Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class $\mathcal{C}_e$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913146183090176 |
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| author | Das, Pradip Sarkar, Nabadwip |
| author_facet | Das, Pradip Sarkar, Nabadwip |
| contents | In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}Γ_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \[ |Γ_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |Γ_2|-|Γ_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \]
Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete--Szegö functional $F_{λ, μ}(f) = \big| a_3(f) - λa_2(f)^2 \big| - μ|a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_ρ$.
The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19614 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class $\mathcal{C}_e$ Das, Pradip Sarkar, Nabadwip Complex Variables 30C45 30C45, 30C50, 30C55 In this paper, we investigate the inverse logarithmic coefficients associated with the class $\mathcal{C}_e$ of analytic and univalent functions satisfying the subordination condition \[ 1+\frac{z f''(z)}{f'(z)} \prec e^z, \quad z\in\mathbb{D}. \] If $F_{f^{-1}}(w) = \log\!\left(\frac{f^{-1}(w)}{w}\right) = 2\sum_{n=1}^{\infty}Γ_n w^n$ denotes the logarithmic expansion corresponding to the inverse function $f^{-1}$, then we establish sharp estimates for the initial inverse logarithmic coefficients and prove that \[ |Γ_n| \le \frac{1}{2n(n+1)}, \qquad n=1,2,3. \] We further derive the sharp coefficient-difference inequality \[ -\frac{1}{2\sqrt7} \le |Γ_2|-|Γ_1| \le \frac1{12}, \] and obtain the sharp bound for the second-order Hankel determinant associated with the inverse logarithmic coefficients: \[ \left| H_{2,1}\!\left(F_{f^{-1}}/2\right) \right| \le \frac{85}{12096}. \] Additionally, we evaluate the sharp lower and upper bounds of the generalized Fekete--Szegö functional $F_{λ, μ}(f) = \big| a_3(f) - λa_2(f)^2 \big| - μ|a_2(f)|$ within this setting and establish relationships associated with the starlike class $\mathcal{S}^{\ast}_ρ$. The extremal functions corresponding to all obtained estimates are explicitly constructed, thereby showing the sharpness of the results. |
| title | Inverse Logarithmic Coefficients, Differences, Hankel Determinant, and Fekete--Szegö Functionals for the Class $\mathcal{C}_e$ |
| topic | Complex Variables 30C45 30C45, 30C50, 30C55 |
| url | https://arxiv.org/abs/2605.19614 |