The second and third Hankel determinants for certain classes of functions

Theorem 3.1. Let $$-1 < B < A \le 1$$ , and define $p(z) = \frac{1 + Az}{1 + Bz}$ . Then $p(z) \prec \varphi(z)$ provided that $\frac{1}{4} \le \frac{1 - A}{1 - B}$ and $\frac{1 + A}{1 + B} \le \frac{9}{4}$ .

Theorem 3.2. Let $f \in \mathcal{S}^*(\varphi)$ . Then $f \in \mathcal{C}_{\gamma}$ in $|z| < r_{\gamma}$ , where $r_{\gamma}$ is the least positive root of the equation $g(r) = \gamma$ , and $$g(r) = \left(1 - r - \frac{r^2}{4}\right) - \frac{r\left(1 + \frac{r}{2}\right)}{\left(1 - \frac{r}{2}\right)^2 (1 - r^2)}, \qquad \gamma \in [0, 1).$$

Lemma 3.1. [8] Let P(z) be a function in the Carathéodory class $\mathcal{P}$ , and $s \in \mathbb{N}$ , then $|p_n - p_{n-s} p_s| \leq 2$ , $n \geq s$ . for n = 1, 2, 3, ...

Theorem 3.3. Let $f(z) = z + \sum_{n=2}^{\infty} a_n z^n \in \mathcal{S}^*(\varphi)$ . Then the following coefficient bounds hold $$|a_2| \le 1$$ , $|a_3| \le \frac{5}{8}$ , and $|a_4| \le 0.338667$ . All these bounds are sharp.

Lemma 4.1. [8] Let $$p \in \mathcal{P}$$ be of the form $p(z) = 1 + \sum_{n=1}^{\infty} p_n z^n$ . Then $$2p_{2} = p_{1}^{2} + \gamma (4 - p_{1}^{2}),$$ $$4p_{3} = p_{1}^{3} + 2(4 - p_{1}^{2})p_{1}\gamma - (4 - p_{1}^{2})p_{1}\gamma^{2} + 2(4 - p_{1}^{2})(1 - |\gamma|^{2})\eta,$$ $$8p_{4} = p_{1}^{4} + (4 - p_{1}^{2})\gamma (p_{1}^{2}(\gamma^{2} - 3\gamma + 3) + 4\gamma)$$ $$-4(4 - p_{1}^{2})(1 - |\gamma|^{2})(p_{1}(\gamma - 1)\eta + \overline{\gamma}\eta^{2} - (1 - |\eta|^{2})\rho)$$ for some complex numbers $\gamma$ , $\eta$ , and $\rho$ such that $|\gamma| \leq 1$ , $|\eta| \leq 1$ , and $|\rho| < 1$ . Next, we recall the following well-known result by Choi et al. [3]

Lemma 4.2. [3] Let $A, B, C \in \mathbb{R}$ and define $$Y(A, B, C) := \max_{z \in \overline{\mathbb{D}}} (|A + Bz + Cz^2| + 1 - |z|^2).$$ (i) If $AC \geq 0$ , then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & \text{if } |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & \text{if } |B| < 2(1 - |C|). \end{cases}$$ (ii) If AC < 0, then $$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & \text{if } -4AC(C^2-1) \leq B^2 \text{ and } |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & \text{if } B^2 < \min\{4(1+|C|)^2, -4AC(C^2-1)\}, \\ R(A,B,C), & \text{otherwise,} \end{cases}$$ where. $$R(A, B, C) = \begin{cases} |A| + |B| + |C|, & \text{if } |C| (|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & \text{if } |AB| \le |C| (|B| - 4|A|), \\ (|A| + |C|) \sqrt{1 - \frac{B^2}{4AC}}, & \text{otherwise.} \end{cases}$$ We now proceed to prove the main results of this paper.

Theorem 4.1. Let $f \in \mathcal{S}^*(\varphi)$ . Then $|H_2(2)| \leq 1/4$ . The result is sharp.

Lemma 4.3. Let $w(z) = \sum_{n=1}^{\infty} c_n z^n$ be a Schwarz function, that is, $\omega$ is analytic in $\mathbb{D}$ , w(0) = 0, and |w(z)| < 1 for all $z \in \mathbb{D}$ . If $c_1 \geq 0$ , then there exist complex numbers $\gamma, \eta, \rho$ with $|\gamma| \leq 1$ , $|\eta| \leq 1$ , $|\rho| \leq 1$ , such that $$c_{2} = (1 - c_{1}^{2})\gamma,$$ $$c_{3} = (1 - c_{1}^{2})(\eta(1 - |\gamma|^{2}) - c_{1}\gamma^{2}),$$ $$c_{4} = (1 - c_{1}^{2})(c_{1}^{2}\gamma^{3} - (1 - |\gamma|^{2})(2c_{1}\gamma\eta + \overline{\gamma}\eta^{2}) + (1 - |\gamma|^{2})(1 - |\eta|^{2})\rho).$$

Theorem 4.2. Let $f \in \mathcal{S}^*(\varphi)$ . Then $|H_3(1)| \leq 1/9$ , where $H_3(1)$ denote the third Hankel determinant. Moreover, the estimate is sharp.