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Maximum-modulus points on concentric circles
Abstract
We give a negative answer to the second question of Erdős Problem #1117: can the number νf(r) of maximum-modulus points of a nonzero, non-monomial entire function on |z|=r tend to infinity as r→∞? Write f(z)=zm g(z) with g(0)≠0, and let k be the first positive exponent with a nonzero coefficient in the Taylor expansion of g. We prove that {r>0:νf(r)>2k} is locally finite in (0,∞), and that the constant 2k is sharp. In particular, liminfr→∞νf(r)≤2k<∞, without a growth assumption. The proof combines finite local parametrizations of a logarithmic-derivative correspondence with analytic continuation and a local description of stationary points.
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BibTeX
@misc{li2026maximummodulus,
title = {{Maximum-modulus points on concentric circles}},
author = {Yunpeng Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15067},
eprint = {2610.15067},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15067},
}
RIS
TY - UNPB
TI - Maximum-modulus points on concentric circles
AU - Yunpeng Li
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.15067
UR - https://arxiv.org/abs/2610.15067
AB - We give a negative answer to the second question of Erdős Problem #1117: can the number ν_f(r) of maximum-modulus points of a nonzero, non-monomial entire function on |z|=r tend to infinity as r→∞? Write f(z)=z^m g(z) with g(0)≠0, and let k be the first positive exponent with a nonzero coefficient in the Taylor expansion of g. We prove that {r>0:ν_f(r)>2k} is locally finite in (0,∞), and that the constant 2k is sharp. In particular, liminf_r→∞ν_f(r)≤2k<∞, without a growth assumption. The proof combines finite local parametrizations of a logarithmic-derivative correspondence with analytic continuation and a local description of stationary points.
ER -