Logarithmic coefficients for exponential classes of starlike and convex functions

Lemma 2.1. [19] If $p \in \mathcal{P}$ be given by (2.1), then $$|c_2 - vc_1^2| \le \begin{cases} -4v + 2 & \text{if } v < 0, \\ 2 & \text{if } 0 \le v \le 1, \\ 4v - 2 & \text{if } v > 1. \end{cases}$$

Lemma 2.2. [1] If $p \in \mathcal{P}$ be given by (2.1) with $0 \le B \le 1$ and $2B(2B-1) \le D \le B$ . Then $|c_3 - 2Bc_1c_2 + Dc_1^3| \le 2$ . .

Lemma 2.3. [27] If $p \in \mathcal{P}$ be given by (2.1). If $\beta$ , $\gamma$ , $\delta$ and $\xi$ satisfy $0 < \beta < 1$ , $0 < \xi < 1$ , and $8\xi(1-\xi)\{(\beta\gamma-2\delta)^2+(\beta(\xi+\beta)-\gamma)^2\}+\beta(1-\beta)(\gamma-2\beta\xi)^2 \le 4\beta^2(1-\beta)^2\xi(1-\xi)$ , then $$\mid \delta c_1^4 + \xi c_2^2 + 2\beta c_1 c_3 - \frac{3}{2} \gamma c_1^2 c_2 - c_4 \mid \leq 2.$$

Lemma 2.4. [9, Lemma 2.4] If $p \in \mathcal{P}$ is of the form (2.1), then <span id="page-5-0"></span> $$c_1 = 2\tau_1, \tag{2.2}$$ <span id="page-5-1"></span> $$c_2 = 2\tau_1^2 + 2(1 - \tau_1^2)\tau_2 \tag{2.3}$$ and <span id="page-5-2"></span> $$c_3 = 2\tau_1^3 + 4(1 - \tau_1^2)\tau_1\tau_2 - 2(1 - \tau_1^2)\tau_1\tau_2^2 + 2(1 - \tau_1^2)(1 - |\tau_2|^2)\tau_3$$ (2.4) for some $\tau_1, \tau_2, \tau_3 \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \le 1\}.$ For $\tau_1 \in \mathbb{T} := \{z \in \mathbb{C} : |z| = 1\}$ , there is a unique function $p \in \mathcal{P}$ with $c_1$ 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 $c_1$ and $c_2$ 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 $c_1$ , $c_2$ and $c_3$ 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 well-known result is due to Choi et al. [10]. <span id="page-5-6"></span>Lemma 2.5. [10] Let A, B, C be real numbers and let $$\Psi(A, B, C) \coloneqq \max_{z \in \overline{\mathbb{D}}} \left\{ |A + Bz + Cz^2| + 1 - |z|^2 \right\}.$$ (i) If $AC \ge 0$ , then $$\Psi(A,B,C) = \begin{cases} |A| + |B| + |C|, & \text{if } |B| \ge 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1-|C|)}, & \text{if } |B| < 2(1-|C|). \end{cases}$$ (ii) If AC < 0, then $$\Psi(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & if \quad -4AC(C^{-2} - 1) \le B^2 \text{ and } |B| < 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & if \quad B^2 < \min\left\{4(1+|C|)^2, -4AC(C^{-2} - 1)\right\}, \\ R(A,B,C), & otherwise, \end{cases}$$ where $$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & \text{if } |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & \text{if } |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & \text{otherwise.} \end{cases}$$

Theorem 3.1. Let $\alpha \in (0,1]$ . For $f \in S_{ex}^*$ and $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ are given in (1.3). Then $$|\gamma_n| \le \frac{\alpha}{2n}, \qquad n = 1, 2, 3, 4.$$ The estimates are sharp.

Theorem 3.2. Let $\alpha \in (0,1]$ . For $f(z) \in \mathcal{C}_{ex}$ and $\gamma_1, \gamma_2, \gamma_3, \gamma_4$ are given in (1.3). Then $$|\gamma_n| \le \frac{\alpha}{2n(n+1)}, n = 1, 2, 3, 4.$$ All inequality are sharp.

Theorem 4.1. Let $\alpha \in (0,1]$ . If $f \in S_{ex}^*$ , then $$|H_{2,1}(F_f/2)| \le \frac{\alpha^2}{16}.$$ The inequality is sharp.

Theorem 4.2. Let $\alpha \in (0,1]$ . If $f \in C_{ex}$ , then $$|H_{2,1}(F_f/2)| \le \frac{\alpha^2}{144}.$$ The inequality is sharp.

Theorem 4.3. Let $\alpha \in (0,1]$ . If $f \in S_{ex}^*$ , then the following inequality hold $$|T_{2,1}(F_f/2)| \le \frac{5\alpha^2}{16}.$$

Theorem 4.4. Let $\alpha \in (0,1]$ . If $f \in \mathcal{C}_{ex}$ , then the following inequality hold $$|T_{2,1}(F_f/2)| \le \frac{5\alpha^2}{72}.$$