指数凸单叶函数的变域
Region of variability for exponentially convex univalent functions
摘要：
对于$α\in\IC\setminus \{0\}$让$\mathcal{E}(α)$表示类的所有单叶函数$f$中的单位盘$\mathbb{D}$和给出的$f(z)=z+a_2z^2+a_3z^3+\cdots$，满足$$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$对于任何固定的$z_0$中的单位盘$\mathbb{D}$和$λ\in\overline{\mathbb{D}}$，我们确定区域的变化$V(z_0,λ)$为$\log f'(z_0)+αf(z_0)$时，$f$的范围超过类$$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$我们几何说明区域的变化$V(z_0,λ)$为几组参数使用Mathematica。在本文的最后一部分，我们提出了一些开放的问题。
英文摘要：
For $α\in\IC\setminus \{0\}$ let $\mathcal{E}(α)$ denote the class of all univalent functions $f$ in the unit disk $\mathbb{D}$ and is given by $f(z)=z+a_2z^2+a_3z^3+\cdots$, satisfying $$ {\rm Re\,} \left (1+ \frac{zf''(z)}{f'(z)}+αzf'(z)\right)>0 \quad {in ${\mathbb D}$}. $$ For any fixed $z_0$ in the unit disk $\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we determine the region of variability $V(z_0,λ)$ for $\log f'(z_0)+αf(z_0)$ when $f$ ranges over the class $$\mathcal{F}_α(λ)=\left\{f\in\mathcal{E}(α) \colon f''(0)=2λ-α%\quad{and} f'''(0)=2[(1-|λ|^2)a+ %(λ-α)^2 -λα] \right\}. $$ We geometrically illustrate the region of variability $V(z_0,λ)$ for several sets of parameters using Mathematica. In the final section of this article we propose some open problems.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
