Second Hankel determinant of logarithmic coefficients for $\mathcal {G}(\alpha )$ and $\mathcal {P}(M)$

期刊论文
发表日期：

2024年5月4日

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Raju Biswas

摘要

The Second Hankel determinant $H_{2,1}\left( F_f/2\right)$ $H_{2, 1} (F_{f} / 2)$ of logarithmic coefficients is defined as $H_{2,1}\left( F_f/2\right) :=\left| \begin{array}{ccc}\gamma _1&{}\gamma _2\\ \gamma _2&{}\gamma _3\end{array}\right| =\gamma _1\gamma _3-\gamma _2^2$ $H_{2, 1} (F_{f} / 2) : = |\begin{matrix} γ_{1} & γ_{2} \\ γ_{2} & γ_{3} \end{matrix}| = γ_{1} γ_{3} - γ_{2}^{2}$ , where $\gamma _i$ $γ_{i}$ ’s are the i-th logarithmic coefficients of function f belonging to the class $\mathcal {S}$ $S$ , where $\mathcal {S}$ $S$ is the class of all normalized analytic and univalent functions in the unit disk $\mathbb {D}$ $D$ . Let $\mathcal {G}(\alpha )$ $G (α)$ and $\mathcal {P}(M)$ $P (M)$ be the classes consisting of normalized analytic functions f satisfying respectively $\text {Re}\left( (1-\alpha )\frac{f(z)}{z}+\alpha f'(z)\right) >0$ $Re ((1 - α) \frac{f (z)}{z} + α f^{'} (z)) > 0$ and $\text {Re}\left( zf''(z)\right) >-M$ $Re (z, f^{''}, (z)) > - M$ for $z\in \mathbb {D}$ $z \in D$ , $M>0$ $M > 0$ and $\alpha \ge 0$ $α \geq 0$ . Then the functions in the class $\mathcal {G}(\alpha )$ $G (α)$ are univalent in $\mathbb {D}$ $D$ for $\alpha \ge 1$ $α \geq 1$ , while the functions in $\mathcal {G}(0)$ $G (0)$ are univalent in $|z| <\sqrt{2}-1$ $| z | < \sqrt{2} - 1$ and the functions in the classs $\mathcal {P}(M)$ $P (M)$ are univalent and starlike for $0 < M \le 1/\log 4$ $0 < M \leq 1 / log 4$ . In this paper, we obtain the sharp bounds of the second Hankel determinant of logarithmic coefficients for functions in the classes $\mathcal {G}(\alpha )$ $G (α)$ and $\mathcal {P}(M)$ $P (M)$

Second Hankel determinant of logarithmic coefficients for \(\mathcal {G}(\alpha )\) and \(\mathcal {P}(M)\)

摘要