具有正实部的函数的变异域
Region of variability for functions with positive real part
摘要：
对于$γ\in\IC$，使$|γ|<π/2$和$0\leqβ<1$，让${\mathcal P}_{γ,β} $表示类的所有解析函数$P$在单位盘$\mathbb{D}$与$P(0)=1$和$$ {\rm Re\,} \left (e^{iγ}P(z)\right)>β\cosγ\quad \mbox{ in ${\mathbb D}$}. $$对于任何固定的$z_0\in\mathbb{D}$和$λ\in\overline{\mathbb{D}}$，我们将确定区域的变化$V_{\mathcal{P}}(z_0,λ)$为$\int_0^{z_0}P(ζ)\,dζ$时，$P$的范围超过类$$ \mathcal{P}(λ) = \left\{ P\in{\mathcal P}_{γ,β} :\, P'(0)=2(1-β)λe^{-iγ}\cosγ\right\}. $$因此，我们提出了区域的变化为一些子类的单叶函数。我们还以图形方式说明了几组参数的变异区域。
英文摘要：
For $γ\in\IC$ such that $|γ|<π/2$ and $0\leqβ<1$, let ${\mathcal P}_{γ,β} $ denote the class of all analytic functions $P$ in the unit disk $\mathbb{D}$ with $P(0)=1$ and $$ {\rm Re\,} \left (e^{iγ}P(z)\right)>β\cosγ\quad \mbox{ in ${\mathbb D}$}. $$ For any fixed $z_0\in\mathbb{D}$ and $λ\in\overline{\mathbb{D}}$, we shall determine the region of variability $V_{\mathcal{P}}(z_0,λ)$ for $\int_0^{z_0}P(ζ)\,dζ$ when $P$ ranges over the class $$ \mathcal{P}(λ) = \left\{ P\in{\mathcal P}_{γ,β} :\, P'(0)=2(1-β)λe^{-iγ}\cosγ\right\}. $$ As a consequence, we present the region of variability for some subclasses of univalent functions. We also graphically illustrate the region of variability for several sets of parameters.
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