Abstract
In this paper, we investigate the sharp bounds of the second Hankel determinant of Logarithmic coefficients for the starlike and convex functions with respect to symmetric points in the open unit disk.
Results & Lemmas (2)
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Lemma 2.1
Lemma 2.1. [9] Let be a Schwarz function. Then,, and. We obtain the following sharp bound for for functions in the class. <span…
Lemma 2.1. [9] Let $w(z) = c_1 z + c_2 z^2 + \cdots$ be a Schwarz function. Then
$$|c_1| \le 1$$
, $|c_2| \le 1 - |c_1|^2$ , and $|c_3| \le 1 - |c_1|^2 - \frac{|c_2|^2}{1 + |c_1|}$ .
We obtain the following sharp bound for $H_{2,1}(F_f/2)$ for functions in the class $\mathcal{S}_S^*$ .
<span id="page-3-3"></span>Theorem 2.2. Let $f \in \mathcal{S}_S^*$ . Then
$$|H_{2,1}(F_f/2)| \le \frac{1}{4}.$$
The inequality is sharp.
Theorem 2.4
Theorem 2.4. Let be of the form (1.1). Then <span id="page-5-0"></span> The inequality is sharp.
Theorem 2.4. Let $f \in \mathcal{K}_S$ be of the form (1.1). Then
<span id="page-5-0"></span>
$$|H_{2,1}(F_f/2)| \le \frac{1}{36}.$$
The inequality is sharp.
Function classes studied:
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