathcal{S}_{l}(\alpha)$ , then $\mathcal{S}_{l}(0) = \mathcal{S}_{l}^{*}$ and $\mathcal{S}_{l}(1) = \mathcal{C}_{l}$ . We obtain the sharp bounds of initial coefficients such as $a_{2}$ , $a_{3}$ , $a_{4}$ and $a_{5}$ , Fekete-Szegö functional, second Hankel determinant for functions in $\mathcal{S}_{l}(\alpha)$ . Further, using these sharp bounds, we estimate the third Hankel determinant bound for the functions in $\mathcal{S}_{l}(\alpha)$ . We need the following lemmas to support our results.

Lemma 1.9. [15] Let $p \in \mathcal{P}$ be of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$|p_2 - vp_1^2| \le \begin{cases} -4v + 2, & v \le 0; \\ 2, & 0 \le v \le 1; \\ 4v - 2, & v \ge 1. \end{cases}$$ When v < 0 or v > 1, the equality holds if and only if p(z) is (1+z)/(1-z) or one of its rotations. If 0 < v < 1, then the equality holds if and only if $p(z) = (1+z^2)/(1-z^2)$ or one of its rotations. If v = 0, the equality holds if and only if $p(z) = (1+\eta)(1+z)/(2(1-z)) + (1-\eta)(1-z)/(2(1+z))(0 \le \eta \le 1)$ or one of its rotations. If v = 1, the equality holds if and only if p is the reciprocal of one of the functions such that the equality holds in the case of v = 0. Though the above upper bound is sharp for 0 < v < 1, still it can be improved as follows: $$|p_2 - vp_1^2| + v|p_1|^2 \le 2$$ $(0 < v \le 1/2)$ and $|p_2 - vp_1^2| + (1 - v)|p_1|^2 \le 2$ $(1/2 \le v < 1)$ . $(1.12)$

Lemma 1.10. [9] Let $p \in \mathcal{P}$ be of the form $1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$2p_2 = p_1^2 + x(4 - p_1^2),$$ $$4p_3 = p_1^3 + 2p_1(4 - p_1^2)x - p_1(4 - p_1^2)x^2 + 2(4 - p_1^2)(1 - |x|^2)y$$ for some x and y such that $|x| \le 1$ and $|y| \le 1$ . The following result is proved in [14]:

Lemma 1.11. Let $p \in \mathcal{P}$ with coefficients $p_n$ as above, then $$|p_3 - 2p_1p_2 + p_1^3| \le 2 \text{ and } |p_1^4 - 3p_1^2p_2 + p_2^2 + 2p_1p_3 - p_4| \le 2.$$ (1.13)

Theorem 2.1. Let $f \in \mathcal{S}_l^*$ . Then we have for |z| = r < 1, $$1 - \log(1+r) \le \operatorname{Re} \frac{zf'(z)}{f(z)} \le 1 - \log(1-r) \tag{2.1}$$ and $$\left| \operatorname{Im} \frac{zf'(z)}{f(z)} \right| \le \tan^{-1} \left( \frac{r}{\sqrt{1 - r^2}} \right). \tag{2.2}$$

Theorem 2.2. Let $f \in \mathcal{S}_{l}^{*}$ . Then the following holds: - (i) f is starlike of order $\alpha$ in $|z| < \exp(1-\alpha) 1$ whenever $1 \log 2 \le \alpha < 1$ . - (ii) $f \in \mathcal{M}(\beta)$ in $|z| < 1 \exp(1 \beta)$ whenever $\beta > 1$ . - (iii) f is convex of order $\alpha$ in $|z| < \tilde{r}(\alpha) < 1$ whenever $0 \le \alpha < 1$ , where $\tilde{r}(\alpha)$ is the smallest positive root of the equation: $$(1-r)(1-\log(1+r))(1-\log(1+r)-\alpha)-r=0, (2.6)$$ for the given value of $\alpha$ . (iv) f is strongly starlike of order $\gamma$ in $|z| < r(\gamma)$ whenever $0 < \gamma \le \gamma_0 \approx 0.514674$ , where $$r(\gamma) = \sqrt{2\left(1 - \frac{1}{\sqrt{1 + \tan^2\left(\tan\frac{\gamma\pi}{2}\right)}}\right)}.$$ (2.7) (v) f is k-starlike function in |z| < r(k) whenever k > 0, where r(k) is the smallest positive root of the equation $$1 + r - e(1 - r)^k = 0, (2.8)$$ for the given value of k. In particular, for k=1, f is parabolic starlike in $|z|<\frac{e-1}{e+1}$ .

Theorem 2.3. Let $f \in A$ . Suppose that $f \ll g$ in $\mathbb{D}$ , where $g \in \mathcal{S}_{l}^{*}$ . Then $$|f'(z)| \le |g'(z)|$$ , for $|z| \le \tilde{r}$ , where $\tilde{r}$ is the smallest positive root of the following equation: $$(1 - r2)(1 - \log(1 + r)) - 2r = 0. (2.12)$$

Theorem 3.1. The class $S_l^*$ satisfies the following results: - (i) $S_l^ \subset S^(\alpha) \subset S^*$ for $0 \le \alpha \le 1 \log 2$ . - (ii) $S_l^ \subset SS^(\gamma) \subset S^*$ for $2\tilde{f}(\theta_0)/\pi \leq \gamma \leq 1$ , where $\theta_0$ is the smallest positive root of the equation $-2 + \log(2(1 + \cos\theta)) + \theta \tan\theta/2 = 0$ and $\tilde{f}(\theta) = \arg(1 \log(1 + e^{i\theta}))$ , $\theta \in [0, \pi)$ . - (iii) $S_l^* \subset \mathcal{ST}(1, \alpha)$ for $\alpha \leq 1 2 \log 2$ . - (iv) $\mathcal{S}^(q_c) \subset \mathcal{S}^_l \subset \mathcal{S}^*$ for $c \leq c_0$ , where $c_0 = \log 2(2 \log 2)$ . The above constants in each part is best possible. The pictorial representation of the result is depicted in the Figure 1.

Theorem 4.2. For $f \in \mathcal{M}_{q,h}(\Phi)$ , we have $$|a_3 - ta_2^2| \le \begin{cases} \frac{C_2}{g_3 - h_3} - \frac{tC_1^2}{(g_2 - h_2)^2} + \frac{(g_2h_2 - h_2^2)C_1^2}{(g_3 - h_3)(g_2 - h_2)^2}, & t \le \kappa_1; \\ \frac{-C_1}{g_3 - h_3}, & \kappa_1 \le t \le \kappa_2; \\ \frac{-C_2}{g_3 - h_3} + \frac{tC_1^2}{(g_2 - h_2)^2} - \frac{(g_2h_2 - h_2^2)C_1^2}{(g_3 - h_3)(g_2 - h_2)^2}, & t \ge \kappa_2, \end{cases}$$ where $$\kappa_1 = \frac{(g_2 - h_2)^2 (C_2 + C_1) + h_2 (g_2 - h_2) C_1^2}{(g_3 - h_3) C_1^2} \text{ and } \kappa_2 = \frac{(g_2 - h_2)^2 (C_2 - C_1) + h_2 (g_2 - h_2) C_1^2}{(g_3 - h_3) C_1^2}.$$ The result is sharp whenever f satisfies: $$\frac{(f g)(z)}{(f h)(z)} = \begin{cases} \Phi(z), & t < \kappa_1 \text{ or } t > \kappa_2; \\ \Phi(z^2), & \kappa_1 < t < \kappa_2; \\ \Phi(\Psi(z)), & t = \kappa_1; \\ \Phi(-\Psi(z)), & t = \kappa_2, \end{cases}$$ where $$\Psi(z) = \frac{z(z+\eta)}{1+\eta z}$$ $(0 \le \eta \le 1)$ . Remark 4.3. We notice that the bound of Fekete-Szegö stated in [18, Theorem 6.1], namely $|a_3 - a_3|$ $|\mu a_2^2| \leq B_1/2(g_3 - h_3)$ when $\sigma_1 \leq \mu \leq \sigma_2$ , is incorrect and should be $|a_3 - \mu a_2^2| \leq B_1/(g_3 - h_3)$ , which is appropriately corrected in Theorem 4.2. In the following example, we establish a Fekete-Szegö result for the class $S_l(\alpha)$ : Example 1. Let $f \in \mathcal{S}_l(\alpha)$ . Then $$|a_3 - ta_2^2| \le \begin{cases} \frac{3}{4(1+2\alpha)} - \frac{t}{(1+\alpha)^2}, & t \le \frac{(1+\alpha)^2}{4(1+2\alpha)} =: \kappa_1; \\ \frac{1}{2(1+2\alpha)}, & \frac{(1+\alpha)^2}{4(1+2\alpha)} \le t \le \frac{5(1+\alpha)^2}{4(1+2\alpha)}; \\ \frac{t}{(1+\alpha)^2} - \frac{3}{4(1+2\alpha)}, & t \ge \frac{5(1+\alpha)^2}{4(1+2\alpha)} =: \kappa_2. \end{cases}$$ The result is sharp. Proof. Since $f \in S_l(\alpha) = \mathcal{M}_{\alpha}(\psi(z))$ , we have $C_1 = -1$ , $C_2 = 1/2$ and $C_3 = -1/3$ . The result follows from Theorem 4.2 by substituting the values of $g_i's$ and $h_i's$ from (1.11). Equality holds whenever f satisfies: $$\frac{(f g)(z)}{(f h)(z)} = \begin{cases} 1 - \log(1+z), & t < \kappa_1 \text{ or } t > \kappa_2; \\ 1 - \log(1+z^2), & \kappa_1 < t < \kappa_2; \\ 1 - \log(1 + \frac{z(z+\eta)}{1+\eta z})), & t = \kappa_1; \\ 1 - \log(1 - \frac{z(z+\eta)}{1+\eta z}), & t = \kappa_2. \end{cases}$$ Example 2. Let $f \in \mathcal{S}_l(\alpha)$ . Then (i) $$|a_3 - a_2^2| \le \frac{1}{2(1+2\alpha)}$$ , (ii) $|a_3| \le \frac{3}{4(1+2\alpha)}$ . These results are sharp. The proof directly follows from Example 1.

Theorem 4.4. Let $f \in \mathcal{M}_{q,h}(\phi)$ and either $$(g_3 - h_3)^2 \le L$$ or $L < (g_3 - h_3)^2 \le 2L$ , (4.2) where $L = (g_2 - h_2)(g_4 - h_4)$ , then (1) $$|a_2a_4 - a_3^2| \le \frac{B_1^2}{(g_3 - h_3)^2},$$ whenever $B_1$ , M and T satisfy the conditions $$|M| - B_1^2 (g_2 - h_2)^4 (g_4 - h_4) \le 0 \text{ and } |T| + B_1 (g_3 - h_3)^2 (g_2 - h_2) - 2B_1 (g_2 - h_2)^2 (g_4 - h_4) \le 0.$$ (4.3) $$|a_2a_4 - a_3^2| \le \frac{|M|}{(g_2 - h_2)^4 (g_3 - h_3)^2 (g_4 - h_4)},$$ whenever $B_1$ , M and T satisfy the conditions $$|T| + B_1(g_3 - h_3)^2(g_2 - h_2) - 2B_1(g_2 - h_2)^2(g_4 - h_4) \ge 0$$ and $$2|M| - B_1|T|(g_2 - h_2)^2 - B_1^2(g_3 - h_3)^2(g_2 - h_2)^3 \ge 0$$ or $$|T| + B_1(g_3 - h_3)^2(g_2 - h_2) - 2B_1(g_2 - h_2)^2(g_4 - h_4) \le 0$$ and $$|M| - B_1^2 (g_2 - h_2)^4 (g_4 - h_4) \ge 0.$$ (3) $$|a_{2}a_{4} - a_{3}^{2}| \leq -\frac{1}{(|M| - B_{1}|T|(g_{2} - h_{2})^{2} - B_{1}^{2}(g_{3} - h_{3})^{2}(g_{2} - h_{2})^{3} + B_{1}^{2}(g_{2} - h_{2})^{4}(g_{4} - h_{4}))} \times \frac{B_{1}^{2}(|T| + B_{1}(g_{3} - h_{3})^{2}(g_{2} - h_{2}) - 2B_{1}(g_{2} - h_{2})^{2}(g_{4} - h_{4}))^{2}}{4(g_{3} - h_{3})^{2}(g_{4} - h_{4})} + \frac{B_{1}^{2}}{(g_{3} - h_{3})^{2}},$$ whenever $B_1$ , M and T satisfy the conditions $$|T| + B_1(g_3 - h_3)^2(g_2 - h_2) - 2B_1(g_2 - h_2)^2(g_4 - h_4) > 0$$ (4.4) and $$2|M| - B_1|T|(g_2 - h_2)^2 - B_1^2(g_3 - h_3)^2(g_2 - h_2)^3 \le 0, (4.5)$$ where $$M = B_1^4 \left( -h_2^2 (g_2 - h_2)^2 (g_4 - h_4) + (g_3 - h_3) \left( g_2 g_3 h_2^2 - g_3 h_2^3 + g_2^2 h_2 h_3 - 3 g_2 h_2^2 h_3 + 2 h_2^3 h_3 \right) + (g_3 - h_3) (-g_2 h_2^2 + h_2^3) \right) - B_2^2 (g_2 - h_2)^4 (g_4 - h_4) + B_1 B_3 (g_3 - h_3)^2 (g_2 - h_2)^3 + B_1^2 B_2 \left( (g_3 - h_3)(g_2 - h_2)^2 \left( g_3 h_2 + g_2 h_3 - 2 h_2 h_3 - 2 h_2 (g_2 - h_2)(g_4 - h_4) \right) \right)$$ $$(4.6)$$ and $$T = 2B_2(g_2 - h_2)^2(g_4 - h_4) + 2B_1^2h_2(g_2 - h_2)(g_4 - h_4) - B_1^2g_3h_2(g_3 - h_3) - B_1^2g_2h_3(g_3 - h_3) + 2B_1^2h_2h_3(g_3 - h_3) - 2B_2(g_3 - h_3)^2(g_2 - h_2).$$ $$(4.7)$$

Corollary 4.9. Let $f \in \mathcal{S}_s^*(\phi)$ . Then we have $$|a_2a_4 - a_3^2| \le \begin{cases} B_1^2/4, & \text{when $A$ holds;} \\ |M|/256, & \text{when $B$ holds;} \\ \frac{B_1^2}{4} - \frac{B_1^2(|T| - 24B_1)^2}{64(|M| - 4B_1|T| + 32B_1^2)}, & \text{when $C$ holds,} \end{cases}$$ where $$M=16B_1^2B_2-64B_2^2+32B_1B_3,\ T=16B_2-4B_1^2\ and$$ $$A:|M|-64B_1^2\leq 0\ and\ |T|-24B_1\leq 0.$$ $$B:|M|-2B_1|T|-16B_1^2\geq 0\ and\ |T|-24B_1\geq 0,\ or$$ $$|M|-64B_1^2\geq 0\ and\ |T|-24B_1\leq 0.$$ $$C:|M|-2B_1|T|-16B_1^2\leq 0\ and\ |T|-24B_1>0.$$

Corollary 4.10. Let $f \in C_s(\phi)$ . Then we have $$|a_2a_4 - a_3^2| \le \begin{cases} B_1^2/36, & \text{when } A \text{ holds}; \\ |M|/147456, & \text{when } B \text{ holds}; \\ \frac{B_1^2}{36} - \frac{B_1^2(|T| - 368B_1)^2}{2304(|M| - 16B_1|T| + 1792B_1^2)}, & \text{when } C \text{ holds}, \end{cases}$$ where $$M = 128(9B_1^2B_2 - 32B_2^2 + 18B_1B_3)$$ , $T = 8(28B_2 - 9B_1^2)$ and, $A: |M| - 4096B_1^2 \le 0$ and $|T| - 368B_1 \le 0$ . $B: |M| - 8B_1|T| - 1152B_1^2 \ge 0$ and $|T| - 368B_1 \ge 0$ , or $|T| - 368B_1 \le 0$ and $|M| - 4096B_1^2 \ge 0$ . $C: |T| - 368B_1 > 0$ and $|M| - 8B_1|T| - 1152B_1^2 < 0$ . Remark 4.11. When $\phi(z) = \frac{1+z}{1-z}$ , the Corollaries 4.9 and 4.10 reduce to the results obtained in [17] for the classes $S_s^*$ and $C_s$ of starlike functions and convex functions with respect to symmetric points respectively. Remark 4.12. Note that the second Hankel determinant bound for the functions in the classes $S_s^*(\Phi)$ and $C_s(\Phi)$ can be obtained from the Corollaries 4.9 and 4.10, respectively by replacing each $B_i$ by $(-1)^i C_i$ . Expressing the fourth coefficient $a_4$ for the function $f \in \mathcal{M}_{g,h}(\phi)$ in terms of the Schwarz function $\omega(z) = 1 + \omega_1 z + \omega_2 z^2 + \cdots$ , we obtain the bound of $a_4$ as follows: $$|a_4| \le \frac{B_1}{q_4 - h_4} H(q_1, q_2),\tag{4.15}$$ where $$q_1 = \frac{2B_2(g_2 - h_2)(g_3 - h_3) + B_1^2(g_3h_2 + g_2h_3 - 2h_2h_3)}{B_1(g_2 - h_2)(g_3 - h_3)}$$ (4.16) and $$q_2 = \frac{B_3(g_2 - h_2)(g_3 - h_3) + B_1^3 h_2 h_3 + B_1 B_2(g_3 h_2 + g_2 h_3 - 2h_2 h_3)}{B_1(g_2 - h_2)(g_3 - h_3)}.$$ (4.17) Remark 4.13. In view of first case of (4.1), for $\phi(z) = \sqrt{1+z}$ , the above result (4.15) reduces to the result obtained in [22]. Remark 4.14. Note that the bound for the fourth coefficient for the functions in the class $\mathcal{M}_{g,h}(\Phi)$ can be obtained from (4.15), (4.16) and (4.17) by replacing each $B_i$ by $(-1)^i C_i$ . Example 4. Let $f \in \mathcal{S}_l(\alpha)$ , then $$|a_4| \le \frac{19}{36(1+3\alpha)}.$$ The result is sharp.

Theorem 4.15. Let $f \in S_l(\alpha)$ . Then, we have $$|a_5| \le \frac{107}{288(1+4\alpha)}.$$ The result is sharp.

Theorem 4.16. Let $f \in S_l(\alpha)$ . Then $$|H_3(1)| \leq g(\alpha),$$ where $$g(\alpha) = \frac{949 + 11388\alpha + 52493\alpha^2 + 114974\alpha^3 + 117180\alpha^4 + 42568\alpha^5}{1728(1+4\alpha)(1+3\alpha)^2(1+2a)^4},$$ when $0 \le \alpha \le \frac{2+\sqrt{15}}{11}$ and $$g(\alpha) = \frac{1}{1728(1+\alpha)(1+4\alpha)(1+3\alpha)^2(1+2\alpha)^3(61\alpha^2-20\alpha-5)} \left(-5069-76035\alpha-385994\alpha^2-619570\alpha^3+831511\alpha^4+3545777\alpha^5+3327024\alpha^6+1298324\alpha^7\right),$$ when $\frac{2+\sqrt{15}}{11} \le \alpha \le 1$ . Remark 4.17. Taking $\alpha = 0$ and 1, we get all the above bounds for the classes $\mathcal{S}_l^*$ and $\mathcal{C}_l$ , respectively. On the similar lines of the estimation of Third Hankel determinant for functions in $\mathcal{SL}$ in [5], we compute the same for $f \in \mathcal{S}_l$ .

Theorem 4.18. Let $f \in \mathcal{S}_l^*$ , then $$|H_3(1)| \le 1/9.$$ The result is sharp.

Theorem 5.1. The set $\mathcal{S}_{l}^{} \subseteq \mathcal{B}$ . Further, if $f \in \mathcal{S}_{l}^{}$ , then $||f||_{\mathcal{B}} \leq x \approx 1.27429$ .

Theorem 5.2. Let $m, n \ge 1$ and $0 \le \lambda \le 1$ . Then, $g(z) = z \exp(\alpha) \in \mathcal{S}_{l}^{*}$ , where $$\alpha = \sum_{k=1}^{\infty} \frac{1}{k^2} \left( \lambda \left( \frac{(-z)^{nk}}{n} - \frac{(-z)^{mk}}{m} \right) + \frac{(-z)^{mk}}{m} \right).$$

Corollary 5.3. Let $n \ge 1$ and $g(z) = z \exp(\alpha)$ , where $$\alpha = \frac{1}{n} \left( \sum_{k=1}^{\infty} \frac{(-z)^{nk}}{k^2} \right).$$ We have $g \in \mathcal{S}_{l}^{*}$ .

Theorem 5.4. The class $S_l^*$ is not a vector space.

Theorem 5.5. Let $f \in \mathcal{S}_l^*$ , then we have $$|f(z)| \le |z| \exp\left(\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}\right) = |z|L \ (z \in \mathbb{D}),$$ where $L \approx 0.822467$ .

Definition 1.1. An analytic univalent function $\phi$ with $\phi'(0) > 0$ , satisfying: A: Re $\phi(z) > 0 \ (z \in \mathbb{D})$ B: $\phi(\mathbb{D})$ symmetric about the real axis and starlike with respect to $\phi(0) = 1$ is called a $Ma\text{-}Minda\ function$ , we denote the class of all such functions by $\mathscr{M}$ . If the condition A above alone is relaxed, the resulting function, we call it a $non\text{-}Ma\text{-}Minda\ of\ type\text{-}A$ , the class of all such functions is denoted by $\widetilde{\mathscr{M}}_{\mathbf{A}}$ . Recently, the classes given in (1.2) were studied extensively for different choices of $\phi$ . Prominently, Aouf et al. [3], studied the class $\mathcal{S}^(q_c)$ , where $q_c = \sqrt{1+cz}$ (0 < $c \le 1$ ), Robertson [23] introduced the class of starlike functions of order alpha $(0 \le \alpha \le 1)$ , denoted by $\mathcal{S}^(\alpha)$ by opting $\phi(z)$ to be $(1 + (1 - 2\alpha)z)/(1 - z)$ and when $\phi(z) = ((1 + z)/(1 - z))^{\eta}$ , $\mathcal{S}^(\phi)$ reduces to the class of strongly starlike functions of order $\eta$ , which can be represented in terms of argument as $\mathcal{S}\mathcal{S}^(\eta) := \{f \in \mathcal{A} : |\arg z f'(z)/f(z)| < \eta \pi/2, \ (0 < \eta \le 1)\}$ . Consider the class $k - \mathcal{S}\mathcal{P}(\alpha, \beta)$ , which is introduced in [25], for k = 1, it reduces to $1 - \mathcal{S}\mathcal{P}(\alpha, \beta) := \{f \in \mathcal{A} : z f'(z)/f(z) \prec \phi(z)\}$ , where $\phi(z) = \alpha + ((2(\alpha - \beta))/\pi^2)(\log((1 + \sqrt{\nu(z)})/(1 - \sqrt{\nu(z)})))^2$ with $\nu(z) = (z + \rho)/(1 + \rho z)$ , $\rho = ((e^A - 1)/(e^A + 1))^2$ and $A = \sqrt{(1 - \alpha)/(2(\alpha - \beta))\pi}$ . The authors in [4,8,10,16] dealt with the radius, inclusion and differential subordination results for the classes involving $\phi(z)$ . Many authors have determined the coefficient bounds for the classes associated with $\phi(z)$ (see [12,13,17,21,22]). In the past, authors considered non-Ma-Minda functions, for instance Kargar et al. [11] and Uralegaddi et al. [27] considered functions in $\widetilde{\mathcal{M}}_{\mathbf{A}}$ to define their classes. We come across the following observations, enlisted below, while examining the geometry of a function defined on $\mathbb{D}$ in general, which are of great use in deriving our results: (1) A function with real coefficients is always symmetric with respect to the real axis, but not conversely, for instance: $$f_1(z) = iz, f_2(z) = 1 + iz, f_3(z) = \frac{1 + iz}{1 - z^2}.$$ The converse holds under special conditions, namely if f is symmetric with respect to the real axis, f(0) = 0 and f'(0) be some non zero real number, then the function f has real coefficients. In fact the functions $f_1$ , $f_2$ and $f_3$ are symmetric with respect to the real axis but do not have real coefficients as $f'_i(0)$ is not a real number for (i = 1, 2, 3). - (2) Let f(z) be an analytic function with real coefficients and f(0) = 0. Then f is typically real if and only if its first coefficient is positive. Thus $\phi'(0) > 0$ implies $\phi 1$ is typically real whereas the function $\Phi 1$ is non typically real due to $\Phi'(0) < 0$ . - (3) Geometrically, it is evident that the real part of a function attains its maximum/ minimum value on the real line if and only if the function is symmetric with respect to the real axis and convex in the direction of imaginary axis. The Ma-Minda function $\phi$ is considered as univalent and therefore $\phi'(0) \neq 0$ . Since $\phi(\mathbb{D})$ is symmetric about the real axis and if $\phi'(0)$ is any non-zero real number, then $\phi$ has real coefficients. Now to address distortion theorem, Ma-Minda perhaps restricted $\phi'(0)$ to be positive instead of any non-zero real number. However, it has no influence in establishing the coefficient, radius, inclusion, subordination, and other results for the classes $\mathcal{C}(\phi)$ and $\mathcal{S}^*(\phi)$ . This very fact, which is under gloom until now, has been brought to daylight in this paper by replacing the condition $\phi'(0) > 0$ with $\phi'(0) < 0$ . Note that $\phi(z)$ and $\Phi(z) := \phi(-z)$ both map unit disk to the same image but different orientation. Thus $\Phi(z)$ differs from its Ma-Minda counterpart by mere a rotation and is therefore non-typically real, but still, image domain invariant and rest all properties are intact. So $\Phi(z)$ can be considered as a special type of Ma-Minda function. We now premise the above notion in the following definition:

Definition 1.2. An analytic univalent function $\Phi$ defined on the unit disk $\mathbb{D}$ is said to be a special type of Ma-Minda if Re $\Phi(\mathbb{D}) > 0$ , $\Phi(\mathbb{D})$ is symmetric with respect to the real axis, starlike with respect to $\Phi(0) = 1$ and $\Phi'(0) < 0$ . Further, it has a power series expansion of the form: $$\Phi(z) = 1 + \sum_{n=1}^{\infty} C_n z^n = 1 + C_1 z + C_2 z^2 + \cdots \quad (C_1 < 0).$$ The class of all such special type of Ma-Minda functions are denoted by $\mathcal{M}^{\circ}$ . Recently, Altinkaya et al. [2] considered a special type of Ma-Minda function $g(z) = \alpha(1-z)/(\alpha-z)$ , $(\alpha > 1)$ ) to define and study their class $P(\alpha)$ . Now the classes $\mathcal{S}^*(\Phi)$ and $\mathcal{C}(\Phi)$ can be defined on the similar lines of (1.2). We introduce here a special type of Ma-Minda function, given by $$\psi(z) := 1 - \log(1+z) = 1 - z + \frac{z^2}{2} - \frac{z^3}{3} + \cdots, \tag{1.4}$$ which maps the unit disk onto a parabolic region, see Figure 1 for its boundary curve $\tau$ . Although $\phi(\mathbb{D}) = \Phi(\mathbb{D})$ , at times considering $\Phi$ is advantageous over its counterpart $\phi$ , which is evident from the example $\Phi(z) = 1 - \log(1+z)$ , dealt here. Another such example is $\cos \sqrt{z}$ . Thus the special type of Ma-Minda functions can now be considered in defining Ma-Minda classes for computational convenience as all results are alike except distortion and growth. We now list in Table 1, a few examples of $\phi \in \mathcal{M}$ and its counter part $\Phi \in \mathcal{M}^{\circ}$ : | $\phi(z)$ | $\Phi(z)$ | |-------------------|-----------------| | $\cos\sqrt{-z}$ | $\cos\sqrt{z}$ | | $\sqrt{1+z}$ | $\sqrt{1-z}$ | | $1 - \log(1 - z)$ | $1 - \log(1+z)$ | Table 1. Examples of Ma-Minda and its counter part Special type of Ma-Minda functions. Distortion and Growth Theorems: Let us define the functions in a similar manner as that in [15]: $d_{\Phi n}(n=1,2,3,\cdots)$ by $d_{\Phi n}(0)=d'_{\Phi n}(0)-1=0$ and $$1 + \frac{zd_{\Phi n}''(z)}{d_{\Phi n}'(z)} = \Phi(z^n), \tag{1.5}$$ which belongs to the class $\mathcal{C}(\Phi)$ and we write $d_{\Phi 1}$ as $d_{\Phi}$ . The structural formula of $d'_{\Phi n}$ is given by: $$d'_{\Phi n}(z) = \exp \int_0^z \frac{\Phi(t^n) - 1}{t} dt, \tag{1.6}$$ which upon simplification, gives the structural formula of $d_{\Phi n}$ . Similarly, we define $t_{\Phi n}(n=1,2,3,\cdots)$ by $t_{\Phi n}(0)=t'_{\Phi n}(0)-1=0$ and $$\frac{zt'_{\Phi n}(z)}{t_{\Phi n}(z)} = \Phi(z^n),$$ which belongs to the class $\mathcal{S}^*(\Phi)$ and we write $t_{\Phi 1}$ as $t_{\Phi}$ . The structural formula for $t_{\Phi n}$ is given by: $$t_{\Phi n}(z) = z \exp \int_0^z \frac{\Phi(t^n) - 1}{t} dt.$$ (1.7) Note that $zd'_{\Phi n}(z) = t_{\Phi n}(z)$ . Ma-Minda [15] proved the distortion and growth theorems for the classes $C(\phi)$ and $S^*(\phi)$ when $\phi \in \mathcal{M}$ . Here, we prove that the result does not remain same in the case of functions in $\mathscr{M}^{\circ}$ . It is examined with an example and which is further generalized. For this, let us consider the class $\mathcal{C}(\psi)$ , the structural formula, given in (1.6) yields: $$d'_{\psi}(z) = \exp \sum_{k=1}^{\infty} \frac{(-z)^k}{k^2}.$$ A numerical computation shows that $d'_{\psi}(1/2) \approx 0.63864$ and $d'_{\psi}(-1/2) \approx 1.79004$ . Let the function f be such that: $$f'(z) = d'_{\psi 2} = \exp \sum_{k=1}^{\infty} \frac{(-1)^k (z)^{2k}}{2k^2},$$ clearly, $f \in \mathcal{C}(\psi)$ . A numerical computation shows that $|f'(1/2)| \approx 0.88874$ . Hence $$d'_{\psi}(r) \le |f'(z_0)| \le d'_{\psi}(-r)$$ , for $z_0 = r = \frac{1}{2}$ . Thus functions in $C(\psi)$ violate distortion theorem, which shows that $\phi'(0) > 0$ is inevitable in obtaining the distortion theorem of [15] for functions in $\mathcal{M}$ . Remark 1.3. Let $\phi \in \mathcal{M}$ and its counter part $\Phi \in \mathcal{M}^{\circ}$ then $\Phi(\mathbb{D}) = \phi(\mathbb{D})$ , which implies $\mathcal{C}(\Phi) = \mathcal{C}(\phi)$ and $\mathcal{S}^(\Phi) = \mathcal{S}^(\phi)$ . Therefore to obtain distortion and growth theorems for functions in $\mathcal{C}(\Phi)$ and $\mathcal{S}^*(\Phi)$ , it is sufficient to replace $\phi(z)$ by $\Phi(-z)$ , in the result [15, Corollary 1, p. 159]. Using the the above Remark and the fact $d'_{\Phi}(z) = d'_{\phi}(-z)$ , we deduce the following result: