零点位于预先给定序列上的复微分方程的解
Solutions of complex differential equation having zeros on pre-given sequences
AI总结：
该复分析研究探讨了单位圆盘内特定解析系数下二阶线性微分方程解的零点序列性质，解答了公开问题，刻画了Carleson测度条件下零点序列的等价特征，并改进了经典的单叶函数相关结果。
摘要：
本文在A是单位圆盘$\u2139$内的解析函数且满足$\u0024\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty\u0024$的假设下，讨论了微分方程$f''+Af=0$解的性质，其中$\u2139$为复平面上的单位圆盘。作为主要结果，本文证明这类微分方程可存在一个非平凡解，其零点序列不满足Blaschke条件，这解答了文献中的一个公开问题。\n本文还证明，当$|A(z)|^2(1-|z|^2)^3\\, dm(z)$为Carleson测度时，$\u039b\subset\u2139$是方程$f''+Af=0$的非平凡解的零点序列当且仅当$\u039b$是一致分离的。作为应用，本文改进了“存在一致局部单叶的非正规函数”这一经典结果。
英文摘要：
Behavior of solutions of $f''+Af=0$ is discussed under the assumption that $A$ is analytic in $\mathbb{D}$ and $\sup_{z\in\mathbb{D}}(1-|z|^2)^2|A(z)|<\infty$, where $\mathbb{D}$ is the unit disc of the complex plane. As a main result it is shown that such differential equation may admit a non-trivial solution whose zero-sequence does not satisfy the Blaschke condition. This gives an answer to an open question in the literature. It is also proved that $Λ\subset\mathbb{D}$ is the zero-sequence of a non-trivial solution of $f''+Af=0$ where $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure if and only if $Λ$ is uniformly separated. As an application an old result, according to which there exists a non-normal function which is uniformly locally univalent, is improved.
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。
