Sharp bounds for second Hankel determinant of logarithmic co

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Abstract

The Hankel determinant $H_{2,2}(F_{f}/2)$ is defined as: \begin{align*} H_{2,2}(F_{f}/2):= \begin{vmatrix} γ_2 & γ_3 γ_3 & γ_4 \end{vmatrix}, \end{align*} where $γ_2, γ_3,$ and $γ_4$ are the second, third, and fourth logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we establish sharp inequalities $|H_{2,2}(F_{f}/2)|\leq (1272 + 113\sqrt{678})/32856$ and $|H_{2,2}(F_{f}/2)| \leq 13/1080$ for the logarithmic coef

Results & Lemmas (3)

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Lemma 2.1 Lemma 2.1. [6] Let be a Schwarz function. Then,, and. 2.1. Second Hankel determinant of logarithmic coefficients of: We obtain the…

Lemma 2.1. [6] Let $w(z) = c_1 z + c_2 z^2 + c_3 z^3 + ...$ be a Schwarz function. Then $$|c_1| \le 1$$ , $|c_2| \le 1 - |c_1|^2$ , $|c_3| \le 1 - |c_1|^2 - \frac{|c_2|^2}{1 + |c_1|}$ and $|c_4| \le 1 - |c_1|^2 - |c_2|^2$ . 2.1. Second Hankel determinant of logarithmic coefficients of $f \in \mathcal{S}_S$ : We obtain the following sharp bound for $H_{2,2}(F_f/2)$ for functions in the class $\mathcal{S}_S$ .

Theorem 2.1 Theorem 2.1. Let. Then The inequality is sharp.

Theorem 2.1. Let $f \in \mathcal{S}_S^*$ . Then $$|H_{2,2}(F_f/2)| \le \frac{(1272 + 113\sqrt{678})}{32856}.$$ The inequality is sharp.

Theorem 2.2 Theorem 2.2. Let. Then The inequality is sharp.

Theorem 2.2. Let $f \in \mathcal{K}_S$ . Then $$|H_{2,2}(F_f/2)| \le \frac{13}{1080}$$ The inequality is sharp.

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Results & Lemmas (3)

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