具有边界正则不动点的单叶函数的参数表示
Parametric representation of univalent functions with boundary regular fixed points
摘要：
给定单位圆盘$\mathbb D$到其自身的共形映射的集合$\mathfrak S$在复合下是闭的，我们讨论了控制函数$t\mapsto G_t$形成凸锥$\mathcal M_{\mathfrak S}$时，$\mathfrak S$是否可以表示为Loewner - Kufarev型常微分方程$\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$，$w_0=\mathsf{id}_{\mathbb D}$的可达集的问题.对于具有$φ(0)=0$的所有共形$φ:\mathbb D\to\mathbb D$的集合，$φ'(0)>0$，对这个问题的肯定回答是Loewner著名的单叶函数的参数表示的本质[Math.Ann.89（1923），103-121]。本文研究了一类由边界正则不动点定义的共形自映射，并在部分情况下建立了它们的Loewner型表示性。
英文摘要：
Given a set $\mathfrak S$ of conformal maps of the unit disk $\mathbb D$ into itself that is closed under composition, we address the question whether $\mathfrak S$ can be represented as the reachable set of a Loewner - Kufarev - type ODE $\mathrm{d}w_t/\mathrm{d}t=G_t\circ w_t$, $w_0=\mathsf{id}_{\mathbb D}$, where the control functions $t\mapsto G_t$ form a convex cone $\mathcal M_{\mathfrak S}$. For the set of all conformal $φ:\mathbb D\to\mathbb D$ with $φ(0)=0$, $φ'(0)>0$, an affirmative answer to this question is the essence of Loewner's well-known Parametric Representation of univalent functions [Math. Ann. 89 (1923), 103-121]. In this paper, we study classes of conformal self-maps defined by their boundary regular fixed points and, in part of the cases, establish their Loewner-type representability.
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