凸函数的圆对称、从属和弧长问题
Circular Symmetrization, Subordination and Arclength problems on Convex Functions
摘要：
本文研究了单位圆盘$\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$中的单叶解析函数类$f$满足\[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \]的$f(z)=z+\sum_{n=2}^{\infty}a_n z^n$，其中$Ω$是${\mathbb C}$的一个真子域，它关于$1 (\in Ω)$是星形的.令$ϕ_Ω$是${\mathbb D}$到$Ω$上的唯一保角映射，其中$ϕ_Ω(0)=1$和$ϕ_Ω'(0) > 0$和$ k_Ω(z) = \int_0^z \exp \left(\int_0^t ζ^{-1} (ϕ_Ω(ζ) -1) \, d ζ\right) \, dt$。令$L_r(f)$表示圆$\{z \in \mathbb{C} : \, |z|=r\}$的像的弧长，$r\in (0,1)$。本文的第一个结果是一个不等式$L_r(f) \leq L_r(k_Ω)$ for $f \in \mathcal{C} (Ω)$，它解决了一般极值问题$\max_{f \in {\mathcal C}(Ω)} L_r(f)$，并包含了许多其他著名的结果作为特例.这篇文章的其他结果涵盖了另一组有关的问题，积分手段的一般设置类${\mathcal C}(Ω)$。
英文摘要：
We study the class ${\mathcal C}(Ω)$ of univalent analytic functions $f$ in the unit disk $\mathbb{D} = \{z \in \mathbb{C} :\,|z|<1 \}$ of the form $f(z)=z+\sum_{n=2}^{\infty}a_n z^n$ satisfying \[ 1+\frac{zf"(z)}{f'(z)} \in Ω, \quad z\in \mathbb{D}, \] where $Ω$ will be a proper subdomain of ${\mathbb C}$ which is starlike with respect to $1 (\in Ω)$. Let $ϕ_Ω$ be the unique conformal mapping of ${\mathbb D}$ onto $Ω$ with $ϕ_Ω(0)=1$ and $ϕ_Ω'(0) > 0$ and $ k_Ω(z) = \int_0^z \exp \left(\int_0^t ζ^{-1} (ϕ_Ω(ζ) -1) \, d ζ\right) \, dt$. Let $L_r(f)$ denote the arclength of the image of the circle $\{z \in \mathbb{C} : \, |z|=r\}$, $r\in (0,1)$. The first result in this paper is an inequality $L_r(f) \leq L_r(k_Ω)$ for $f \in \mathcal{C} (Ω)$, which solves the general extremal problem $\max_{f \in {\mathcal C}(Ω)} L_r(f)$, and contains many other well-known results of the previous authors as special cases. Other results of this article cover another set of related problems about integral means in the general setting of the class ${\mathcal C}(Ω)$.
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