尖锐从属半径的统一方法

中文摘要 AI 辅助

设$\varphi$和$\psi$为单位圆盘$\mathbb D$内满足$\varphi(0)=\psi(0)=1$与$\varphi\not\prec\psi$的单叶函数，$\mathcal P_{\varphi}$表示$\mathbb D$内满足$p\prec\varphi$的解析函数$p$类。在对$\varphi^{-1}$的分支满足$\varphi^{-1}(1)=0$的适当解析延拓、单叶性及边界性态假设下，确定$\mathcal P_{\varphi}$的尖锐$\mathcal P_{\psi}$半径为\\( \mathcal R(\varphi,\psi) = \min_{|\zeta|=1} \left| \varphi^{-1}\bigl(\psi(\zeta)\bigr) \right| \\)。对于满足这些假设的Ma–Minda函数，该半径对关联类$\mathcal{ST}(\varphi)$和$\mathcal{CV}(\varphi)$分别关于$\mathcal{ST}(\psi)$与$\mathcal{CV}(\psi)$是尖锐的。将该一般结果应用于$ \varphi_{\mathrm L}(z)=\sqrt{1+z}, \varphi_{\mathrm{Lune}}(z)=z+\sqrt{1+z^2}, \varphi_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $，选取若干Ma–Minda函数$\psi$，得到对应广义族的参数依赖扩展，并解析计算所得边界最小值。

英文摘要

Let $φ$ and $ψ$ be univalent functions in the unit disk $\mathbb D$ satisfying $φ(0)=ψ(0)=1$ and $φ\not\precψ$, and let $\mathcal P_φ$ denote the class of analytic functions $p$ in $\mathbb D$ such that $p\precφ$. Under suitable analytic continuation, univalence, and boundary-behavior hypotheses on the branch of $φ^{-1}$ satisfying $φ^{-1}(1)=0$, we determine the sharp $\mathcal P_ψ$-radius of $\mathcal P_φ$ as \[ \mathcal R(φ,ψ) = \min_{|ζ|=1} \left| φ^{-1}\bigl(ψ(ζ)\bigr) \right|. \] For Ma--Minda functions satisfying these hypotheses, the same radius is sharp for the associated classes $\mathcal{ST}(φ)$ and $\mathcal{CV}(φ)$ with respect to $\mathcal{ST}(ψ)$ and $\mathcal{CV}(ψ)$, respectively. We apply the general result to $ φ_{\mathrm L}(z)=\sqrt{1+z}, φ_{\mathrm{Lune}}(z)=z+\sqrt{1+z^2}, φ_{\mathrm{Lim}}(z)=\left(1+{z}/{\sqrt2}\right)^2, $ for several choices of the Ma--Minda function $ψ$, and obtain parameter-dependent extensions for the corresponding generalized families. The resulting boundary minima are evaluated analytically.

发表机构

• Department of Mathematics \ Institute of Technology\ -620015, India

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