双曲测地线、Krzyz猜想及其他
Hyperbolic geodesics, Krzyz's conjecture and beyond
摘要：
1968年，Krzyz证明了对于单位圆盘中的非零全纯函数$f(z) = c_0 + c_1 z + \dots$，$|f(z)| \leq 1$，我们对所有$n \geq 1$都有$|c_n| \leq 2/e$的尖界，只有函数$f(z) = exp [(z^n - 1)/(z^n + 1)]$和它的旋转相等。这个猜想被许多研究者所考虑，但只有部分结果已经建立。仅对$n \leq 5$证明了所需的估计。我们在这里提供了两个不同的证明这个猜想和它的推广基于完全不同的想法。
英文摘要：
In 1968, Krzyz conjectured that for non-vanishing holomorphic functions $f(z) = c_0 + c_1 z + \dots$ in the unit disk with $|f(z)| \leq 1$, we have the sharp bound $|c_n| \leq 2/e$ for all $n \geq 1$, with equality only for the function $f(z) = exp [(z^n - 1)/(z^n + 1)]$ and its rotations. This conjecture was considered by many researchers, but only partial results have been established. The desired estimate has been proved only for $n \leq 5$. We provide here two different proofs of this conjecture and its generalizations based on completely different ideas.
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