For \(0<s<1\), let \(\Lambda \subset {\mathbb {D}}\) be a separated sequence such that \(\sum _{z_n\in \Lambda }(1-|z_n|)^s \delta _{z_n}\) is an s-Carleson measure. In this paper, we show that there exist certain analytic functions A such that the second order complex differential equation \(f''+Af=0\) admits a non-trivial solution f whose zero-sequence is \(\Lambda \), where the solution f belongs to some Möbius invariant function spaces. We strengthen a previous result from the literature.

中文翻译：

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具有某些给定零的二阶复微分方程的解

对于\（0 <s <1 \），令\（\ Lambda \ subset {\ mathbb {D}} \）是一个分隔的序列，使得\（\ sum _ {z_n \ in \ Lambda}（1- | z_n |）^ s \ delta _ {z_n} \）是s -Carleson度量。在本文中，我们证明存在某些解析函数A，使得二阶复数微分方程\（f''+ Af = 0 \）允许一个零序列为\（\ Lambda \）的非平凡解f，其中解f属于一些Möbius不变函数空间。我们加强了文献的先前结果。