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Abstract
The Hankel determinant $H_{2,1}(F_{f^{-1}}/2)$ of logarithmic coefficients is defined as: \begin{align*} H_{2,1}(F_{f^{-1}}/2):= \begin{vmatrix} Γ_1 & Γ_2 Γ_2 & Γ_3 \end{vmatrix}=Γ_1Γ_3-Γ^2_2, \end{align*} where $Γ_1, Γ_2,$ and $Γ_3$ are the first, second and third logarithmic coefficients of inverse functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,1}(F_{f^{-1}}/2)|\leq 19/288$, $|H_{2,1}(F_{f^{-1}}/2)| \leq
Results & Lemmas (5)
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Lemma 2.1
Lemma 2.1. [16,17] If is of the form (2.1) with, then and for some and For, there is a unique function with as in (2.2), namely For and,…
Lemma 2.1. [16,17] If $p \in \mathcal{P}$ is of the form (2.1) with $c_1 \geq 0$ , then
$$(2.2) c_1 = 2\tau_1,$$
$$(2.3) c_2 = 2\tau_1^2 + 2(1 - \tau_1^2)\tau_2$$
and
$$(2.4) c_3 = 2\tau_1^3 + 4(1-\tau_1^2)\tau_1\tau_2 - 2(1-\tau_1^2)\tau_1\tau_2^2 + 2(1-\tau_1^2)(1-|\tau_2|^2)\tau_3$$
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 (2.2), 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
$$p(z) = \frac{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 + \tau_1)z + (\overline{\tau_1}\tau_3 + \tau_1\overline{\tau_2}\tau_3 + \tau_2)z^2 + \tau_3z^3}{1 + (\overline{\tau_2}\tau_3 + \overline{\tau_1}\tau_2 - \tau_1)z + (\overline{\tau_1}\tau_3 - \tau_1\overline{\tau_2}\tau_3 - \tau_2)z^2 - \tau_3z^3}, \quad z \in \mathbb{D}.$$
Lemma 2.2
Lemma 2.2. [7] Let A, B, C be real numbers and (i) If, then (ii) If AC < 0, then (ii) If, then where where To ensure a clear presentation,…
Lemma 2.2. [7] 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 $AC \geq 0$ , 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
(ii) If
$$AC < 0$$
, then
$$Y(A,B,C) = \begin{cases} 1 - |A| + \frac{B^2}{4(1-|C|)}, & -4AC(C^{-2}-1) \le B^2 \wedge |B| < 2(1-|C|), \\ 1 + |A| + \frac{B^2}{4(1+|C|)}, & B^2 < \min\{4(1+|C|)^2, -4AC(C^{-2}-1)\}, \\ R(A,B,C), & otherwise, \end{cases}$$
where
where
$$R(A,B,C) := \begin{cases} |A| + |B| - |C|, & |C|(|B| + 4|A|) \le |AB|, \\ -|A| + |B| + |C|, & |AB| \le |C|(|B| - 4|A|), \\ (|C| + |A|)\sqrt{1 - \frac{B^2}{4AC}}, & otherwise. \end{cases}$$
To ensure a clear presentation, we have divided the content of Hankel determinants of logarithmic coefficients of inverse functions into three sections, each focusing on different classes of functions belonging to A. Our main results for starlike, convex, and bounded turning functions are demonstrated separately within these sections.
3. Sharp bound of
$$|H_{2,1}(F_{f^{-1}}/2)|$$
for the class $\mathcal{S}^*(1/2)$
We obtain the following result finding the sharp bound of $|H_{2,1}(F_{f^{-1}}/2)|$ for functions in the class $S^*(1/2)$ .
Theorem 3.1
Theorem 3.1. Let. Then The inequality is sharp.
Theorem 3.1. Let $f \in \mathcal{S}^*(1/2)$ . Then
$$|H_{2,1}(F_{f^{-1}}/2)| \le \frac{19}{288}.$$
The inequality is sharp.
Theorem 4.1
Theorem 4.1. Let. Then The inequality is sharp.
Theorem 4.1. Let $f \in \mathcal{S}^c(1/2)$ . Then
$$(4.1) |H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{144}.$$
The inequality is sharp.
Theorem 5.1
Theorem 5.1. Let. Then The inequality is sharp.
Theorem 5.1. Let $f \in \mathcal{R}(1/2)$ . Then
$$(5.1) |H_{2,1}(F_{f^{-1}}/2)| \le \frac{1}{36}.$$
The inequality is sharp.
Function classes studied:
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