Sharp Estimates of Logarithmic Coefficients for a Certain Class of Starlike Functions

Lemma 2.1. [27, 28] If $p \in \mathcal{P}$ is of the form (2.4) with $c_1 \geq 0$ , then $$(2.5) c_1 = 2\tau_1.$$ $$(2.6) c_2 = 2\tau_1^2 + 2(1 - \tau_1^2)\tau_2$$ and $$(2.7) c_3 = 2\tau_1^3 + 4(1-\tau_1^2)\tau_1\tau_2 - 2(1-\tau_1^2)\tau_1\tau_2^2 + 2(1-\tau_1^2)(1-|\tau_2|^2)\tau_3$$ for some $\tau_1 \in [0,1]$ and $\tau_2, \tau_3 \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \le 1\}.$ 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 (2.5), 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 (2.5) and (2.6), 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 (2.5)–(2.7), namely $$p(z) = \frac{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 + \tau_1)z + (\overline{\tau_1}\tau_3 + \tau_1\overline{\tau_2}\tau_3 + \tau_2)z^2 + \tau_3z^3}{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 - \tau_1)z + (\overline{\tau_1}\tau_3 - \tau_1\overline{\tau_2}\tau_3 - \tau_2)z^2 - \tau_3z^3}, \quad z \in \mathbb{D}.$$ Lemma 2.2. [9] Let A, B, C be real numbers and $$Y(A, B, C) := \max\{|A + Bz + Cz^2| + 1 - |z|^2 : z \in \overline{\mathbb{D}}\}.$$ (i) If AC ≥ 0, then $$Y(A, B, C) = \begin{cases} |A| + |B| + |C|, & |B| \ge 2(1 - |C|), \\ 1 + |A| + \frac{B^2}{4(1 - |C|)}, & |B| < 2(1 - |C|). \end{cases}$$ (ii) If AC < 0, then $$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & -4AC(C^{-2}-1) \le B^2 \land |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\{4(1+|C|)^2, -4AC(C^{-2}-1)\}, \\ R(A,B,C), & otherwise, \end{cases}$$ where $$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$ Lemma 2.3. [30] Let p ∈ P be given by (2.1). Then $$|c_2 - vc_1^2| \le \begin{cases} -4v + 2 & v < 0, \\ 2 & 0 \le v \le 1, \\ 4v - 2 & v > 1. \end{cases}$$ For v < 0 or v > 1, the equality holds if, and only if, $$h(z) = \frac{1+z}{1-z}$$ or one of its rotations. If 0 < v < 1, then the equality is true if, and only if, $$h(z) = \frac{1+z^2}{1-z^2}$$ or one of its rotations. Lemma 2.4. [3] Let p ∈ P be given by (2.1) with 0 ≤ B ≤ 1 and B(2B − 1) ≤ D ≤ B. Then $$|c_3 - 2Bc_1c_2 + Dc_1^3| \le 2.$$ The significance of logarithmic coefficients in geometric function theory has led to a growing interest in finding sharp bound of logarithmic coefficients and the Hankel determinants with these coefficients. We obtain the following sharp bound of initial logarithmic coefficients γ1, γ2, γ³ and H2,1(F^f /2) for functions in the class S ∗ ch. Theorem 2.1. Let f(z) = z + a2z ² + a3z ³ + · · · ∈ S^∗ ch and γ1, γ2, γ³ are given by (2.2). Then we have $$|\gamma_n| \le \frac{1}{2n}$$ for $n = 1, 2, 3$ . The inequality is sharp for the following functions: $$(2.8) f_1(z) = z \exp\left(\int_0^z \frac{t + \cosh(t) - 1}{t} dt\right) = z + z^2 + \frac{3}{4}z^3 + \frac{5}{12}z^4 + \cdots, \ z \in \mathbb{D}.$$ (2.9) $$f_2(z) = z \exp\left(\int_0^z \frac{t^2 + \cosh(t^2) - 1}{t} dt\right) = z + \frac{1}{2}z^3 + \frac{1}{4}z^5 + \cdots, \quad z \in \mathbb{D}.$$ (2.10) $$f_3(z) = z \exp\left(\int_0^z \frac{t^3 + \cosh(t^3) - 1}{t} dt\right) = z + \frac{1}{3}z^4 + \frac{5}{36}z^7 + \dots, \ z \in \mathbb{D}.$$ Proof of Theorem 2.1. Let $f \in \mathcal{S}_{ch}^*$ . Then there exists an analytic function $\omega \in \mathcal{H}$ with $\omega(0) = 0$ and $|\omega(z)| < 1$ for $z \in \mathbb{D}$ , such that (2.11) $$\frac{zf'(z)}{f(z)} = \varphi_0(\omega(z)) = \omega(z) + \cosh(\omega(z)), \quad z \in \mathbb{D}.$$ Let $p \in \mathcal{P}$ . Then, by using the definition of subordination, we have $$p(z) = \frac{1 + \omega(z)}{1 - \omega(z)} = 1 + c_1 z + c_2 z^2 + \cdots, \ z \in \mathbb{D}.$$ Hence, it is evident that $$\omega(z) = \frac{p(z) - 1}{p(z) + 1}$$ $$= \frac{c_1}{2}z + \frac{1}{2}\left(c_2 - \frac{c_1^2}{2}\right)z^2 + \frac{1}{2}\left(c_3 - c_1c_2 + \frac{c_1^3}{4}\right)z^3$$ $$+ \frac{1}{2}\left(c_4 - c_1c_3 + \frac{3c_1^2c_2}{4} - \frac{c_2^2}{2} - \frac{c_1^4}{8}\right)z^4 + \cdots$$ (2.12) in $\mathbb{D}$ . Then p is analytic in $\mathbb{D}$ with p(0) = 1 and has positive real part in $\mathbb{D}$ . In view of (2.11) together with $\varphi_0(\omega(z))$ , a tedious computation shows that $$\omega(z) + \cosh(\omega(z)) = 1 + \frac{1}{2}c_1z + \left(-\frac{1}{8}c_1^2 + \frac{1}{2}c_2\right)z^2 + \left(-\frac{1}{4}c_1c_2 + \frac{1}{2}c_3\right)z^3 + \left(-\frac{1}{4}c_1c_3 + \frac{13}{384}c_1^4 - \frac{1}{8}c_2^2 + \frac{1}{2}c_4\right)z^4 + \cdots$$ (2.13) and (2.14) $$\frac{zf'(z)}{f(z)} = 1 + a_2z + (2a_3 - a_2^2)z^2 + (3a_4 - 3a_2a_3 + a_2^3)z^3 + (4a_5 - 2a_3^2 - 4a_2a_4 + 4a_2^2a_3 - a_2^4)z^4 + \cdots$$ Thus, using (2.13) and (2.14), we compute from (2.11) that (2.15) $$\begin{cases} a_2 = \frac{1}{2}c_1, \\ a_3 = \frac{1}{16}c_1^2 + \frac{1}{4}c_2, \\ a_4 = -\frac{1}{96}c_1^3 + \frac{1}{24}c_1c_2 + \frac{1}{6}c_3. \end{cases}$$ Sharp bounds of $\gamma_1$ : By using (2.2) and (2.15), it is easy to that $$|\gamma_1| = \left| \frac{1}{2} a_2 \right| = \frac{1}{4} |c_1| \le \frac{1}{2}.$$ The desired inequality is thus obtained. The function $f_1$ , which is defined in (2.8) gives the sharpness of the inequality (2.16). Sharp bounds of $\gamma_2$ : From (2.2) and (2.15), we see that $$|\gamma_2| = \left| \frac{1}{2} \left( a_3 - \frac{1}{2} a_2^2 \right) \right|$$ $$= \left| \frac{1}{2} \left( \frac{1}{16} c_1^2 + \frac{1}{4} c_2 - \frac{1}{2} \left( \frac{1}{2} c_1 \right)^2 \right) \right|$$ $$= \frac{1}{8} \left| c_2 - \frac{1}{4} c_1^2 \right|.$$ By using Lemma 2.3, we obtain the desired inequality $$(2.17) |\gamma_2| \le \frac{1}{4}.$$ The function $f_2$ , which is defined in (2.9) gives the sharpness of the inequality (2.17). Sharp bounds of $\gamma_3$ : By using (2.2) and (2.15), we obtain $$\begin{aligned} |\gamma_3| &= \left| \frac{1}{2} \left( a_4 - a_2 a_3 + \frac{1}{3} a_2^3 \right) \right| \\ &= \left| \frac{1}{2} \left( -\frac{1}{96} c_1^3 + \frac{1}{24} c_1 c_2 + \frac{1}{6} c_3 - \frac{1}{32} c_1^3 - \frac{1}{8} c_1 c_2 + \frac{1}{24} c_1^3 \right) \right| \\ &= \frac{1}{12} \left| c_3 - 2Bc_1 c_2 + Dc_1^3 \right|, \end{aligned}$$ where $B = \frac{1}{4}$ and D = 0. Therefore, it is easy to see that $0 \le B \le 1$ and the inequality $B(2B-1) \le D \le B$ . By Lemma 2.4, we have the inequality $$(2.18) |\gamma_3| \le \frac{1}{6}.$$ The function $f_3$ , defined in (2.10), establishes the sharpness of the inequality (2.18). This completes the proof. Next, we obtain a result finding the sharp bound of the second Hankel determinant $H_{2,1}(F_f/2)$ with logarithmic coefficients for functions in the class $\mathcal{S}_{ch}^*$ .

Theorem 2.2. Let $f \in \mathcal{S}_{ch}^*$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\gamma_1, \gamma_2, \gamma_3$ are given by (2.2). Then we have $$|H_{2,1}(F_f/2)| \le \frac{1}{16}.$$ The inequality is sharp for the function $f_2$ , which is defined in (2.9). Proof of Theorem 2.2. Let $f \in \mathcal{S}_{ch}^*$ . Then there exists an analytic function $\omega \in \mathcal{H}$ with $\omega(0) = 0$ and $|\omega(z)| < 1$ for $z \in \mathbb{D}$ , such that (2.19) $$\frac{zf'(z)}{f(z)} = \varphi_0(\omega(z)) = \omega(z) + \cosh(\omega(z)), \quad z \in \mathbb{D}.$$ Let $p \in \mathcal{P}$ . Then, in view of the definition of subordination, we have $$p(z) = \frac{1 + \omega(z)}{1 - \omega(z)} = 1 + c_1 z + c_2 z^2 + \cdots, \ z \in \mathbb{D}.$$ It is easy to see that $$\omega(z) = \frac{p(z) - 1}{p(z) + 1}$$ $$= \frac{c_1}{2}z + \frac{1}{2}\left(c_2 - \frac{c_1^2}{2}\right)z^2 + \frac{1}{2}\left(c_3 - c_1c_2 + \frac{c_1^3}{4}\right)z^3$$ $$+ \frac{1}{2}\left(c_4 - c_1c_3 + \frac{3c_1^2c_2}{4} - \frac{c_2^2}{2} - \frac{c_1^4}{8}\right)z^4 + \cdots$$ (2.20) in $\mathbb{D}$ . Then p is analytic in $\mathbb{D}$ with p(0) = 1 and has positive real part in $\mathbb{D}$ . In view of (2.19) together with $\varphi_0(\omega(z))$ , a tedious computation shows that (2.21) $$\omega(z) + \cosh(\omega(z)) = 1 + \frac{1}{2}c_1z + \left(-\frac{1}{8}c_1^2 + \frac{1}{2}c_2\right)z^2 + \left(-\frac{1}{4}c_1c_2 + \frac{1}{2}c_3\right)z^3 + \left(-\frac{1}{4}c_1c_3 + \frac{13}{384}c_1^4 - \frac{1}{8}c_2^2 + \frac{1}{2}c_4\right)z^4 + \cdots$$ and (2.22) $$\frac{zf'(z)}{f(z)} = 1 + a_2z + (2a_3 - a_2^2)z^2 + (3a_4 - 3a_2a_3 + a_2^3)z^3 + (4a_5 - 2a_3^2 - 4a_2a_4 + 4a_2^2a_3 - a_2^4)z^4 + \cdots$$ Thus, using (2.21) and (2.22), we see from (2.19) that (2.23) $$\begin{cases} a_2 = \frac{1}{2}c_1, \\ a_3 = \frac{1}{16}c_1^2 + \frac{1}{4}c_2, \\ a_4 = -\frac{1}{96}c_1^3 + \frac{1}{24}c_1c_2 + \frac{1}{6}c_3, \\ a_5 = \frac{1}{192}c_1^4 - \frac{5}{192}c_2c_1^2 + \frac{1}{48}c_1c_3 + \frac{1}{8}c_4. \end{cases}$$ A simple computation by using (2.3) and (2.14), shows that $$H_{2,1}(F_f/2) = \frac{1}{48} \left( a_2^4 - 12a_3^2 + 12a_2a_4 \right)$$ $$= \frac{1}{3072} \left( -3c_1^4 - 8c_1^2c_2 - 48c_2^2 + 64c_1c_3 \right).$$ By Lemma 2.1 and (2.15), we obtain $$H_{2,1}(F_f/2) = \frac{1}{192} \left( -3\tau_1^4 + 4\tau_1^2 \tau_2 \left( 1 - \tau_1^2 \right) - 4\tau_2^2 (3 + \tau_1^2) (1 - \tau_1^2) + 16\tau_1 \tau_3 \left( 1 - \tau_1^2 \right) \left( 1 - |\tau_2|^2 \right) \right).$$ (2.25) We now explore three possible cases involving $\tau_1$ . Case-I. Let $\tau_1 = 1$ . Then, from (2.25) we see that $$|H_{2,1}(F_f/2)| = \frac{1}{64} \approx 0.015625.$$ Case-II. Let $\tau_1 = 0$ . Then, from (2.25) we get $$|H_{2,1}(F_f/2)| = \left| \frac{1}{192} \left( -12\tau_2^2 \right) \right| \le \frac{1}{16} \approx 0.0625.$$ Case-III. Let $\tau_1 \in (0,1)$ . Applying triangle inequality in (2.25) and using the fact that $|\tau_3| \leq 1$ , we obtain $$|H_{2,1}(F_f/2)| \leq \frac{1}{192} \left( \left| -3\tau_1^4 + 4\tau_1^2 \tau_2 (1 - \tau_1^2) - 4\tau_2^2 (1 - \tau_1^2)(3 + \tau_1^2) \right| + 16\tau_1 (1 - \tau_1^2)(1 - |\tau_2|^2) \right)$$ $$= \frac{1}{12} \tau_1 (1 - \tau_1^2) \left( |A + B\tau_2 + C\tau_2^2| + 1 - |\tau_2|^2 \right)$$ $$:= \frac{1}{12} \tau_1 (1 - \tau_1^2) Y(A, B, C),$$ $$(2.26)$$ where $$A = \frac{-3\tau_1^3}{16(1-\tau_1^2)}, \ B = \frac{\tau_1}{4}, \ \text{and} \ C = -\frac{(3+\tau_1^2)}{4\tau_1}.$$ We note that AC > 0. Hence, we can apply case (i) of Lemma 2.2 and discuss the following cases. A simple computation shows that $$|B| - 2(1 - |C|) = \frac{\tau_1}{4} - 2\left(1 - \frac{3 + \tau_1^2}{4\tau_1}\right) = \frac{6 - 8\tau_1 + 3\tau_1^2}{4\tau_1} > 0$$ for all $\tau_1 \in (0,1)$ . i.e., |B| > 2(1-|C|). Thus from Lemma 2.2, we see that $$Y(A, B, C) = |A| + |B| + |C| = \frac{12 - 4\tau_1^2 - 5\tau_1^4}{16\tau_1 (1 - \tau_1^2)}.$$ In view of the inequality (2.26) it follows that $$|H_{2,1}(F_f/2)| = \frac{1}{12}\tau_1(1-\tau_1^2)(|A|+|B|+|C|)$$ $$= \frac{1}{192}(12-4\tau_1^2-5\tau_1^4)$$ $$= \frac{1}{192}\phi(\tau_1),$$ (2.27) where $\phi(t) = 12 - 4t^2 - 5t^4$ for $t \in [0,1]$ . A simple computation shows that $\phi'(t) = -8t - 20t^3 < 0$ for all $t \in [0,1]$ which shows that $\phi$ is a decreasing function on [0,1]. Hence, the maximum of $\phi(t)$ is attained at t = 0, and the maximum value is 12. Hence, from (2.27), we see that $$|H_{2,1}(F_{f/2})| \le \frac{1}{16}.$$ Thus the desired inequality is established. By summarizing Cases I, II, and III, we obtain the desired inequality of the result. The function $f_2$ , which is defined in (2.9) gives the sharpness of the desired inequality. This completes the

Theorem 3.1. Let $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots \in \mathcal{S}_{ch}^*$ and $\Gamma_1, \Gamma_2$ are given by (3.4). Then we have $$|\Gamma_1| \le \frac{1}{2}$$ and $|\Gamma_2| \le \frac{3}{8}$ . The inequalities are sharp for the functions $f_1$ , defined in (2.8). Proof of Theorem 3.1. From (2.23) and (3.4), we have $$|\Gamma_1| = \left| -\frac{1}{2}a_2 \right| = \left| -\frac{1}{4}c_1 \right| \le \frac{1}{2}.$$ By using (2.23) and (3.4), we have $$|\Gamma_2| = \left| -\frac{1}{2} \left( a_3 - \frac{3}{2} a_2^2 \right) \right|$$ $$= \left| -\frac{1}{2} \left( \frac{1}{16} c_1^2 + \frac{1}{4} c_2 - \frac{3}{8} c_1^2 \right) \right|$$ $$= \frac{1}{8} \left| c_2 - \frac{5}{4} c_1^2 \right|.$$ By Lemma 2.3, we obtain $$|\Gamma_2| \le \frac{3}{8}.$$ The function $f_1$ , which is defined in (2.8) gives the sharpness of inequality for $\Gamma_1$ and $\Gamma_2$ . This completes the proof. Next, we obtain a result finding the sharp bound of the second Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ with logarithmic coefficients for inverse functions in the class $\mathcal{S}_{ch}^*$ .

Theorem 3.2. Let $f \in \mathcal{S}_{ch}^*$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\Gamma_1, \Gamma_2, \Gamma_3$ are given by (3.4). Then we have $$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{3}{44}.$$ The inequality is sharp. Proof of Theorem 3.2. Let $f \in \mathcal{S}_{ch}^*$ . Then there exists an analytic function $\omega \in \mathcal{H}$ with $\omega(0) = 0$ and $|\omega(z)| < 1$ for $z \in \mathbb{D}$ , such that (3.6) $$\frac{zf'(z)}{f(z)} = \varphi_0(\omega(z)) = \omega(z) + \cosh(\omega(z)), \quad z \in \mathbb{D}.$$ Let $p \in \mathcal{P}$ . By the definition of subordination, we have $$p(z) = \frac{1 + \omega(z)}{1 - \omega(z)} = 1 + c_1 z + c_2 z^2 + \cdots, \ z \in \mathbb{D}.$$ A simple computation shows that $$\omega(z) = \frac{p(z) - 1}{p(z) + 1}$$ $$= \frac{c_1}{2}z + \frac{1}{2}\left(c_2 - \frac{c_1^2}{2}\right)z^2 + \frac{1}{2}\left(c_3 - c_1c_2 + \frac{c_1^3}{4}\right)z^3$$ $$+ \frac{1}{2}\left(c_4 - c_1c_3 + \frac{3c_1^2c_2}{4} - \frac{c_2^2}{2} - \frac{c_1^4}{8}\right)z^4 + \cdots$$ (3.7) in $\mathbb{D}$ . Then p is analytic in $\mathbb{D}$ with p(0) = 1 and has positive real part in $\mathbb{D}$ . In view of (3.6) together with $\varphi_0(\omega(z))$ , a tedious computation shows that (3.8) $$\omega(z) + \cosh(\omega(z)) = 1 + \frac{1}{2}c_1z + \left(-\frac{1}{8}c_1^2 + \frac{1}{2}c_2\right)z^2 + \left(-\frac{1}{4}c_1c_2 + \frac{1}{2}c_3\right)z^3 + \left(-\frac{1}{4}c_1c_3 + \frac{13}{384}c_1^4 - \frac{1}{8}c_2^2 + \frac{1}{2}c_4\right)z^4 + \cdots$$ and (3.9) $$\frac{zf'(z)}{f(z)} = 1 + a_2z + (2a_3 - a_2^2)z^2 + (3a_4 - 3a_2a_3 + a_2^3)z^3 + (4a_5 - 2a_3^2 - 4a_2a_4 + 4a_2^2a_3 - a_2^4)z^4 + \cdots$$ Thus, using (3.8) and (3.9), we see from (3.6) that (3.10) $$\begin{cases} a_2 = \frac{1}{2}c_1, \\ a_3 = \frac{1}{16}c_1^2 + \frac{1}{4}c_2, \\ a_4 = -\frac{1}{96}c_1^3 + \frac{1}{24}c_1c_2 + \frac{1}{6}c_3. \end{cases}$$ A simple computation by using (3.5) and (3.10), shows that $$H_{2,1}(F_{f^{-1}}/2) = \frac{1}{48} \left( 13a_2^4 - 12a_2^2a_3 - 12a_3^2 + 12a_2a_4 \right)$$ $$= \frac{1}{3072} \left( 33c_1^4 - 56c_1^2c_2 - 48c_2^2 + 64c_1c_3 \right).$$ By Lemma 2.1 and (3.11), we obtain $$H_{2,1}(F_{f^{-1}}/2) = \frac{1}{192} \left( 9\tau_1^4 - 20\tau_1^2 \tau_2 \left( 1 - \tau_1^2 \right) - 4\tau_2^2 (3 + \tau_1^2) (1 - \tau_1^2) + 16\tau_1 \tau_3 \left( 1 - \tau_1^2 \right) \left( 1 - |\tau_2|^2 \right) \right).$$ (3.12) We now explore three possible cases involving $\tau_1$ : Case-I. Let $\tau_1 = 1$ . Then, from (3.12) we see that $$|H_{2,1}(F_{f^{-1}}/2)| = \frac{3}{64} \approx 0.046875.$$ Case-II. Let $\tau_1 = 0$ . Then, from (3.12) we get $$|H_{2,1}(F_{f^{-1}}/2)| = \left|\frac{1}{192}\left(-12\tau_2^2\right)\right| \le \frac{1}{16} \approx 0.0625.$$ Case-III. Let $\tau_1 \in (0,1)$ . Applying triangle inequality in (3.12) and using the fact that $|\tau_3| \leq 1$ , we obtain $$|H_{2,1}(F_f/2)| \leq \frac{1}{192} \left( \left| 9\tau_1^4 - 20\tau_1^2 \tau_2 \left( 1 - \tau_1^2 \right) - 4\tau_2^2 (3 + \tau_1^2) (1 - \tau_1^2) \right| + 16\tau_1 (1 - \tau_1^2) (1 - |\tau_2|^2) \right)$$ $$= \frac{1}{12} \tau_1 (1 - \tau_1^2) \left( \left| A + B\tau_2 + C\tau_2^2 \right| + 1 - |\tau_2|^2 \right)$$ $$:= \frac{1}{12} \tau_1 (1 - \tau_1^2) Y(A, B, C),$$ $$(3.13)$$ where $$A = \frac{9\tau_1^3}{16(1-\tau_1^2)}, \ B = -\frac{5\tau_1}{4} \ \text{and} \ C = -\frac{(3+\tau_1^2)}{4\tau_1}.$$ Observe that AC < 0. Hence, we can apply case (ii) of Lemma 2.2. Next, we check all the conditions of case (ii). 3(a) We note the inequality $$-4AC\left(\frac{1}{C^2} - 1\right) - B^2 = \frac{9\tau_1^2(3 + \tau_1^2)}{16(1 - \tau_1^2)} \left(\frac{16\tau_1^2}{(3 + \tau_1^2)^2} - 1\right) - \frac{25\tau_1^2}{16} \le 0$$ is equivalent to $$-\frac{(39+4\tau_1^2)\tau_1^2}{4(3+\tau_1^2)} \le 0$$ which is evidently holds for $\tau_1 \in (0,1)$ . However, the inequality |B| < 2(1-|C|) is equivalent to $7\tau_1^2 - 8\tau_1 + 6 < 0$ , which is false for $\tau_1 \in (0,1)$ . 3(b) Since $$4(1+|C|)^2 = \frac{(3+4\tau_1+\tau_1^2)^2}{4\tau_1^2} > 0$$ and $$-4AC\left(\frac{1}{C^2} - 1\right) = -\frac{9\tau_1^2(9 - 10\tau_1^2 + \tau_1^4)}{16(1 - \tau_1^2)(3 + \tau_1^2)} < 0,$$ a simple computation shows that the inequality $$\frac{25\tau_1^2}{16} = B^2 < \min\left\{4(1+|C|)^2, -4AC\left(\frac{1}{C^2} - 1\right)\right\} = -\frac{9\tau_1^2(9 - 10\tau_1^2 + \tau_1^4)}{16(1 - \tau_1^2)(3 + \tau_1^2)} < 0$$ is false for $\tau_1 \in (0,1)$ . 3(c) Next note that the inequality $$|C|(|B|+4|A|) - |AB| = \frac{(3+\tau_1^2)}{4\tau_1} \left(\frac{5\tau_1}{4} + \frac{9\tau_1^3}{4(1-\tau_1^2)}\right) - \frac{45\tau_1^4}{64(1-\tau_1^2)} \le 0$$ is equivalent to $60 + 68\tau_1^2 - 29\tau_1^4 \le 0$ , which is false for $\tau_1 \in (0,1)$ . 3(d) Note that the inequality $$|AB| - |C|(|B| - 4|A|) = \frac{45\tau_1^4}{64(1 - \tau_1^2)} - \frac{(3 + \tau_1^2)}{4\tau_1} \left(\frac{5\tau_1}{4} - \frac{9\tau_1^3}{4(1 - \tau_1^2)}\right) \le 0$$ is equivalent to $101\tau_1^4 + 148\tau_1^2 - 60 \le 0$ which is true for $$0 < \tau_1 \le \tau_1^{"} = \sqrt{\frac{4\sqrt{721}}{101} - \frac{74}{101}} \approx 0.575109.$$ Applying Lemma 2.2 for $0 < \tau_1 \le \tau_1''$ , we obtain $$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{12}\tau_1(1-\tau_1^2)(-|A|+|B|+|C|)$$ $$= \frac{1}{192}(12+12\tau_1^2-33\tau_1^4)$$ $$= \frac{1}{192}\Psi(\tau_1),$$ (3.14) where $$\Psi(\tau_1) := 12 + 12\tau_1^2 - 33\tau_1^4, \quad 0 < \tau_1 \le \tau_1''$$ Since $\Psi'(\tau_1) = 0$ for $\tau_1 \in (0, \tau_1'']$ holds only for $t_0 = \sqrt{2/11} < \tau_1''$ . By a simple calculation, the maximum of the function $\Psi(\tau_1)$ for $0 < \tau_1 \le \tau_1''$ occurs at the point $t_0 = \sqrt{2/11}$ . Therefore, in $0 < \tau_1 \le \tau_1''$ , we obtain $$\Psi(\tau_1) \le \Psi(t_0) = \frac{144}{11}.$$ Hence, in view of (3.14), an easy computation shows that $$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{192}\Psi(\tau_1) \le \frac{1}{192}\Psi(t_0) = \frac{3}{44} \approx 0.068182.$$ 3(e) Applying Lemma 2.2 for $\tau_1'' < \tau_1 < 1$ , we obtain $$|H_{2,1}(F_{f^{-1}}/2)| \leq \frac{1}{12}\tau_1(1-\tau_1^2)(|C|+|A|)\sqrt{1-\frac{B^2}{4AC}}$$ $$= \frac{(12-8\tau_1^2+5\tau_1^4)}{192}\sqrt{\frac{52-16\tau_1^2}{9(3+\tau_1^2)}}$$ $$= \frac{1}{192}\Phi(\tau_1),$$ (3.15) where $$\Phi(t) := (12 - 8t^2 + 5t^4)\sqrt{\frac{52 - 16t^2}{9(3 + t^2)}}, \quad \tau_1'' < t < 1.$$ A simple computation shows that $$\Phi'(t) = -\frac{2t(924 - 964t^2 + 41t^4 + 80t^6)\sqrt{3 + t^2}}{3(3 + t^2)^2\sqrt{13 - 4t^2}} < 0, \quad \tau_1'' < t < 1,$$ hence, the function $\Phi$ is decreasing on $(\tau'', 1)$ . Therefore, we have $\Phi(t) \leq \Phi(\tau_1'')$ for $\tau_1'' < t < 1$ . Thus it follows from (3.8) that $$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{192}\Phi(\tau_1) \le \frac{1}{192}\Phi(\tau_1'') = \frac{254\sqrt{721} - 5507}{20402} \approx 0.0643695.$$ Summarizing Cases 1, 2, and 3, the desired inequality is established. The proof can be concluded by establishing the sharpness of the bound. In order to show that the equality holds when $$\tau_1 = \sqrt{\frac{2}{11}}, \quad \tau_3 = 1$$ and $$(3.16) |A + B\tau_2 + C\tau_2^2| + 1 - |\tau_2|^2 = -|A| + |B| + |C|,$$ where $$A = \frac{1}{8}\sqrt{\frac{2}{11}}, \quad B = -\frac{5}{4}\sqrt{\frac{2}{11}} \quad \text{and} \quad C = -\frac{35}{8}\sqrt{\frac{2}{11}}.$$ Indeed, we can easily verify that one of the solutions of (3.16) is $\tau_2 = 1$ . It follows that $$c_1 = 2\sqrt{\frac{2}{11}}$$ , $c_2 = 2$ and $c_3 = 2\sqrt{\frac{2}{11}}$ . In view of (3.5) and (3.10), a simple computations shows that $$|\Gamma_1 \Gamma_3 - \Gamma_2^2| = \frac{3}{44}.$$ Hence, the extremal function $f \in \mathcal{S}_{ch}^*$ is obtained from (3.6) and (3.7) when $$p(z) = \frac{1 + 2\sqrt{\frac{2}{11}}z + z^2}{1 - z^2} = 1 + 2\sqrt{\frac{2}{11}}z + 2z^2 + 2\sqrt{\frac{2}{11}}z^3 + 2z^4 + \cdots$$ This shows that the bound in the result is sharp. This completes the proof. Remark 3.1. We obtain the sharp bounds of $H_{2,1}(F_f/2)$ and $H_{2,1}(F_{f^{-1}}/2)$ for the class $S_{ch}$ are 1/16 and 3/44 repectively. It is observed that the sharp bound of $H_{2,1}(F_f/2)$ differs from that of $H_{2,1}(F_{f^{-1}}/2)$ for the class $S_{ch}$ . This ensures that the results do not exhibit the invariance properties associated with the Hankel determinants of logarithmic coefficients for f and $f^{-1}$ in the class $S_{ch}^*$ .

Lemma 4.1. [43] Let $B_1$ , $B_2$ and $B_3$ be numbers such that $B_1 > 0$ , $B_2 \in \mathbb{C}$ and $B_3 \in \mathbb{R}$ . Let $p \in \mathcal{P}$ of the form (2.4). Define $\Psi_+(c_1, c_2)$ and $\Psi_-(c_1, c_2)$ by $$\Psi_{+}(c_1, c_2) = |B_2 c_1^2 + B_3 c_2| - |B_1 c_1|,$$ and $$\Psi_{-}(c_1, c_2) = -\Psi_{+}(c_1, c_2).$$ Then (4.1) $$\Psi_{+}(c_{1}, c_{2}) \leq \begin{cases} |4B_{2} + 2B_{3}| - 2B_{1}, & \text{if } |2B_{2} + B_{3}| \geq |B_{3}| + B_{1}, \\ 2|B_{3}|, & \text{otherwise,} \end{cases}$$ and (4.2) $$\Psi_{-}(c_{1}, c_{2}) \leq \begin{cases} 2B_{1} - B_{4}, & \text{if } B_{1} \geq B_{4} + 2|B_{3}|, \\ 2B_{1}\sqrt{\frac{2|B_{3}|}{B_{4} + 2|B_{3}|}}, & \text{if } B_{1}^{2} \leq 2|B_{3}|(B_{4} + 2|B_{3}|), \\ 2|B_{3}| + \frac{B_{1}^{2}}{B_{4} + 2|B_{3}|}, & \text{otherwise,} \end{cases}$$ where $B_4 = |4B_2 + 2B_3|$ . All inequalities in (4.1) and (4.2) are sharp. We have established the following result on the sharp inequality for the moduli differences of logarithmic coefficients in the class $\mathcal{S}_{ch}^*$ .

Theorem 4.1. Let $f \in \mathcal{S}_{ch}^*$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\gamma_1, \gamma_2$ are given by (2.2). Then we have $$-\frac{1}{\sqrt{6}} \le |\gamma_2| - |\gamma_1| \le \frac{1}{4}.$$ Both inequalities are sharp. Proof of Theorem 4.1. In view of (2.2) and (2.15), we see that $$|\gamma_2| - |\gamma_1| = \left| \frac{1}{2} \left( a_3 - \frac{1}{2} a_2^2 \right) \right| - \left| \frac{1}{2} a_2 \right|$$ $$= \left| -\frac{1}{32} c_1^2 + \frac{1}{8} c_2 \right| - \left| \frac{1}{4} c_1 \right|$$ $$:= \Psi_+(c_1, c_2),$$ (4.3) where $$B_1 = \frac{1}{4}$$ , $B_2 = -\frac{1}{32}$ and $B_3 = \frac{1}{8}$ . Estimate of the upper bound: For the upper bound, we see that $|2B_2 + B_3| = \frac{1}{16}$ and $|B_3| + B_1 = \frac{3}{8}$ . It follows that $|2B_2 + B_3| \ge |B_3| + B_1$ . Hence, applying Lemma 4.1, we obtain $$\Psi_+(c_1, c_2) \le 2|B_3| = \frac{1}{4}.$$ As a result, applying (4.3), we obtain the upper bound $$(4.4) |\gamma_2| - |\gamma_1| \le \frac{1}{4}.$$ The function $f_2$ , which is defined in (2.9) gives the sharpness of the inequality (4.4). Estimate of the lower bound: For the lower bound, we see that $B_4 = |4B_2 + 2B_3| = \frac{1}{8}$ , $B_4 + 2|B_3| = \frac{3}{8}$ . It follows that $B_1 \not\geq B_4 + 2|B_3|$ . Also, the equality $2|B_3|(B_4 + 2|B_3|) = \frac{3}{32}$ ensures that the condition $B_1^2 \leq 2|B_3|(B_4 + 2|B_3|)$ holds. Thus, by Lemma 4.1, we have $$\Psi_{-}(c_1, c_2) \le 2B_1 \sqrt{\frac{2|B_3|}{B_4 + 2|B_3|}} = \frac{1}{\sqrt{6}}.$$ Clearly, we observe that $$\Psi_{+}(c_1, c_2) = -\Psi_{-}(c_1, c_2) \ge -\frac{1}{\sqrt{6}}.$$ Hence, from (4.3), we conclude $$(4.5) |\gamma_2| - |\gamma_1| \ge -\frac{1}{\sqrt{6}}.$$ The inequality (4.5) is sharp for the function $f \in \mathcal{A}$ given by (2.11) with $$p(z) = \frac{1 - z^2}{1 - 2\sqrt{\frac{2}{3}}z + z^2},$$ which completes the proof. We have established the following result on the sharp inequality for the moduli differences of logarithmic inverse coefficients in the class $\mathcal{S}_{ch}^*$ .

Theorem 4.2. Let $f \in \mathcal{S}_{ch}^*$ and has the series representation $f(z) = z + a_2 z^2 + a_3 z^3 + \cdots$ , and $\Gamma_1, \Gamma_2$ are given by (3.4). Then we have $$-\frac{1}{\sqrt{10}} \le |\Gamma_2| - |\Gamma_1| \le \frac{1}{4}.$$ Both inequalities are sharp. Proof of Theorem 4.2. In view of (2.15) and (3.4), we see that $$|\Gamma_{2}| - |\Gamma_{1}| = \left| -\frac{1}{2} \left( a_{3} - \frac{3}{2} a_{2}^{2} \right) \right| - \left| -\frac{1}{2} a_{2} \right|$$ $$= \left| \frac{5}{32} c_{1}^{2} - \frac{1}{8} c_{2} \right| - \left| \frac{1}{4} c_{1} \right|$$ $$= \Psi_{+}(c_{1}, c_{2}),$$ $$(4.6)$$ where $$B_1 = \frac{1}{4}$$ , $B_2 = \frac{5}{32}$ and $B_3 = -\frac{1}{8}$ . Estimate of the upper bound: With regard to the lower bound, we calculate $B_4 = |4B_1 + 2B_3| = \frac{3}{8}$ and $B_4 + 2|B_3| = \frac{5}{8}$ , from which it is evident that the condition $B_1 \ge B_4 + 2|B_3|$ is not met. Moreover, the relation $2|B_3|(B_4 + 2|B_3|) = \frac{5}{32}$ confirms that the inequality $B_1^2 \le 2|B_3|(B_4 + 2|B_3|)$ is satisfied. Consequently, by virtue of Lemma 4.1, we obtain $$\Psi_+(c_1, c_2) \le 2|B_3| = \frac{1}{4}.$$ Thus, it follows from (4.6) that $$|\Gamma_2| - |\Gamma_1| \le \frac{1}{4}.$$ The function $f_2$ , which is defined in (2.12) gives the sharpness of the inequality (4.7). Estimate of the lower bound: Regarding the lower bound, we observe that $B_4 = |4B_1 + 2B_3| = \frac{3}{8}$ and $B_4 + 2|B_3| = \frac{5}{8}$ . It follows that the condition $B_1 \ge B_4 + 2|B_3|$ is not satisfied. Furthermore, the equality $2|B_3|(B_4 + 2|B_3|) = \frac{5}{32}$ ensures that the inequality $B_1^2 \le 2|B_3|(B_4 + 2|B_3|)$ holds. Consequently, by applying Lemma 4.1, we obtain $$\Psi_{-}(c_1, c_2) \le 2B_1 \sqrt{\frac{2|B_3|}{B_4 + 2|B_3|}} = \frac{1}{\sqrt{10}}.$$ A simple computation leads to $$\Psi_{+}(c_1, c_2) = -\Psi_{-}(c_1, c_2) \ge -\frac{1}{\sqrt{10}}.$$ Consequently, from (4.6), we obtain $$|\Gamma_2| - |\Gamma_1| \ge -\frac{1}{\sqrt{10}}.$$ The inequality (4.8) is sharp for the function $f \in \mathcal{A}$ given by (2.11) with $$p(z) = \frac{1 + 2\sqrt{\frac{2}{5}}z + z^2}{1 - z^2},$$ which completes the

Definition 1.1. Let f and g be two analytic functions in the unit disk $\mathbb{D}$ . Then f is said to be subordinate to g, written as $f \prec g$ or $f(z) \prec g(z)$ , if there exists a function $\omega$ , analytic in $\mathbb{D}$ with w(0) = 0, |w(z)| < 1 such that f(z) = g(w(z)) for $z \in \mathbb{D}$ . Moreover, if g is univalent in $\mathbb{D}$ and f(0) = g(0), then $f(\mathbb{D}) \subseteq g(\mathbb{D})$ . In 1994, Ma and Minda [30] gave a unified presentation of various subclasses of starlike and convex functions by replacing the subordinate function (1+z)/(1-z) by a more general analytic function $\varphi$ with positive real part and normalized by the conditions $\varphi(0) = 1$ , $\varphi'(0) > 0$ and $\varphi$ maps $\mathbb D$ onto univalently a starlike region with respect to 1 and symmetric with respect to the real axis. Ma and Minda [30] have introduced the following general class that envelopes several well-known classes as special cases $$\mathcal{S}^*[\varphi] = \left\{ f \in \mathcal{A} : \frac{zf'(z)}{f(z)} \prec \varphi(z) \right\}.$$ File: Aha-San-GFTP8-07-03-2025.tex, printed: 2026-3-18, $0.01\,$ ²⁰²⁰ Mathematics Subject Classification. Primary 30C45; Secondary 30C50, 30C80. Key words and phrases. Starlike functions; Hyperbolic cosine; Logarithmic coefficients; Hankel determinant; Inverse coefficients; Sharp bounds. In the literature, functions belonging to the class $\mathcal{S}^[\varphi]$ are known as the Ma-Minda starlike functions. For $-1 \leq B < A \leq 1$ , the class $\mathcal{S}^[(1+Az)/(1+Bz)] := \mathcal{S}^[A,B]$ is called the class of Janowski starlike functions, introduced by Janowski [20]. The class $\mathcal{S}^[\beta]$ of starlike functions of order $\beta$ , where $0 \leq \beta < 0$ , is defined by taking $\varphi(z) = (1+(1-2\beta)z)/(1-z)$ . Note that $\mathcal{S}^ = \mathcal{S}^[0]$ is the classical class of starlike functions. By taking $$\varphi(z) = 1 + \frac{2}{\pi^2} \left( \log \left( (1 + \sqrt{z}) / (1 - \sqrt{z}) \right) \right)^2,$$ we obtain the class $S^[\varphi] = S_p$ of parabolic starlike functions, introduced by Rønning [41]. The coefficient problem and many other geometric properties for functions in the class $S^[\varphi]$ have been studied extensively in (see [12,17,21,22,24]). For example, Deniz [12] has studied the sharp coefficient problem for the function $$\varphi(z) = e^{z + \frac{\lambda}{2}z^2} \quad (z \in \mathbb{C}, \lambda \ge 1),$$ which is a generated function of generalized telephone numbers. In 2015, Mendiratta et al. [35] obtained the structural formula, inclusion relations, coefficient estimates, growth and distortion results, subordination theorems and various radii constants for the exponential function $\varphi(z) = e^z$ . In 2019, Cho et al. [10] studied the class $S_S^ := S^(1 + \sin z)$ and the class $S_{\cos}^ := S^(\cos z)$ has been studied by [34] and [7]. In the recent years, the class of starlike functions associated with cosine hyperbolic functions has been discussed by Riaz et al. [40] which is defined as follows:

Definition 1.2. Let $f \in \mathcal{A}$ . Then $f \in \mathcal{S}_{ch}^*$ if, and only if, $$\frac{zf'(z)}{f(z)} \prec \varphi_0(z) \quad z \in \mathbb{D},$$ where $$\mathbb{C}\ni z\mapsto \varphi_0(z):=z+\cosh(z)=z+\frac{1}{2}\left(\exp(z)+\exp(-z)\right).$$  FIGURE 1. The geometrical representation of $z + \cosh(z)$ . It is important to note that the function $\varphi$ need not be univalent in $\mathbb{D}$ (see [40, Observation 1.2]). A function $f \in \mathcal{S}_{ch}^*$ if, and only if, there exists an analytic function $\varphi \in \mathcal{H}$ , satisfying $\varphi \prec \varphi_0$ such that $$f(z) = z \exp\left(\int_0^z \frac{\varphi(t) - 1}{t} dt\right), \quad z \in \mathbb{D}.$$ 1.1. New problem formulations for the class $\mathcal{S}_{ch}$ . Establishing the upper bounds for coefficients has remained a central theme in geometric function theory, as these estimates reveal fundamental structural properties of analytic functions. A primary challenge in this pursuit is the identification of suitable extremal functions that demonstrate the sharpness of such bounds. We observe that while coefficient problems involving the Hankel determinant for the class $\mathcal{S}_{ch}$ have been extensively explored, the corresponding determinants composed of logarithmic coefficients for this class have not yet garnered significant research attention. Furthermore, the sharp bounds for individual logarithmic coefficients for the class $\mathcal{S}_{ch}^*$ remain largely unaddressed. This gap in the literature serves as the primary motivation for the present study. The primary motivation of this study is to provide a comprehensive characterization of the logarithmic functionals for functions subordinated to the hyperbolic cosine function. To this end, we formulate and resolve the following key problems. Problem 1.1. Let $f \in \mathcal{S}_{ch}^*$ be a starlike function associated with the hyperbolic cosine function. Determine the sharp upper bounds for the modulus of the logarithmic coefficients $\gamma_n$ for $n \in \mathbb{N}$ . Problem 1.2. For $f \in \mathcal{S}_{ch}^*$ , find the sharp bound for the second Hankel determinant $H_{2,1}(F_f/2) = \gamma_1 \gamma_3 - \gamma_2^2$ consisting of logarithmic coefficients. Problem 1.3. Let $f^{-1}$ be the inverse of a function f in $\mathcal{S}_{ch}^*$ . Determine the sharp bounds of the modulus of the inverse logarithmic coefficients $\Gamma_n$ . Problem 1.4. Establish the sharp bound for the second Hankel determinant $|H_{2,1}(F_{f^{-1}}/2)|$ for the inverse functions belonging to the class $S_{ch}^*$ . Problem 1.5. Analyze the difference between the moduli of consecutive logarithmic coefficients $|\gamma_{n+1}| - |\gamma_n|$ and their inverse counterparts. Determine if a uniform sharp bound exists for these differences for functions in the class $\mathcal{S}_{ch}^*$ . In this article, our aim is give affirmative answers to the Problems 1.1 to 1.5. These problems include finding the sharp bound of the Hankel determinant of logarithmic coefficients, the sharpness of $\gamma_1, \gamma_2$ , and $\gamma_3$ . In the subsequent sections, we will discuss our findings and provide a background study on these topics. The organization of this paper is as follows: In Section 2, we establish the sharp bounds for $\gamma_1, \gamma_2$ , and $\gamma_3$ and $|H_{2,1}(F_f/2)|$ for functions $f \in \mathcal{S}_{ch}$ . In Section 3, we establish the sharp bound of $\Gamma_1$ , $\Gamma_2$ and $|H_{2,1}(F_{f^{-1}}/2)|$ for function in the class $\mathcal{S}_{ch}$ . In Section 4, we establish results finding sharp bounds of the moduli differences of logarithmic coefficients and inverse counterpart for functions in the class $\mathcal{S}_{ch}^*$ . The proofs of the results are discussed in detail in each respective section. 2. Sharp bound of logarithmic coefficients for functions in the class $\mathcal{S}_{ch}^*$ : For $f \in \mathcal{S}$ , we define the logarithmic coefficients $\gamma_n(f)$ by (2.1) $$F_f(z) := \log \frac{f(z)}{z} = 2 \sum_{n=1}^{\infty} \gamma_n(f) z^n, \ z \in \mathbb{D}, \ \log 1 := 0.$$ The logarithmic coefficients $\gamma_n$ for functions in the class $\mathcal{S}$ play a vital role in Milin's conjecture ([36], see also [13, p.155]). Milin conjectured that for $f \in \mathcal{S}$ and $n \geq 2$ , $$\sum_{m=1}^{n} \sum_{k=1}^{m} \left( k |\gamma_k|^2 - \frac{1}{k} \right) \le 0,$$ where the equality holds if, and only if, f is a rotation of the Koebe function. De Branges [8] has proved Milin conjecture which confirmed the famous Bieberbach conjecture. On the other hand, one of reasons for more attention has been given to the logarithmic coefficients is that the sharp bound for the class S is known only for $\gamma_1$ and $\gamma_2$ , namely $$|\gamma_1| \le 1, \ |\gamma_2| \le \frac{1}{2} (1 + 2e^{-2}) = 0.635...$$ It is still an open problem to find the sharp upper bounds for absolute value of $\gamma_n$ , $n \geq 3$ , for functions in the class $\mathcal{S}$ . Estimating the modulus of logarithmic coefficients for functions $f \in \mathcal{S}$ and various sub-classes has been considered recently by several authors. For more information on logarithmic coefficients, we refer to [1, 2, 4, 5, 11, 14, 31, 31, 32, 42, 46] and references therein. The evaluation of Hankel determinants has been a major concern in geometric function theory, where these determinants are formed by employing coefficients of analytic functions f that are represented by (1.1) in the unit disk $\mathbb{D}$ . Hankel matrices (and determinants) have emerged as fundamental elements in different areas of mathematics, finding a wide range of applications (see [47]). The primary objective of this study is to determine the sharp bound of logarithmic coefficients and the Hankel determinants involving the logarithmic coefficients. To begin, we present the definitions of Hankel determinants in situations where $f \in \mathcal{A}$ . The Hankel determinant $H_{q,n}(f)$ of Taylor's coefficients of functions $f \in \mathcal{A}$ represented by (1.1) is defined for $q, n \in \mathbb{N}$ as follows: $$H_{q,n}(f) := \begin{vmatrix} a_n & a_{n+1} & \cdots & a_{n+q-1} \\ a_{n+1} & a_{n+2} & \cdots & a_{n+q} \\ \vdots & \vdots & \vdots & \vdots \\ a_{n+q-1} & a_{n+q} & \cdots & a_{n+2(q-1)} \end{vmatrix}.$$ The extensive exploration of the sharp bounds of the Hankel determinants for starlike, convex, and other function classes have been undertaken in various studies (see [23,31,39]), and their precise bounds have been successfully established. Differentiating (2.1) and then using (1.1), a simple computation shows that (2.2) $$\begin{cases} \gamma_1 = \frac{1}{2}a_2, \\ \gamma_2 = \frac{1}{2}\left(a_3 - \frac{1}{2}a_2^2\right), \\ \gamma_3 = \frac{1}{2}\left(a_4 - a_2a_3 + \frac{1}{3}a_2^3\right). \end{cases}$$ In 2022, Kowalczyk and Lecko [23] proposed a Hankel determinant $H_{q,n}(F_f/2)$ whose elements are the logarithmic coefficients of $f \in \mathcal{S}$ , realizing the extensive use of these coefficients. It follows that (2.3) $$H_{2,1}(F_f/2) := \gamma_1 \gamma_3 - \gamma_2^2 = \frac{1}{48} \left( a_2^4 - 12a_3^2 + 12a_2 a_4 \right).$$ Furthermore, $H_{2,1}(F_f/2)$ is invariant under rotation, since for $f_{\theta}(z) := e^{-i\theta} f(e^{i\theta}z)$ , $\theta \in \mathbb{R}$ when $f \in \mathcal{S}$ , we have $$H_{2,1}(F_{f_{\theta}}/2) = \frac{e^{4i\theta}}{48} \left( a_2^4 - 12a_3^2 + 12a_2a_4 \right) = e^{4i\theta} H_{2,1}(F_f/2).$$ Let $\mathcal{P}$ be the class of all analytic functions p in the unit disk $\mathbb{D}$ satisfying p(0) = 1 and Re p(z) > 0 for $z \in \mathbb{D}$ . Therefore, every $p \in \mathcal{P}$ can be represented as (2.4) $$p(z) = 1 + \sum_{n=1}^{\infty} c_n z^n, \ z \in \mathbb{D}.$$ Elements of the class $\mathcal{P}$ are called Carathéodory functions. It is known that $|c_n| \leq 2$ , $n \geq 1$ for a functions $p \in \mathcal{P}$ (see [13]). The Carathéodory class $\mathcal{P}$ and it's coefficients bound play a significant role in establishing the bound of Hankel determinants. Now, we state some lemmas, which will be useful to establish our main results. Parametric representations of the coefficients are often useful in finding the bound for Hankel determinants, and in this regard, Libera and Zlotkiewicz (see [27,28]) obtained the parameterizations of possible values of $c_2$ and $c_3$ , which are Taylor coefficients for functions with positive real part.