Logarithmic Coefficients Problems of Geometric Subclass of Closed-to-convex Functions

Lemma 2.1. [23] Let $p \in \mathcal{P}$ be of the form (2.1). Then for a real number v, $$|p_2 - vp_1^2| \le \begin{cases} -4v + 2, & \text{if } v \le 0, \\ 2, & \text{if } 0 \le v \le 1, \\ 4v - 2, & \text{if } v \ge 1. \end{cases}$$ Moreover, equality holds under the following conditions: (i) If v < 0 or v > 1, equality holds if and only if $$p(z) = \frac{1+z}{1-z}$$ or one of its rotations. (ii) If 0 < v < 1, equality holds if and only if $$p(z) = \frac{1+z^2}{1-z^2}$$ or one of its rotations. (iii) If v=0, equality holds if and only if $$p(z) = \left(\frac{1}{2} + \frac{\lambda}{2}\right) \frac{1+z}{1-z} + \left(\frac{1}{2} - \frac{\lambda}{2}\right) \frac{1-z}{1+z}, \quad 0 \le \lambda \le 1,$$ or one of its rotations. (iv) If v = 1, equality holds only when p is the reciprocal of one of the functions that guarantees equality in the case v = 0, and only in that case. <span id="page-4-2"></span>lemma 2.2. [\[29\]](#page-18-14) Let p ∈ P be of the form [\(2.1\)](#page-3-1) and B1, B2, and B³ be numbers such that B¹ ≥ 0, B² ∈ C, and B³ ∈ R. Define Ψ+(p1, p2) and Ψ−(p1, p2) by $$\Psi_{+}(p_1, p_2) = \left| B_2 p_1^2 + B_3 p_2 \right| - \left| B_1 p_1 \right|,$$ and $$\Psi_{-}(p_1, p_2) = -\Psi_{+}(p_1, p_2).$$ Then $$\Psi_{+}(p_1, p_2) \leq \begin{cases} |4B_2 + 2B_3| - 2B_1, & when |2B_2 + B_3| \geq |B_3| + B_1, \\ 2|B_3|, & otherwise, \end{cases}$$ and $$\Psi_{-}(p_1, p_2) \leq \begin{cases} 2B_1 - B_4, & when \ B_1 \geq B_4 + 2|B_3|, \\ \frac{2B_1\sqrt{2|B_3|}}{\sqrt{B_4 + 2|B_3|}}, & when \ B_1^2 \leq 2|B_3|(B_4 + 2|B_3|), \\ 2|B_3| + \frac{B_1^2}{B_4 + 2|B_3|}, & otherwise, \end{cases}$$ where $$B_4 = |4B_2 + 2B_3|.$$ All inequalities are sharp. Let B denote the class of analytic functions φ : D → D such that φ(0) = 0. Functions in B are known as the Schwarz functions. A function φ ∈ B can be written as a power series <span id="page-4-0"></span> $$\varphi(z) = \sum_{n=1}^{\infty} c_n z^n. \tag{2.2}$$ For f, g ∈ H, we say that f is subordinate to g, denoted by f ≺ g, if there exists a Schwarz function φ such that $$f(z) = g(\varphi(z)) \ z \in \mathbb{D}.$$ In particular, if g is univalent, then f ≺ g if, and only if, f(0) = g(0) and f(D) ⊂ g(D). Next we recall the following well-known result for Schwarz functions. <span id="page-4-3"></span>lemma 2.3. [\[9\]](#page-17-13) Let φ(z) = c1z + c2z ² + c3z ³ + · · · be a Schwarz function. Then |c1| ≤ 1, |c2| ≤ 1 − |c1| 2 , |c3| ≤ 1 − |c1| ² − |c2| 2 1 + |c1| . In 1981, Prokhorov et al. [\[11\]](#page-17-14) obtained a coefficient inequality for functions in B. <span id="page-4-1"></span>lemma 2.4. [\[11\]](#page-17-14) Let φ(z) ∈ B be of the form [\(2.2\)](#page-4-0). Then for any real µ and ν,we have $$|c_3 + \mu c_1 c_2 + \nu c_1^3| = \begin{cases} 1, & \text{if } (\mu, \nu) \in D_1 \cup D_2 \cup \{(2, 1)\}, \\ \\ |\nu|, & \text{if } (\mu, \nu) \in \bigcup_{k=3}^7 D_k, \end{cases}$$ The regions D^k for k = 1, . . . , 7 are defined as follows: $$D_{1} = \{(\mu, \nu) : |\mu| \leq \frac{1}{2}, |\nu| \leq 1\},$$ $$D_{2} = \{(\mu, \nu) : \frac{1}{2} \leq |\mu| \leq 2, \frac{4}{27}(|\mu| + 1)^{3} - (|\mu| + 1) \leq |\nu| \leq 1\},$$ $$D_{3} = \{(\mu, \nu) : |\mu| \leq \frac{1}{2}, \nu \leq -1\},$$ $$D_{4} = \{(\mu, \nu) : |\mu| \geq \frac{1}{2}, \nu \leq -\frac{2}{3}(|\mu| + 1)\},$$ $$D_{5} = \{(\mu, \nu) : |\mu| \leq 2, |\nu| \geq 1\},$$ $$D_{6} = \{(\mu, \nu) : 2 \leq |\mu| \leq 4, |\nu| \geq \frac{1}{12}\sqrt{\mu^{2} + 8},$$ $$D_{7} = \{(\mu, \nu) : |\mu| \geq 4, |\nu| \geq \frac{2}{3}(|\mu| - 1)\}.$$ Moreover, all inequalities are sharp.

Theorem 3.2. Let $f \in W(\alpha)$ for $\alpha \geq 0$ , be of the form (1.1) and F be the inverse function of f have the form (1.4). Then the first three logarithmic inverse coefficients of f satisfy the sharp estimates $$|\Gamma_{1}| \leq \frac{1}{2(1+\alpha)}, \ \alpha \geq 0,$$ $$|\Gamma_{2}| \leq \begin{cases} \frac{5+10\alpha-4\alpha^{2}}{12(1+2\alpha)(1+\alpha)^{2}}, & 0 \leq \alpha < 1/2, \\ \frac{1}{3(1+2\alpha)}, & \alpha \geq 1/2, \end{cases}$$ $$|\Gamma_{3}| \leq \begin{cases} \frac{1}{4(1+3\alpha)} - \frac{4}{3(1+\alpha)(1+2\alpha)} + \frac{5}{3(1+\alpha)^{3}}, & 0 \leq \alpha \leq 0.8090, \\ \frac{1}{4(1+3\alpha)}, & \alpha > 0.8090. \end{cases}$$

Theorem 4.1. Let $f \in \mathcal{W}(\alpha)$ be of the form (1.1). Then $$|\gamma_2| - |\gamma_1| \le \frac{1}{3(1+2\alpha)}$$ for $\alpha \ge 0$ and $$|\gamma_2| - |\gamma_1| \ge \begin{cases} -\frac{1}{\sqrt{8\alpha^2 + 10\alpha + 5}} & 0 \le \alpha \le 2.232\\ -\frac{1}{3(1+2\alpha)} - \frac{3(1+2\alpha)}{4(8\alpha^2 + 10\alpha + 5)} & \alpha > 2.232. \end{cases}$$ All these inequalities are sharp.

Theorem 4.2. Let $f(z) \in W(\alpha)$ be of the form (1.1) and $\Gamma_1, \Gamma_2$ logarithmic inverse coefficients of f. Then $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{3(1+2\alpha)}$$ , for $\alpha \ge 0$ , and $$\left| \Gamma_{2} \right| - \left| \Gamma_{1} \right| \geq \begin{cases} -\frac{1}{\sqrt{3(1+2\alpha)}} & \text{for } \alpha \in [0, 1.5], \\ -\frac{1}{2(1+\alpha)} + \left| \frac{3}{4(1+\alpha)^{2}} - \frac{1}{3(1+2\alpha)} \right| & \text{for } \alpha \in [2, 5.84], \\ M(\alpha) & \text{for } \alpha \in (1.5, 2) \cup (5.84, \infty), \end{cases}$$ where $$M(\alpha) = -\frac{1}{3(1+2\alpha)} - \frac{3(1+2\alpha)}{4|4\alpha^2 - 10\alpha - 5| + 16(1+\alpha)^2}.$$ All these inequalities are sharp

Theorem 5.1. Let $f \in \mathcal{W}(1)$ be of the form (1.1). Then $$\left| H_{2,1}\left(\frac{F_f}{2}\right) \right| \le \frac{1}{81}.$$ The bound is sharp.

Theorem 5.2. If $f \in \mathcal{W}(1)$ be of the form (1.1). Then $$\left| H_{2,1}\left(\frac{F_{f^{-1}}}{2}\right) \right| \le \frac{1}{81}.$$ The inequality is sharp.