单位圆盘内有限对数测度集合外的Wiman-Valiron不等式
Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure
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中文总结 AI 辅助
本文肯定回答了Grosse-Erdmann关于单位圆盘内Wiman-Valiron不等式的两个问题，在有限对数测度例外集外建立迭代对数不等式，并给出Rosenbloom复合估计的圆盘类比，关键利用Khinchin族方差界，且指数1/2不可改进。
中文摘要 AI 辅助
我们针对Grosse-Erdmann（2025）提出的关于单位圆盘内Wiman-Valiron不等式的Question 2.6的两个部分给出了肯定回答。对于圆盘内每个无界解析函数，我们在有限对数测度的例外集合之外建立了所提出的迭代对数不等式。相应的幂估计是推论。这两个结论均源于Khinchin族的方差界及其最大原子的经典估计。主要不等式中的乘法常数可以选取为绝对常数。我们还获得了Rosenbloom复合估计的圆盘类比，并带有显式的边界前因子。关键步骤结合了边界变量替换与一个单调辅助函数，其导数恰好是Khinchin族重标成员的方差。一个经典例子表明，主导对数指数$1/2$不能减小。
英文摘要
We give affirmative answers to both parts of Question 2.6 posed by Grosse-Erdmann (2025) concerning Wiman-Valiron inequalities in the unit disk. For every unbounded analytic function in the disk, we establish the proposed iterated-logarithm inequalities outside exceptional sets of finite logarithmic measure. The corresponding power estimate is a corollary. Both conclusions follow from a variance bound for Khinchin families and a classical estimate for their largest atom. The multiplicative constants in the main inequalities can be chosen absolute. We also obtain a disk analogue of Rosenbloom's composition estimate, with an explicit boundary prefactor. The key step combines a boundary change of variable with a monotone auxiliary function whose derivative is exactly the variance of a rescaled member of the Khinchin family. A classical example shows that the leading logarithmic exponent $1/2$ cannot be decreased.
发表机构
- Universidad de La Laguna（拉帕鲁尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。
