关于\(\mathcal{S}^*_{q_1}\)类的系数问题与行列式的精确估计
Coefficient Problems and Sharp Determinant Estimates for the Class $\mathcal{P}^{*}$
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中文总结 AI 辅助
研究归一化解析函数子类\(\mathcal{S}^*_{q_1}\)的系数问题，通过从属条件定义该子类，得到对数和逆对数系数及其绝对差的精确界，以及相关二阶行列式的精确估计，扩展和细化了已知解析函数子类的界。
中文摘要 AI 辅助
本文研究了归一化解析函数子类\(\mathcal{S}^*_{q_1}\)的系数相关问题，它由从属条件\(\frac{f'(z)}{q_1(z)} \prec 1+\sin z\)和\(q_1(z) \prec e^z\)（\(z\in\mathbb{D}\)）定义。得到了对数和逆对数系数及其绝对差的精确界。此外，还导出了与这些系数相关的二阶汉克尔和埃尔米特 - 托普利兹行列式的精确估计。这些结果扩展并细化了一些已知的解析函数相关子类的界。
英文摘要
We study coefficient problems for the class $\mathcal{P}^{*}$ of normalized analytic functions $f$ in the unit disk satisfying the subordination condition \[ f'(z) \prec e^z(1+\sin z), \qquad z\in\mathbb{D}. \] For functions in this class, we determine sharp bounds for the logarithmic coefficients $γ_n$ for $n=1,2,3,4$ and for the inverse logarithmic coefficients $Γ_n$ for $n=1,2,3$. In particular, we prove \[ |γ_n| \le \frac{1}{n+1} \quad (n=1,2,3,4), \qquad |Γ_1|,|Γ_2| \le \frac12,\quad |Γ_3|\le \frac{35}{48}, \] with equality cases explicitly identified. Moreover, we establish sharp estimates for the moduli of the differences $|γ_2|-|γ_1|$ and $|Γ_2|-|Γ_1|$, yielding \[ -\frac12 \le |γ_2|-|γ_1| \le \frac13, \qquad -\frac{1}{\sqrt{10}} \le |Γ_2|-|Γ_1| \le \frac13. \] Turning to determinant problems, we obtain sharp upper bounds for the second-order Hankel determinants associated with both the logarithmic and inverse logarithmic coefficients: \[ \bigl|H_{2,1}(F_f/2)\bigr| \le \frac19,\qquad \bigl|H_{2,1}(F_{f^{-1}}/2)\bigr| \le \frac{11}{96}. \] Finally, we derive the sharp two-sided estimate \[ -\frac{9}{35} \le T_{3,1}(f) \le 1 \] for the third-order Hermitian--Toeplitz determinant. All bounds are sharp, and the extremal functions are given explicitly. Our results extend and refine several known coefficient estimates for related subclasses of starlike and convex functions.
发表机构
- Raiganj University（拉甘杰大学）
- Amity University Mumbai（孟买阿米提大学）
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