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A bootstrap proof of Bohr's theorem
Abstract
Let f be an analytic function in the unit disc whose modulus is bounded by $1$. We present a new proof of Bohr's theorem, which asserts that its majorant function Mf(r) is bounded by $1$ for 0≤ r ≤ 1/3 and that 1/3 is optimal. The argument also yields a sharp upper bound for Mf(r) in terms of lvert f(0) rvert, complementing a result due to Bombieri.
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BibTeX
@misc{brevig2026bootstrap,
title = {{A bootstrap proof of Bohr's theorem}},
author = {Ole Fredrik Brevig},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26850},
eprint = {2609.26850},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26850},
}
RIS
TY - UNPB
TI - A bootstrap proof of Bohr's theorem
AU - Ole Fredrik Brevig
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26850
UR - https://arxiv.org/abs/2609.26850
AB - Let f be an analytic function in the unit disc whose modulus is bounded by $1$. We present a new proof of Bohr's theorem, which asserts that its majorant function Mf(r) is bounded by $1$ for 0≤ r ≤ 1/3 and that 1/3 is optimal. The argument also yields a sharp upper bound for Mf(r) in terms of lvert f(0) rvert, complementing a result due to Bombieri.
ER -