双曲距离、非零全纯函数与Krzyz猜想
Hyperbolic distances, nonvanishing holomorphic functions and Krzyz's conjecture
摘要：
本文的目的是证明Krzyz在1968年提出的猜想：对于单位圆上的非零全纯函数$f(z) = c_0 + c_1 z + ...$，如果$f(z) = c_0 + c_1 z + ...$是$|f(z)| \le 1$，则对所有$f(z) = c_0 + c_1 z + ...$都有$|c_n| \le 2/e$的尖界，且只有函数$f(z) = \exp [(z^n - 1)/(z^n + 1)]$及其旋转是相等的.许多研究者都考虑过这个问题，但只得到了部分结果。仅对$n \le 5$证明了所需的估计。我们的方法是完全不同的，并依赖于复杂的几何和多功能的凸区域在复杂的巴拿赫空间。
英文摘要：
The goal of this paper is to prove the conjecture of Krzyz posed in 1968 that for nonvanishing holomorphic functions $f(z) = c_0 + c_1 z + ...$ in the unit disk with $|f(z)| \le 1$, we have the sharp bound $|c_n| \le 2/e$ for all $n \ge 1$, with equality only for the function $f(z) = \exp [(z^n - 1)/(z^n + 1)]$ and its rotations. The problem was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \le 5$. Our approach is completely different and relies on complex geometry and pluripotential features of convex domains in complex Banach spaces.
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。
