Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure
Abstract
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.
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BibTeX
@misc{macia2026wimanvaliron,
title = {{Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure}},
author = {Víctor J. Maciá},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28814},
eprint = {2609.28814},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28814},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure
AU - Víctor J. Maciá
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.28814
UR - https://arxiv.org/abs/2609.28814
AB - 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.
ER -