On the length of lemniscates
Complex Variables
2024-03-28 v2
Abstract
We show that for a monic polynomial p of degree d, the length of the level set {z: |p(z)|=1} is at most 9.2 d, which improves an earlier estimate due to P. Borwein. For d=2 we show that the extremal level set is the Bernoullis' Lemniscate. One ingredient of our proofs is the fact that for an extremal polynomial this level set is connected.
Cite
@article{arxiv.0805.2295,
title = {On the length of lemniscates},
author = {Alexandre Eremenko and Walter Hayman},
journal= {arXiv preprint arXiv:0805.2295},
year = {2024}
}