论文标题
平面域的内在几何形状和边界结构
Intrinsic geometry and boundary structure of plane domains
论文作者
论文摘要
对于合适的子域$ω$，对于复杂飞机的$ω$，我们表示直径为$ e $，分别为$ e $的直径到$ e $到$ d（e）$和$ d（e，\partialΩ）的范围。数量$ d（e）/d（e，\partialΩ）$在相似性下是不变的，并且在几何函数理论中起着重要作用。在本文中，当$ω$具有双曲线距离$h_Ω（z，w）时，$，我们考虑$h_Ω（e）/\ log（1+d（e）/d（e）/d（e，\partialΩ $ E. $我们用$ \ mathbb {h} $表示上半平面。我们的主要结果声称，$κ（ω）$是正面的，并且仅当$ω$的边界均匀地完美，并且不平等$κ（ω）\leqκ（\ mathbb {h}）$都可以满足所有$ω，而当$ω$ convex convex convex时，则$均等$均等。
For a non-empty compact set $E$ in a proper subdomain $Ω$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Ω$ by $d(E)$ and $d(E,\partialΩ),$ respectively. The quantity $d(E)/d(E,\partialΩ)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Ω$ has the hyperbolic distance $h_Ω(z,w),$ we consider the infimum $κ(Ω)$ of the quantity $h_Ω(E)/\log(1+d(E)/d(E,\partialΩ))$ over compact subsets $E$ of $Ω$ with at least two points, where $h_Ω(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $κ(Ω)$ is positive if and only if the boundary of $Ω$ is uniformly perfect and that the inequality $κ(Ω)\leqκ(\mathbb{H})$ holds for all $Ω,$ where equality holds precisely when $Ω$ is convex.
