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Abstract
In this paper, we obtain sharp bounds for the third Hankel determinants of the coefficients of the inverse of bounded turning functions. Thus answering a negatively to a conjecture recently posed regarding these functions. Additionally, we offer a positive response for the Hankel determinant related to a class of bounded turning functions.
Results & Lemmas (3)
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Lemma 1.1
Lemma 1.1. [2, Lemma 2.1] Let be the closed unit disk, and be given by (1.5). Then, for some for.
Lemma 1.1. [2, Lemma 2.1] Let $\overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \leq 1\}$ be the closed unit disk, and $p \in \mathcal{P}$ be given by (1.5). Then,
$$\begin{split} c_1 &= 2t_1, \\ c_2 &= 2t_1^2 + 2\left(1 - |t_1|^2\right)t_2, \\ c_3 &= 2t_1^3 + 4\left(1 - |t_1|^2\right)t_1t_2 - 2\left(1 - |t_1|^2\right)\overline{t_1}t_2^2 + 2\left(1 - |t_1|^2\right)\left(1 - |t_2|^2\right)t_3, \\ and \\ c_4 &= 2t_1^4 + 2(1 - |t_1|^2)\left(3t_1^2 + \overline{t_1}^2t_2^2 - 3|t_1|^2t_2 + t_2\right)t_2 \\ &\quad + 2(1 - |t_1|^2)(1 - |t_2|^2)\left(2t_1 - 2\overline{t_1}t_2 - \overline{t_2}t_3\right)t_3 \\ &\quad + 2(1 - |t_1|^2)(1 - |t_2|^2)(1 - |t_3|^2)t_4, \end{split}$$
for some $t_k \in \overline{\mathbb{D}}$ for $k \in \{1, 2, 3, 4\}$ .
Theorem 2.1
Theorem 2.1. If has the form (1.1). Then (2.1) This result is sharp for the function.
Theorem 2.1. If $f \in \mathcal{S}^*$ has the form (1.1). Then
$$|H_3(1)(f^{-1})| \le \frac{44}{135}.$$
(2.1)
This result is sharp for the function $f_0(z) = -z + 2\operatorname{arctanh}(z)$ .
Theorem 2.2
Theorem 2.2. Let has the form (1.1). Then (2.10) The result is sharp for the function given by
Theorem 2.2. Let $f \in \mathcal{R}_1$ has the form (1.1). Then
$$|H_3(1)(f^{-1})| \le \frac{1}{64}.$$
(2.10)
The result is sharp for the function $f_*$ given by
$$(zf'_*(z))' = \frac{1+z^3}{1-z^3}.$$
Function classes studied:
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