Coefficient bounds for starlike functions associated with Gregory coefficients
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929627337850880 |
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| author | Ahamed, Molla Basir Mandal, Sanju |
| author_facet | Ahamed, Molla Basir Mandal, Sanju |
| contents | It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ of the form
\begin{align*}
f(z)=z+\sum_{n=2}^{\infty}a_nz^n\; \mbox{for}\; z\in\mathbb{D}
\end{align*}
and suppose that
\begin{align*}
F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}γ_{n}(f)z^n, \;\; z\in\mathbb{D},\;\;\log 1:=0,
\end{align*}
where $ γ_{n}(f) $ is the logarithmic coefficients. The second Hankel determinant of logarithmic coefficients $H_{2,1}(F_{f}/2)$ is defined as: $H_{2,1}(F_{f}/2) :=γ_{1}γ_{3} -γ^2_{2}$, where $γ_1, γ_2,$ and $γ_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we first establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/64$ with logarithmic coefficients for the classes of starlike functions associated with Gregory coefficients. In addition, we establish the sharpness of Fekete-Szeg$ \ddot{o} $ inequality, Zalcman functional and generalized Zalcman functional for the class starlike functions associated with Gregory coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Coefficient bounds for starlike functions associated with Gregory coefficients Ahamed, Molla Basir Mandal, Sanju Complex Variables Primary 30C45, Secondary 30C50, 30C80 It is of interest to know the sharp bounds of the Hankel determinant, Zalcman functionals, Fekete-Szeg$ \ddot{o} $ inequality as a part of coefficient problems for different classes of functions. Let $\mathcal{H}$ be the class of functions $ f $ which are holomorphic in the open unit disk $\mathbb{D}=\{z\in\mathbb{C}: |z|<1\}$ of the form \begin{align*} f(z)=z+\sum_{n=2}^{\infty}a_nz^n\; \mbox{for}\; z\in\mathbb{D} \end{align*} and suppose that \begin{align*} F_{f}(z):=\log\dfrac{f(z)}{z}=2\sum_{n=1}^{\infty}γ_{n}(f)z^n, \;\; z\in\mathbb{D},\;\;\log 1:=0, \end{align*} where $ γ_{n}(f) $ is the logarithmic coefficients. The second Hankel determinant of logarithmic coefficients $H_{2,1}(F_{f}/2)$ is defined as: $H_{2,1}(F_{f}/2) :=γ_{1}γ_{3} -γ^2_{2}$, where $γ_1, γ_2,$ and $γ_3$ are the first, second and third logarithmic coefficients of functions belonging to the class $\mathcal{S}$ of normalized univalent functions. In this article, we first establish sharp inequalities $|H_{2,1}(F_{f}/2)|\leq 1/64$ with logarithmic coefficients for the classes of starlike functions associated with Gregory coefficients. In addition, we establish the sharpness of Fekete-Szeg$ \ddot{o} $ inequality, Zalcman functional and generalized Zalcman functional for the class starlike functions associated with Gregory coefficients. |
| title | Coefficient bounds for starlike functions associated with Gregory coefficients |
| topic | Complex Variables Primary 30C45, Secondary 30C50, 30C80 |
| url | https://arxiv.org/abs/2412.09127 |