Hankel and Toeplitz Determinants of Logarithmic Coefficients of Inverse Functions for the Subclass of Starlike Functions with Respect to Symmetric Conjugate Points

Lemma 1. ([6]) For a function $p(z) \in P$ of the form (6), the sharp inequality $|p_n| \leq 2$ holds for each $n \geq 1$ . Equality holds for the function $p(z) = \frac{1+z}{1-z}$ .

Lemma 2. ([7]) Let $p(z) \in P$ be a function of the form (6) and $\mu \in \mathbb{C}$ . Then $$|p_n - \mu p_k p_{n-k}| \le 2 \max\{1, |2\mu - 1|\}, \ 1 \le k \le n - 1.$$ If $|2\mu - 1| \ge 1$ , then the inequality is sharp for the function $p(z) = \frac{1+z}{1-z}$ or its rotations. If $|2\mu - 1| < 1$ , then the inequality is sharp for the function $p(z) = \frac{1+z^n}{1-z^n}$ or its rotations.

Lemma 3. ([10]) Let $p(z) \in P$ be a function of the form (6) and $\alpha, \beta, \gamma \in \Re$ . Then $$|\alpha p_1^3 - \beta p_1 p_2 + \gamma p_3| \le 2 |\alpha| + 2 |\beta - 2\alpha| + 2 |\alpha - \beta + \gamma|.$$

Theorem 1. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$|a_2| \le \frac{1}{2},$$ $|a_3| \le \frac{1}{2},$ $|a_4| \le \frac{25}{96},$ and $$|a_5| \le \frac{7}{24}.$$

Theorem 2. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$|\Gamma_1| \le \frac{1}{4},$$ $$|\Gamma_2| \le \frac{1}{4},$$ $$|\Gamma_3| \le \frac{41}{192},$$ and $$|\Gamma_4| \le \frac{197}{256}.$$

Theorem 3. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$\left| H_{2,1} \left( \Gamma_{f^{-1}} \right) \right| \le \frac{95}{768}.$$

Theorem 4. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$|H_{2,2}(\Gamma_{f^{-1}})| \le \frac{7691}{36864}.$$

Theorem 5. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$\left|T_{2,1}\left(\Gamma_{f^{-1}}\right)\right| \le \frac{9}{32}.$$

Theorem 6. Let $f(z) \in S_{SC}^*(e^z)$ . Then $$\left| T_{2,2} \left( \Gamma_{f^{-1}} \right) \right| \le \frac{7165}{9216}.$$

Definition 1. Let $S_{SC}^*(e^z)$ be the class of functions defined by $$\frac{zf'(z)}{h(z)} \prec \phi(z), z \in E,$$ where $\phi\left(z\right)=e^{z},$ is an analytic univalent function and $h\left(z\right)=\frac{f\left(z\right)-\overline{f\left(-z\right)}}{2}.$ Remark 1. Changing the function $\phi(z)$ in Definition 1 gives us more subclasses of starlike functions with respect to symmetric conjugate points: - (i) For $\phi(z) = \frac{1+z}{1-z}$ , which has been introduced and studied in [8]. - (ii) For $\phi(z) = \frac{1+Az}{1+Bz}$ , $-1 \le B < A \le 1$ , which has been introduced and studied in [14]. - (iii) For $\phi(z) = \frac{1+Az}{1+Bz}$ , $-1 \le B < A \le 1$ and considering the tilted factor $e^{i\alpha}$ , $|\alpha| < \frac{\pi}{2}$ , which has been defined and investigated in [25]. - (iv) For $\phi(z) = 1 + \sin z$ , which has been defined and studied in [26]. It is observed that there have been few studies on Hankel and Toeplitz determinants, whose entries are logarithmic coefficients of inverse functions for the subclass of univalent functions, particularly starlike functions with respect to other points, i.e., symmetric points, conjugate points, and symmetric conjugate points. We can refer the reader to [8, 9], who were among the early researchers who investigated these subclasses. Some researchers, including [14, 20, 21, 25–27], and references therein, have also carried out comprehensive studies related to these subclasses, which may provide diverse insights. Thus, inspired by the ideas of [11, 12, 18], in this paper, we aim to estimate the upper bounds of the initial Taylor coefficients $|a_n|$ , n=2,3,4,5, logarithmic coefficients of inverse functions $|\Gamma_n|$ , n=1,2,3,4, and the second order Hankel and Toeplitz determinants whose entries are logarithmic coefficients of inverse functions belonging to the new subclass $S_{SC}^*(e^z)$ , i.e., $|H_{2,1}(\Gamma_{f^{-1}})|$ , $|H_{2,2}(\Gamma_{f^{-1}})|$ , $|T_{2,1}(\Gamma_{f^{-1}})|$ , and $|T_{2,2}(\Gamma_{f^{-1}})|$ .