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On the Second Hankel determinant of Logarithmic Coefficients for certain univalent functions

Vasudeavarao Allu, Vibhuti Arora, Amal Shaji · 2021 📄 arXiv →

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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 (3)

Each result is auto-extracted from the paper via OCR - the math is shown verbatim and may carry occasional transcription artifacts. Click to expand the full statement.

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.
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 $\mathcal{S}_S^*$ .
Theorem 2.2 Theorem 2.2. Let. Then The inequality is sharp.
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 The inequality is sharp.
Theorem 2.4. Let $f \in \mathcal{K}_S$ be of the form (1.1). Then $$|H_{2,1}(F_f/2)| \le \frac{1}{36}.$$ The inequality is sharp.
Function classes studied:
close-to-convex

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