Sharp Coefficient Estimates for the Exponential Starlike class

Lemma 2.1. [6, Lemma 2.4] If $p \in \mathcal{P}$ is of the form (2.1), then $$p_1 = 2\tau_1, \tag{2.2}$$ $$p_2 = 2\tau_1^2 + 2(1 - |\tau_1|^2)\tau_2 \tag{2.3}$$ and $$p_3 = 2\tau_1^3 + 4(1 - |\tau_1|^2)\tau_1\tau_2 - 2(1 - |\tau_1|^2)\overline{\tau_1}\tau_2^2 + 2(1 - \tau_1^2)(1 - |\tau_2|^2)\tau_3$$ (2.4) for some $\tau_1, \tau_2, \tau_3 \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \leq 1\}$ . For $\tau_1 \in \mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$ , there is a unique function $p \in \mathcal{P}$ with $p_1$ as in (2.2), namely, $$p(z) = \frac{1 + \tau_1 z}{1 - \tau_1 z}, \quad z \in \mathbb{D}.$$ For $\tau_1 \in \mathbb{D}$ and $\tau_2 \in \mathbb{T}$ , there is a unique function $p \in \mathcal{P}$ with $p_1$ and $p_2$ as in (2.2) and (2.3), namely, $$p(z) = \frac{1 + (\overline{\tau}_1 \tau_2 + \tau_1)z + \tau_2 z^2}{1 + (\overline{\tau}_1 \tau_2 - \tau_1)z - \tau_2 z^2}, \quad z \in \mathbb{D}.$$ For $\tau_1, \tau_2 \in \mathbb{D}$ and $\tau_3 \in \mathbb{T}$ , there is a unique function $p \in \mathcal{P}$ with $p_1, p_2$ , and $p_3$ as in (2.2-2.4), namely, $$p(z) = \frac{1 + (\overline{\tau}_2 \tau_3 + \overline{\tau}_1 \tau_2 + \tau_1)z + (\overline{\tau}_1 \tau_3 + \tau_1 \overline{\tau}_2 \tau_3 + \tau_2)z^2 + \tau_3 z^3}{1 + (\overline{\tau}_2 \tau_3 + \overline{\tau}_1 \tau_2 - \tau_1)z + (\overline{\tau}_1 \tau_3 - \tau_1 \overline{\tau}_2 \tau_3 - \tau_2)z^2 - \tau_3 z^3}, \quad z \in \mathbb{D}$$ Following a well-known result due to Choi et al. [7].

Lemma 2.2. [7] Let A, B, C be real numbers and let $$Y(A, B, C) := \max_{z \in \overline{\mathbb{D}}} \{ |A + Bz + Cz^2| + 1 - |z|^2 \}.$$ (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 $$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & \text{if } -4AC(C^{-2}-1) \leq B^2 \text{ and } |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & \text{if } B^2 < \min\left\{4(1+|C|)^2, -4AC(C^{-2}-1)\right\}, \\ R(A,B,C), & \text{otherwise}, \end{cases}$$ where $$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & \text{if } |C|(|B| + 4|A|) \leq |AB|, \\ -|A| + |B| + |C|, & \text{if } |AB| \leq |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & \text{otherwise.} \end{cases}$$

Lemma 2.3. [34] Let $p \in \mathcal{P}$ be given by (2.1). Then $$|p_2 - vp_1^2| \le \begin{cases} -4v + 2, & v < 0, \\ 2, & 0 \le v \le 1, \\ 4v - 2, & v > 1. \end{cases}$$ Moreover, for v < 0 or v > 1, equality holds if and only if $$h(z) = \frac{1+z}{1-z}$$ or one of its rotations. For 0 < v < 1, equality holds if and only if $$h(z) = \frac{1+z^2}{1-z^2}$$ or one of its rotations.

Lemma 2.4. [42] Let J, K, and L be numbers such that $J \geq 0$ , $K \in \mathbb{C}$ , and $L \in \mathbb{R}$ . Let $p \in \mathcal{P}$ be of the form (2.1) and define a function by $$\Phi(p_1, p_2) = |Kp_1^2 + Lp_2| - |Jp_1|.$$ Then $$\Phi(p_1, p_2) \le \begin{cases} |4K + 2L| - 2J, & \text{if } |2K + L| \ge |L| + J, \\ 2|L|, & \text{otherwise.} \end{cases}$$ and $$-\Phi(p_1, p_2) \le \begin{cases} 2J - M, & when \ J \ge M + 2|L|, \\ 2J\sqrt{\frac{2|L|}{M + 2|L|}}, & when \ J^2 \le 2|L|(M + 2|L|), \\ 2|L| + \frac{J^2}{M + 2|L|}, & otherwise \end{cases}$$ where M = |4K + 2L|.

Theorem 3.1. Let $0 < \alpha \le 1$ and define the parameter $\alpha_0 = \sqrt{\frac{12}{29}} \approx 0.6432$ . For any function $f \in \mathcal{S}_{ex}^*$ , the initial inverse logarithmic coefficients satisfy the following sharp bounds: $$|\Gamma_1| \le \frac{\alpha}{2},\tag{3.1}$$ $$|\Gamma_2| \le \begin{cases} \frac{\alpha}{4}, & \text{if } 0 < \alpha \le \frac{2}{3}, \\ \frac{3\alpha^2}{8}, & \text{if } \frac{2}{3} < \alpha \le 1, \end{cases}$$ $$(3.2)$$ $$|\Gamma_3| \le \begin{cases} \frac{\alpha}{6}, & \text{if } 0 < \alpha \le \alpha_0, \\ \frac{29\alpha^3}{72}, & \text{if } \alpha_0 < \alpha \le 1. \end{cases}$$ $$(3.3)$$ Equality is attained in each case for the extremal functions $f_{Koebe}(z)$ , $f_{odd}(z)$ , or $f_{tri}(z)$ .

Theorem 4.1. Let $0 < \alpha \le 1$ . If $f \in \mathcal{S}_{ex}^*$ and the inverse logarithmic coefficients $\Gamma_n$ (n = 1, 2) are defined by (1.8), then $$-\frac{\alpha}{2}\sqrt{\frac{2}{3\alpha+2}} \le |\Gamma_2| - |\Gamma_1| \le \frac{\alpha}{4}.\tag{4.1}$$ These inequalities are sharp for all $\alpha \in (0,1]$ .

Theorem 5.1. Let $0 < \alpha \le 1$ . If $f \in \mathcal{S}_{ex}^*$ and the inverse logarithmic coefficients $\Gamma_n$ (n = 1, 2, 3) defined by (1.8), then the second-order inverse logarithmic Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ satisfies the sharp inequalities: $$\left| H_{2,1} \left( F_{f^{-1}} / 2 \right) \right| \le \begin{cases} \frac{\alpha^2}{16}, & \text{if } 0 < \alpha \le \frac{2}{5}, \\ \frac{\alpha^2 (15\alpha^2 + 10\alpha + 4)}{4(35\alpha^2 + 60\alpha + 12)}, & \text{if } \frac{2}{5} < \alpha \le 1. \end{cases}$$ (5.1) These inequalities are sharp for all $\alpha \in (0,1]$ .

Theorem 6.1. Let $0 < \alpha \le 1$ . If $f \in \mathcal{S}_{ex}^*$ is given by (1.2), then the third-order Hermitian–Toeplitz determinant satisfies the sharp inequalities: $$1 - 2\alpha^2 + \frac{15\alpha^4}{16} \le T_{3,1}(f) \le 1. \tag{6.3}$$ Both bounds are sharp for all $\alpha \in (0,1]$ .

Theorem 7.1. Let $0 < \alpha \le 1$ , $\lambda \in \mathbb{C}$ , and $\mu > 0$ . If $f \in \mathcal{S}_{ex}^*$ is given by (1.2), then the generalized Fekete–Szegő functional satisfies the sharp inequalities $$\mathcal{B}_L \le |a_3 - \lambda a_2^2| - \mu |a_2| \le \mathcal{B}_U, \tag{7.2}$$ where the upper bound $\mathcal{B}_U$ is given by $$\mathcal{B}_{U} = \begin{cases} \frac{\alpha}{2}, & \text{if } |\alpha(2-\lambda)+1| < 1+2\mu, \\ \frac{\alpha|\alpha(2-\lambda)+1| - 2\alpha\mu}{2}, & \text{if } |\alpha(2-\lambda)+1| \ge 1+2\mu, \end{cases}$$ (7.3) and the lower bound $\mathcal{B}_L$ is given by $$\mathcal{B}_{L} = \begin{cases} -\frac{\alpha}{2}(2\mu - |\alpha(2-\lambda) + 1|), & \text{if } \mu - 1 \ge |\alpha(2-\lambda) + 1|, \\ -\alpha\mu\sqrt{\frac{1}{|\alpha(2-\lambda) + 1| + 1}}, & \text{if } \mu^{2} - 1 \le |\alpha(2-\lambda) + 1|, \\ -\frac{\alpha}{2}\left(\frac{|\alpha(2-\lambda) + 1| + \mu^{2} + 1}{|\alpha(2-\lambda) + 1| + 1}\right), & \text{if } \mu - 1 < |\alpha(2-\lambda) + 1| < \mu^{2} - 1. \end{cases}$$ $$(7.4)$$ Both bounds are sharp.