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Krzyż's conjecture for the seventh coefficient
Abstract
A holomorphic function bounded in modulus by one in the unit disk has Taylor coefficients of modulus at most one. If the function has no zeros, Krzyż's conjecture predicts the stronger bound 2/e for every coefficient of positive order. We establish this bound for the seventh coefficient: |a7|≤2/e, with equality precisely for the functions ηexp[−(1−ξz7)/(1+ξz7)], where |η|=|ξ|=1. Thus the largest possible value of |f(7)(0)| in this class is 7! 2/e, and the extremal functions are uniquely determined up to rotations of the variable and multiplication by a constant of modulus one.
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BibTeX
@misc{xie2026krzyżs,
title = {{Krzyż's conjecture for the seventh coefficient}},
author = {Xiaohui Xie},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00397},
eprint = {2611.00397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00397},
}
RIS
TY - UNPB
TI - Krzyż's conjecture for the seventh coefficient
AU - Xiaohui Xie
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00397
UR - https://arxiv.org/abs/2611.00397
AB - A holomorphic function bounded in modulus by one in the unit disk has Taylor coefficients of modulus at most one. If the function has no zeros, Krzyż's conjecture predicts the stronger bound 2/e for every coefficient of positive order. We establish this bound for the seventh coefficient: |a_7|≤2/e, with equality precisely for the functions ηexp[−(1−ξz^7)/(1+ξz^7)], where |η|=|ξ|=1. Thus the largest possible value of |f^(7)(0)| in this class is 7! 2/e, and the extremal functions are uniquely determined up to rotations of the variable and multiplication by a constant of modulus one.
ER -